Lie groups pqrFrom Wikipedia, the free encyclopedia

Contents

1 p-compact group 11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Pansu derivative 22.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Pauli matrices 33.1 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.1.1 Eigenvectors and eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.1.2 Pauli vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.1.3 Commutation relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.1.4 Relation to dot and cross product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.1.5 Exponential of a Pauli vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.1.6 Completeness relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1.7 Relation with the permutation operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.1 SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.2 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3.1 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3.2 Quantum information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Pin group 114.1 General denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Denite form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 Indenite form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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4.4 As topological group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.5 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.5.1 Low dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.6 Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.7 Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Ping-pong lemma 145.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 Formal statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.2.1 Ping-pong lemma for several subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2.2 The Ping-pong lemma for cyclic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.3.1 Special linear group example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.3.2 Word-hyperbolic group example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.4 Applications of the ping-pong lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6 Poincar group 196.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.3 Poincar symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7 PoissonLie group 227.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.2 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

8 Polar decomposition 248.1 Matrix polar decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.2 Bounded operators on Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.3 Unbounded operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.4 Quaternion polar decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.5 Alternative planar decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.6 Numerical determination of the matrix polar decomposition . . . . . . . . . . . . . . . . . . . . . 278.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.9 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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8.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

9 Pre-Lie algebra 299.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

10 Principal homogeneous space 3110.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.3 Other usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

11 Projective linear group 3511.1 Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

11.1.1 Collineations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.2 Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

11.3.1 Fractional linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.4 Finite elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

11.4.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.4.2 Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.4.3 Exceptional isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.4.4 Mathieu groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.4.5 Hurwitz surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.4.6 Modular group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

11.5 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.5.1 Covering groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

11.6 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.7 Low dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.9 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.10Larger groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

12 Projective orthogonal group 4712.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

12.1.1 Odd and even dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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12.1.2 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.2 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.3 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.4 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

12.4.1 Homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.4.2 Homology groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5012.4.3 Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

12.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5012.5.1 Finite elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

12.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5112.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5112.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5112.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

13 Projective unitary group 5213.1 Projective special unitary group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5213.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5313.3 Finite elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5313.4 The topology of PU(H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

13.4.1 PU(H) is a classifying space for circle bundles . . . . . . . . . . . . . . . . . . . . . . . . 5313.4.2 The homotopy and (co)homology of PU(H) . . . . . . . . . . . . . . . . . . . . . . . . . 54

13.5 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5413.5.1 The adjoint representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5413.5.2 Projective representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

13.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5513.6.1 Twisted K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5513.6.2 Pure YangMills gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

13.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5513.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

14 Pseudogroup 5614.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5714.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

15 Quaternion-Khler symmetric space 5815.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5815.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

16 Ratners theorems 5916.1 Short description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

16.3.1 Expositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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16.3.2 Selected original articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

17 Real form (Lie theory) 6117.1 Real forms for Lie groups and algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.2 Classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

17.2.1 Split real form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6217.2.2 Compact real form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

17.3 Construction of the compact real form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6217.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6217.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6217.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

18 Reductive group 6418.1 Lie group case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6518.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

19 Regular element of a Lie algebra 6619.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

20 Representation of a Lie group 6720.1 Representations on a complex nite-dimensional vector space . . . . . . . . . . . . . . . . . . . . 6720.2 Representations on a nite-dimensional vector space over an arbitrary eld . . . . . . . . . . . . . 6720.3 Representations on Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6820.4 Classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6820.5 Formulaic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6820.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6920.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

21 Representation ring 7021.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7021.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7021.3 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7021.4 -ring and Adams operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7121.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

22 Restricted root system 7222.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7222.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

23 Root system 7323.1 Denitions and rst examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

vi CONTENTS

23.1.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7323.1.2 Rank two examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

23.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7523.3 Elementary consequences of the root system axioms . . . . . . . . . . . . . . . . . . . . . . . . . 7523.4 Positive roots and simple roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

23.4.1 The root poset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7623.5 Dual root system and coroots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7623.6 Classication of root systems by Dynkin diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 7623.7 Properties of the irreducible root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7823.8 Explicit construction of the irreducible root systems . . . . . . . . . . . . . . . . . . . . . . . . . 78

23.8.1 An . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7823.8.2 Bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7823.8.3 Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7923.8.4 Dn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7923.8.5 E6, E7, E8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7923.8.6 F4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8023.8.7 G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

23.9 Root systems and Lie theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8123.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8223.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8223.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8323.13Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8323.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

24 Semilinear transformation 8424.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8424.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8524.3 General semilinear group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

24.3.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8624.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

24.4.1 Projective geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8624.4.2 Mathieu group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

24.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8624.6 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 87

24.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8724.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8824.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Chapter 1

p-compact group

Inmathematics, in particular algebraic topology, a p-compact group is (roughly speaking) a space that is a homotopicalversion of a compact Lie group, but with all the structure concentrated at a single prime p. This concept was intro-duced by Dwyer and Wilkerson.[1] Subsequently the name homotopy Lie group has also been used.

1.1 ExamplesExamples include the p-completion of a compact and connected Lie group, and the Sullivan spheres, i.e. the p-completion of a sphere of dimension

2n 1,

if n divides p 1.

1.2 ClassicationThe classication of p-compact groups states that there is a 1-1 correspondence between connected p-compact groups,and root data over the p-adic integers. This is analogous to the classical classication of connected compact Lie groups,with the p-adic integers replacing the rational integers.

1.3 References Homotopy Lie Groups: A Survey (PDF) Homotopy Lie Groups and Their Classication (PDF)

1.4 Notes[1] W. G. Dwyer and C. W. Wilkerson, Homotopy xed-point methods for Lie groups and nite loop spaces, Ann. of Math.

(2) 139 (1994), no. 2, 395442.

1

Chapter 2

Pansu derivative

In mathematics, the Pansu derivative is a derivative on a Carnot group, introduced by Pierre Pansu (1989).

2.1 References Pansu, Pierre (1989), Mtriques de Carnot-Carathodory et quasiisomtries des espaces symtriques de rangun, Annals of Mathematics. Second Series 129 (1): 160, doi:10.2307/1971484, ISSN 0003-486X, MR979599

2

Chapter 3

Pauli matrices

In mathematical physics and mathematics, the Pauli matrices are a set of three 2 2 complex matrices which areHermitian and unitary.[1] Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau ()when used in connection with isospin symmetries. They are

1 = x =

0 11 0

2 = y =

0 ii 0

3 = z =

1 00 1

:

These matrices are named after the physicist Wolfgang Pauli. In quantummechanics, they occur in the Pauli equationwhich takes into account the interaction of the spin of a particle with an external electromagnetic eld.Each Pauli matrix is Hermitian, and together with the identity matrix I (sometimes considered as the zeroth Paulimatrix 0), the Pauli matrices (multiplied by real coecients) span the full vector space of 2 2 Hermitian matrices.In the language of quantum mechanics, Hermitian matrices are observables, so the Pauli matrices span the spaceof observables of the 2-dimensional complex Hilbert space. In the context of Paulis work, k is the observablecorresponding to spin along the kth coordinate axis in three-dimensional Euclidean space 3.The Pauli matrices (after multiplication by i to make them anti-Hermitian), also generate transformations in the senseof Lie algebras: the matrices i1, i2, i3 form a basis for su(2), which exponentiates to the special unitary groupSU(2). The algebra generated by the three matrices 1, 2, 3 is isomorphic to the Cliord algebra of 3, called thealgebra of physical space.

3.1 Algebraic propertiesAll three of the Pauli matrices can be compacted into a single expression:

a =

a3 a1 ia2

a1 + ia2 a3

where i = 1 is the imaginary unit, and ab is the Kronecker delta, which equals +1 if a = b and 0 otherwise. Thisexpression is useful for selecting any one of the matrices numerically by substituting values of a = 1, 2, 3, in turnuseful when any of the matrices (but no particular one) is to be used in algebraic manipulations.The matrices are involutory:

21 = 22 =

23 = i123 =

1 00 1

= I

3

4 CHAPTER 3. PAULI MATRICES

where I is the identity matrix.

The determinants and traces of the Pauli matrices are:

deti = 1;Tri = 0:

From above we can deduce that the eigenvalues of each i are 1.

Together with the 2 2 identity matrix I (sometimes written as 0), the Pauli matrices form an orthogonalbasis, in the sense of HilbertSchmidt, for the real Hilbert space of 2 2 complex Hermitian matrices, or thecomplex Hilbert space of all 2 2 matrices.

3.1.1 Eigenvectors and eigenvalues

Each of the (Hermitian) Pauli matrices has two eigenvalues, +1 and 1. The corresponding normalized eigenvectorsare:

x+ =1p2

11

; x =

1p2

11;

y+ =1p2

1i

; y =

1p2

1i;

z+ =

10

; z =

01

:

3.1.2 Pauli vector

The Pauli vector is dened by[nb 1]

~ = 1x^+ 2y^ + 3z^

and provides a mapping mechanism from a vector basis to a Pauli matrix basis[2] as follows,

~a ~ = (aix^i) (j x^j)= aij x^i x^j= aijij

= aii =

a3 a1 ia2

a1 + ia2 a3

using the summation convention. Further,

det~a ~ = ~a ~a = j~aj2;

and also (see completeness, below)

1

2tr[(~a ~)~] = ~a:

3.1. ALGEBRAIC PROPERTIES 5

3.1.3 Commutation relations

The Pauli matrices obey the following commutation relations:

[a; b] = 2i"abc c ;

and anticommutation relations:

fa; bg = 2ab I:

where abc is the Levi-Civita symbol, Einstein summation notation is used, ab is the Kronecker delta, and I is the2 2 identity matrix.For example,

[1; 2] = 2i3

[2; 3] = 2i1

[3; 1] = 2i2

[1; 1] = 0

f1; 1g = 2If1; 2g = 0 :

3.1.4 Relation to dot and cross product

Pauli vectors elegantly map these commutation and anticommutation relations to corresponding vector products.Adding the commutator to the anticommutator gives

[a; b] + fa; bg = (ab ba) + (ab + ba)2iXc

"abc c + 2abI = 2ab

so that, cancelling the factors of 2,

Contracting each side of the equation with components of two 3-vectors ap and bq (which commute with the Paulimatrices, i.e., apq = qap) for each matrix q and vector component ap (and likewise with bq), and relabelingindices a, b, c p, q, r, to prevent notational conicts, yields

apbqpq = apbq

iXr

"pqr r + pqI

!appbqq = i

Xr

"pqr apbqr + apbqpqI :

Finally, translating the index notation for the dot product and cross product results in

3.1.5 Exponential of a Pauli vector

For

6 CHAPTER 3. PAULI MATRICES

~a = an^; jn^j = 1;

one has, for even powers,

(n^ ~)2n = I

which can be shown rst for the n = 1 case using the anticommutation relations.Thus, for odd powers,

(n^ ~)2n+1 = n^ ~ :

Matrix exponentiating, and using the Taylor series for sine and cosine,

eia(n^~) =1Xn=0

in [a(n^ ~)]nn!

=1Xn=0

(1)n(an^ ~)2n(2n)!

+ i1Xn=0

(1)n(an^ ~)2n+1(2n+ 1)!

= I

1Xn=0

(1)na2n(2n)!

+ i(n^ ~)1Xn=0

(1)na2n+1(2n+ 1)!

and, in the last line, the rst sum is the cosine, while the second sum is the sine; so, nally,which is analogous to Eulers formula. Note

det[ia(n^ ~)] = a2

while the determinant of the exponential itself is just 1, which makes it the generic group element of SU(2).A more abstract version of formula (2) for a general 2 2 matrix can be found in the article on matrix exponentials.

The group composition law of SU(2)

A straightforward application of this formula provides a parameterization of the composition law of the groupSU(2).[nb 2] One may directly solve for c in

eia(n^~)eib(m^~) = I(cos a cos b n^ m^ sin a sin b) + i(n^ sin a cos b+ m^ sin b cos a n^ m^ sin a sin b) ~= I cos c+ i(k^ ~) sin c= eic(k^~);

which species the generic group multiplication, where, manifestly,

cos c = cos a cos b n^ m^ sin a sin b ;

the spherical law of cosines. Given c, then,

k^ =1

sin c (n^ sin a cos b+ m^ sin b cos a n^ m^ sin a sin b) :

3.1. ALGEBRAIC PROPERTIES 7

Consequently, the composite rotation parameters in this group element (a closed form of the respective BCH expan-sion in this case) simply amount to[3]

eick^~ = expic

sin c (n^ sin a cos b+ m^ sin b cos a n^ m^ sin a sin b) ~:

(Of course, when n is parallel to m, so is k, and c = a + b.)The fact that any 2 2 complex Hermitian matrices can be expressed in terms of the identity matrix and the Paulimatrices also leads to the Bloch sphere representation of 2 2 mixed states' density matrix, (2 2 positive semidef-inite matrices with trace 1). This can be seen by simply rst writing an arbitrary Hermitian matrix as a real linearcombination of {0, 1, 2, 3} as above, and then imposing the positive-semidenite and trace 1 conditions.See also: Rotation formalisms in three dimensions Rodrigues parameters and Gibbs representation

3.1.6 Completeness relationAn alternative notation that is commonly used for the Pauli matrices is to write the vector index i in the superscript,and the matrix indices as subscripts, so that the element in row and column of the i-th Pauli matrix is i.In this notation, the completeness relation for the Pauli matrices can be written

~ ~ 3X

i=1

ii

= 2 :

Proof

The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the complex Hilbertspace of all 2 2 matrices means that we can express any matrix M as

M = cI +Xi

aii

where c is a complex number, and a is a 3-component complex vector. It is straightforward to show, using theproperties listed above, that

trij = 2ijwhere tr denotes the trace, and hence that c = 12 trM and ai = 12 triM .This therefore gives

2M = ItrM +Xi

itriM

which can be rewritten in terms of matrix indices as

2M = M

+Xi

ii

M

where summation is implied over the repeated indices and . Since this is true for any choice of the matrix M, thecompleteness relation follows as stated above.As noted above, it is common to denote the 2 2 unit matrix by 0, so 0 = . The completeness relation cantherefore alternatively be expressed as

8 CHAPTER 3. PAULI MATRICES

3Xi=0

ii

= 2

3.1.7 Relation with the permutation operatorLet Pij be the transposition (also known as a permutation) between two spins i and j living in the tensor productspace 2 2,

Pij jiji = jjii :

This operator can also be written more explicitly as Diracs spin exchange operator,

Pij =12 (~i ~j + 1) :

Its eigenvalues are therefore[4] 1 or 1. It may thus be utilized as an interaction term in a Hamiltonian, splitting theenergy eigenvalues of its symmetric versus antisymmetric eigenstates.

3.2 SU(2)The group SU(2) is the Lie group of unitary 22 matrices with unit determinant; its Lie algebra is the set of all 22anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that the Lie algebra su2 is the 3-dimensionalreal algebra spanned by the set {ij}. In compact notation,

su(2) = spanfi1; i2; i3g:

As a result, each ij can be seen as an innitesimal generator of SU(2). The elements of SU(2) are exponentialsof linear combinations of these three generators, and multiply as indicated above in discussing the Pauli vector.Although this suces to generate SU(2), it is not a proper representation of su(2), as the Pauli eigenvalues are scaledunconventionally. The conventional normalization is = 1/2, so that

su(2) = spani12;i22;i32

:

As SU(2) is a compact group, its Cartan decomposition is trivial.

3.2.1 SO(3)The Lie algebra su(2) is isomorphic to the Lie algebra so(3), which corresponds to the Lie group SO(3), the groupof rotations in three-dimensional space. In other words, one can say that the ij are a realization (and, in fact, thelowest-dimensional realization) of innitesimal rotations in three-dimensional space. However, even though su(2) andso(3) are isomorphic as Lie algebras, SU(2) and SO(3) are not isomorphic as Lie groups. SU(2) is actually a doublecover of SO(3), meaning that there is a two-to-one group homomorphism from SU(2) to SO(3), see relationshipbetween SO(3) and SU(2).

3.2.2 QuaternionsMain article: versor

3.3. PHYSICS 9

The real linear span of {I, i1, i2, i3} is isomorphic to the real algebra of quaternions . The isomorphism from to this set is given by the following map (notice the reversed signs for the Pauli matrices):

1 7! I; i 7! i1; j 7! i2; k 7! i3:Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,[5]

1 7! I; i 7! i3; j 7! i2; k 7! i1:As the quaternions of unit norm is group-isomorphic to SU(2), this gives yet another way of describing SU(2) viathe Pauli matrices. The two-to-one homomorphism from SU(2) to SO(3) can also be explicitly given in terms of thePauli matrices in this formulation.Quaternions form a division algebraevery non-zero element has an inversewhereas Pauli matrices do not. Fora quaternionic version of the algebra generated by Pauli matrices see biquaternions, which is a venerable algebra ofeight real dimensions.

3.3 Physics

3.3.1 Quantum mechanicsIn quantummechanics, each Pauli matrix is related to an angularmomentumoperator that corresponds to an observabledescribing the spin of a spin particle, in each of the three spatial directions. As an immediate consequence of theCartan decomposition mentioned above, ij are the generators of a projective representation (spin representation)of the rotation group SO(3) acting on non-relativistic particles with spin . The states of the particles are representedas two-component spinors. In the same way, the Pauli matrices are related to the isospin operatorAn interesting property of spin particles is that they must be rotated by an angle of 4 in order to return to theiroriginal conguration. This is due to the two-to-one correspondence between SU(2) and SO(3) mentioned above,and the fact that, although one visualizes spin up/down as the north/south pole on the 2-sphere S 2, they are actuallyrepresented by orthogonal vectors in the two dimensional complex Hilbert space.For a spin particle, the spin operator is given by J=/2, the fundamental representation of SU(2). By takingKronecker products of this representation with itself repeatedly, one may construct all higher irreducible representa-tions. That is, the resulting spin operators for higher spin systems in three spatial dimensions, for arbitrarily large j,can be calculated using this spin operator and ladder operators. They can be found in Rotation group SO(3)#A noteon representations. The analog formula to the above generalization of Eulers formula for Pauli matrices, the groupelement in terms of spin matrices, is tractable, but less simple.[6]

Also useful in the quantum mechanics of multiparticle systems, the general Pauli group Gn is dened to consist of alln-fold tensor products of Pauli matrices.

3.3.2 Quantum information In quantum information, single-qubit quantum gates are 2 2 unitary matrices. The Pauli matrices are some ofthe most important single-qubit operations. In that context, the Cartan decomposition given above is called theZY decomposition of a single-qubit gate. Choosing a dierent Cartan pair gives a similar XY decompositionof a single-qubit gate.

3.4 See also For higher spin generalizations of the Pauli matrices, see spin (physics) Higher spins Gamma matrices Angular momentum

10 CHAPTER 3. PAULI MATRICES

Gell-Mann matrices Poincar group Generalizations of Pauli matrices Bloch sphere Eulers four-square identity

3.5 Remarks[1] The Pauli vector is a formal device. It may be thought of as an element ofM2() 3, where the tensor product space is

endowed with a mapping : 3 M2() 3 M2().

[2] N.B. The relation among a, b, c, n, m, k derived here in the 2 2 representation holds for all representations of SU(2),being a group identity.

3.6 Notes[1] Pauli matrices. Planetmath website. 28 March 2008. Retrieved 28 May 2013.

[2] See the spinor map.

[3] cf. J W Gibbs (1884). Elements of Vector Analysis, New Haven, 1884, p. 67

[4] Explicitly, in the convention of right-space matrices into elements of left-space matrices, it is

1 0 0 00 0 1 00 1 0 00 0 0 1

:

[5] Nakahara, Mikio (2003). Geometry, topology, and physics (2nd ed.). CRC Press. ISBN 978-0-7503-0606-5, pp. xxii.

[6] Curtright, T L; Fairlie, D B; Zachos, C K (2014). A compact formula for rotations as spin matrix polynomials. SIGMA10: 084. doi:10.3842/SIGMA.2014.084.

3.7 References Libo, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5. Schi, Leonard I. (1968). Quantum Mechanics. McGraw-Hill. ISBN 978-0070552876. Leonhardt, Ulf (2010). Essential Quantum Optics. Cambridge University Press. ISBN 0-521-14505-8.

Chapter 4

Pin group

For the postal services company, see PIN Group.Pinor redirects here. For the Spanish municipality, see Pior.

In mathematics, the pin group is a certain subgroup of the Cliord algebra associated to a quadratic space. It maps2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group.In general the map from the Pin group to the orthogonal group is not onto or a universal covering space, but if thequadratic form is denite (and dimension is greater than 2), it is both.The non-trivial element of the kernel is denoted 1, which should not be confused with the orthogonal transform ofreection through the origin, generally denoted I.

4.1 General denitionSee also: Cliord algebra Spin_and_Pin_groups

4.2 Denite form

11

12 CHAPTER 4. PIN GROUP

The pin group of a denite form maps onto the orthogonal group, and each component is simply connected: it doublecovers the orthogonal group. The pin groups for a positive denite quadratic form Q and for its negative Q are notisomorphic, but the orthogonal groups are.[note 1]

In terms of the standard forms, O(n, 0) = O(0,n), but Pin(n, 0) and Pin(0, n) are not isomorphic. Using the "+" signconvention for Cliord algebras (where v2 = Q(v) 2 C`(V;Q) ), one writes

Pin+(n) := Pin(n; 0) Pin(n) := Pin(0; n)

and these both map onto O(n) = O(n, 0) = O(0, n).By contrast, we have the natural isomorphism[note 2] Spin(n, 0) Spin(0, n) and they are both the (unique) doublecover of the special orthogonal group SO(n), which is the (unique) universal cover for n 3.

4.3 Indenite formThere are as many as eight dierent double covers of O(p, q), for p, q 0, which correspond to the extensions ofthe center (which is either C2 C2 or C4) by C2. Only two of them are pin groupsthose that admit the Cliordalgebra as a representation. They are called Pin(p, q) and Pin(q, p) respectively.

4.4 As topological groupEvery connected topological group has a unique universal cover as a topological space, which has a unique groupstructure as a central extension by the fundamental group. For a disconnected topological group, there is a uniqueuniversal cover of the identity component of the group, and one can take the same cover as topological spaces on theother components (which are principal homogeneous spaces for the identity component) but the group structure onother components is not uniquely determined in general.The Pin and Spin groups are particular topological groups associated to the orthogonal and special orthogonal groups,coming from Cliord algebras: there are other similar groups, corresponding to other double covers or to other groupstructures on the other components, but they are not referred to as Pin or Spin groups, nor studied much.Recently, Andrzej Trautman[note 3] found the set of all 32 inequivalent double covers of O(p) x O(q), the maximalcompact subgroup of O(p, q) and an explicit construction of 8 double covers of the same group O(p, q).

4.5 ConstructionThe two pin groups correspond to the two central extensions

1! f1g ! Pin(V )! O(V )! 1:

The group structure on Spin(V) (the connected component of determinant 1) is already determined; the group struc-ture on the other component is determined up to the center, and thus has a 1 ambiguity.The two extensions are distinguished by whether the preimage of a reection squares to 1 Ker (Spin(V) SO(V)),and the two pin groups are named accordingly. Explicitly, a reection has order 2 in O(V), r2 = 1, so the square of thepreimage of a reection (which has determinant one) must be in the kernel of Spin(V) SO(V), so ~r2 = 1 , andeither choice determines a pin group (since all reections are conjugate by an element of SO(V), which is connected,all reections must square to the same value).Concretely, in Pin, ~r has order 2, and the preimage of a subgroup {1, r} is C2 C2: if one repeats the same reectiontwice, one gets the identity.In Pin, ~r has order 4, and the preimage of a subgroup {1, r} is C4: if one repeats the same reection twice, one getsa rotation by 2"the non-trivial element of Spin(V) SO(V) can be interpreted as rotation by 2" (every axisyields the same element).

4.6. CENTER 13

4.5.1 Low dimensionsIn 2 dimensions, the distinction between Pin and Pin mirrors the distinction between the dihedral group of a 2n-gonand the dicyclic group of the cyclic group Cn.In Pin, the preimage of the dihedral group of an n-gon, considered as a subgroup Dihn < O(2), is the dihedral groupof an 2n-gon, Dihn < Pin(2), while in Pin, the preimage of the dihedral group is the dicyclic group Dicn < Pin(2).The resulting commutative square of subgroups for Spin(2), Pin(2), SO(2), O(2) namely Cn, Dihn, Cn, Dihn is also obtained using the projective orthogonal group (going down from O by a 2-fold quotient, instead of up by a2-fold cover) in the square SO(2), O(2), PSO(2), PO(2), though in this case it is also realized geometrically, as theprojectivization of a 2n-gon in the circle is an n-gon in the projective line.In 1 dimension, the pin groups are congruent to the rst dihedral and dicyclic groups:

Pin+(1) = C2 C2 = Dih1Pin(1) = C4 = Dic1:

4.6 CenterThe center is either (C2 C2 or C4) by C2.

4.7 NameThe name was introduced in (Atiyah, Bott & Shapiro 1964, page 3, line 17), where they state This joke is due to J-P.Serre". It is a back-formation from Spin: Pin is to O(n) as Spin is to SO(n)", hence dropping the S from Spinyields Pin.

4.8 Notes[1] In fact, they are equal as subsets of GL(V), not just isomorphic as abstract groups: an operator preserves a form if and only

if it preserves the negative form.[2] They are subalgebras of the dierent algebras C`(n; 0) 6= C`(0; n) , but they are equal as subsets of the vector spaces

C`(n; 0) = C`(0; n) = Rn , and carry the same algebra structure, hence they are naturally identied.[3] A. Trautman (2001). Double Covers of Pseudo-orthogonal Groups. Cliord Analysis and Its Applications, NATO Science

Series, 25: 377388. doi:10.1007/978-94-010-0862-4_32.

4.9 References Atiyah, M.F.; Bott, R.; Shapiro, A. (1964), Cliord modules, Topology, 3, suppl. 1: 338 M. Karoubi (1968). Algbres de Cliord et K-thorie. Ann. Sci. c. Norm. Sup. 1 (2): 161270. Dabrowski, L. (1988), Group Actions on Spinors, Bibliopolis, ISBN 88-7088-205-5 Carlip, S.; DeWitt-Morette, C. (1988), Where the sign of the metric makes a dierence, Phys. Rev. Lett.60: 15991601

Chamblin, A. (1994), On the obstructions to non-Cliordian pin structures, Comm. Math. Phys. 164: 6585 Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. ISBN978-0-691-08542-5.

Karoubi, Max (2008). K-Theory. Springer. pp. 212214. ISBN 978-3-540-79889-7.

Chapter 5

Ping-pong lemma

In mathematics, the ping-pong lemma, or table-tennis lemma, is any of several mathematical statements that ensurethat several elements in a group acting on a set freely generates a free subgroup of that group.

5.1 HistoryThe ping-pong argument goes back to late 19th century and is commonly attributed[1] to Felix Klein who used it tostudy subgroups of Kleinian groups, that is, of discrete groups of isometries of the hyperbolic 3-space or, equivalentlyMbius transformations of the Riemann sphere. The ping-pong lemma was a key tool used by Jacques Tits in his1972 paper[2] containing the proof of a famous result now known as the Tits alternative. The result states that a nitelygenerated linear group is either virtually solvable or contains a free subgroup of rank two. The ping-pong lemma andits variations are widely used in geometric topology and geometric group theory.Modern versions of the ping-pong lemma can be found in many books such as Lyndon&Schupp,[3] de la Harpe,[1]Bridson&Haeiger[4] and others.

5.2 Formal statements

5.2.1 Ping-pong lemma for several subgroups

This version of the ping-pong lemma ensures that several subgroups of a group acting on a set generate a free product.The following statement appears in,[5] and the proof is from.[1]

Let G be a group acting on a set X and let H1, H2,...., Hk be nontrivial subgroups of G where k2, such that at leastone of these subgroups has order greater than 2. Suppose there exist disjoint nonempty subsets X1, X2,....,Xk of Xsuch that the following holds:

For any is and for any hHi, h1 we have h(Xs)Xi.

Then

hH1; : : : ;Hki = H1 Hk:

Proof

By the denition of free product, it suces to check that a given reduced word is nontrivial. Let w be such a word,and let

14

5.3. EXAMPLES 15

w =

mYi=1

wi;i :

Where ws,k Hs for all such k, and since w is fully reduced i i for any i. We then let w act on an element ofone of the sets Xi. As we assume for at least one subgroup H has order at least 3, without loss we may assume thatH1 is at least 3. We rst make the assumption that 1 and are both 1. From here we consider w acting on X2. Weget the following chain of containments and note that since the X are disjoint that w acts nontrivially and is thus notthe identity element.

w(X2) Qm1

i=1 wi;i(X1) Qm2

i=1 wi;i(Xm1) w1;1w2;2(X3) w1;1(X2) X1

To nish the proof we must consider the three cases:

If 1 = 1;m 6= 1 , then let h 2 H1 n fw11;1 ; 1g If 1 6= 1;m = 1 , then let h 2 H1 n fw1;m ; 1g And if 1 6= 1;m 6= 1 , then let h 2 H1 n f1g

In each case, hwh1 is a reduced word with 1' and m '' both 1, and thus is nontrivial. Finally, hwh1 is not 1, andso neither is w. This proves the claim.

5.2.2 The Ping-pong lemma for cyclic subgroupsLetG be a group acting on a set X. Let a1,...,ak be elements ofG, where k 2. Suppose there exist disjoint nonemptysubsets

X1+,...,Xk+ and X1,...,Xk

of X with the following properties:

ai(X Xi) Xi+ for i = 1, ..., k;

ai1(X Xi+) Xi for i = 1, ..., k.

Then the subgroup H = G generated by a1, ..., ak is free with free basis {a1, ..., ak}.

Proof

This statement follows as a corollary of the version for general subgroups if we let Xi= Xi+Xi and let Hi = ai.

5.3 Examples

5.3.1 Special linear group exampleOne can use the ping-pong lemma to prove[1] that the subgroup H = SL(2,Z), generated by the matrices

A=

0@1 20 1

1A and B=0@1 02 1

1A

is free of rank two.

16 CHAPTER 5. PING-PONG LEMMA

Proof

Indeed, let H1 = and H2 = be cyclic subgroups of SL(2,Z) generated by A and B accordingly. It is not hardto check that A and B are elements of innite order in SL(2,Z) and that

H1 = fAnjn 2 Zg =

1 2n0 1

: n 2 Z

and

H2 = fBnjn 2 Zg =

1 02n 1

: n 2 Z

:

Consider the standard action of SL(2,Z) on R2 by linear transformations. Put

X1 =

xy

2 R2 : jxj > jyj

and

X2 =

xy

2 R2 : jxj < jyj

:

It is not hard to check, using the above explicitly descriptions of H1 and H2 that for every nontrivial g H1 we haveg(X2) X1 and that for every nontrivial g H2 we have g(X1) X2. Using the alternative form of the ping-ponglemma, for two subgroups, given above, we conclude thatH =H1H2. Since the groupsH1 andH2 are innite cyclic,it follows that H is a free group of rank two.

5.3.2 Word-hyperbolic group exampleLet G be a word-hyperbolic group which is torsion-free, that is, with no nontrivial elements of nite order. Let g, h G be two non-commuting elements, that is such that gh hg. Then there existsM1 such that for any integers n M, m M the subgroup H = G is free of rank two.

Sketch of the proof[6]

The groupG acts on its hyperbolic boundary G by homeomorphisms. It is known that if a G is a nontrivial elementthen a has exactly two distinct xed points, a and a in G and that a is an attracting xed point while a is arepelling xed point.Since g and h do not commute, the basic facts about word-hyperbolic groups imply that g, g, h and h are fourdistinct points in G. Take disjoint neighborhoods U, U, V and V of g, g, h and h in G respectively.Then the attracting/repelling properties of the xed points of g and h imply that there exists M 1 such that for anyintegers n M, m M we have:

gn(G U) U gn(G U) U hm(G V) V hm(G V) V

The ping-pong lemma now implies that H = G is free of rank two.

5.4. APPLICATIONS OF THE PING-PONG LEMMA 17

5.4 Applications of the ping-pong lemma The ping-pong lemma is used in Kleinian groups to study their so-called Schottky subgroups. In the Kleiniangroups context the ping-pong lemma can be used to show that a particular group of isometries of the hyperbolic3-space is not just free but also properly discontinuous and geometrically nite.

Similar Schottky-type arguments are widely used in geometric group theory, particularly for subgroups ofword-hyperbolic groups[6] and for automorphism groups of trees.[7]

Ping-pong lemma is also used for studying Schottky-type subgroups of mapping class groups of Riemannsurfaces, where the set on which the mapping class group acts is the Thurston boundary of the Teichmllerspace.[8] A similar argument is also utilized in the study of subgroups of the outer automorphism group of afree group.[9]

One of the most famous applications of the ping-pong lemma is in the proof of Jacques Tits of the so-calledTits alternative for linear groups.[2] (see also [10] for an overview of Tits proof and an explanation of the ideasinvolved, including the use of the ping-pong lemma).

There are generalizations of the ping-pong lemma that produce not just free products but also amalgamatedfree products and HNN extensions.[3] These generalizations are used, in particular, in the proof of MaskitsCombination Theorem for Kleinian groups.[11]

There are also versions of the ping-pong lemma which guarantee that several elements in a group generate afree semigroup. Such versions are available both in the general context of a group action on a set,[12] and forspecic types of actions, e.g. in the context of linear groups,[13] groups acting on trees[14] and others.[15]

5.5 References[1] Pierre de la Harpe. Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press,

Chicago. ISBN 0-226-31719-6; Ch. II.B The table-Tennis Lemma (Kleins criterion) and examples of free products"; pp.2541.

[2] J. Tits. Free subgroups in linear groups. Journal of Algebra, vol. 20 (1972), pp. 250270

[3] Roger C. Lyndon and Paul E. Schupp. Combinatorial Group Theory. Springer-Verlag, New York, 2001. Classics inMathematics series, reprint of the 1977 edition. ISBN 978-3-540-41158-1; Ch II, Section 12, pp. 167169

[4] Martin R. Bridson, and Andr Haeiger. Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wis-senschaften [Fundamental Principles of Mathematical Sciences], 319. Springer-Verlag, Berlin, 1999. ISBN 3-540-64324-9; Ch.III., pp. 467468

[5] Andrij Olijnyk and Vitaly Suchchansky. Representations of free products by innite unitriangular matrices over nite elds.International Journal of Algebra and Computation. Vol. 14 (2004), no. 56, pp. 741749; Lemma 2.1

[6] M. Gromov. Hyperbolic groups. Essays in group theory, pp. 75263, Mathematical Sciiences Research Institute Publica-tions, 8, Springer, New York, 1987; ISBN 0-387-96618-8; Ch. 8.2, pp. 211219.

[7] Alexander Lubotzky. Lattices in rank one Lie groups over local elds. Geometric and Functional Analysis, vol. 1 (1991),no. 4, pp. 406431

[8] Richard P. Kent, and Christopher J. Leininger. Subgroups of mapping class groups from the geometrical viewpoint. In thetradition of Ahlfors-Bers. IV, pp. 119141, Contemporary Mathematics series, 432, American Mathematical Society,Providence, RI, 2007; ISBN 978-0-8218-4227-0; 0-8218-4227-7

[9] M. Bestvina, M. Feighn, and M. Handel. Laminations, trees, and irreducible automorphisms of free groups. Geometric andFunctional Analysis, vol. 7 (1997), no. 2, pp. 215244.

[10] Pierre de la Harpe. Free groups in linear groups. L'Enseignement Mathmatique (2), vol. 29 (1983), no. 1-2, pp. 129144

[11] Bernard Maskit. Kleinian groups. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathe-matical Sciences], 287. Springer-Verlag, Berlin, 1988. ISBN 3-540-17746-9; Ch. VII.C and Ch. VII.E pp.149156 andpp. 160167

18 CHAPTER 5. PING-PONG LEMMA

[12] Pierre de la Harpe. Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press,Chicago. ISBN 0-226-31719-6; Ch. II.B The table-Tennis Lemma (Kleins criterion) and examples of free products"; pp.187188.

[13] Alex Eskin, Shahar Mozes and Hee Oh. On uniform exponential growth for linear groups. Inventiones Mathematicae. vol.60 (2005), no. 1, pp.14321297; Lemma 2.2

[14] Roger C. Alperin and Guennadi A. Noskov. Uniform growth, actions on trees and GL2. Computational and StatisticalGroup Theory:AMS Special Session Geometric Group Theory, April 2122, 2001, Las Vegas, Nevada, AMS SpecialSession Computational Group Theory, April 2829, 2001, Hoboken, New Jersey. (Robert H. Gilman, Vladimir Shpilrain,Alexei G. Myasnikov, editors). American Mathematical Society, 2002. ISBN 978-0-8218-3158-8; page 2, Lemma 3.1

[15] Yves de Cornulier and Romain Tessera. Quasi-isometrically embedded free sub-semigroups. Geometry & Topology, vol.12 (2008), pp. 461473; Lemma 2.1

5.6 See also Free group Free product Kleinian group Tits alternative Word-hyperbolic group Schottky group

Chapter 6

Poincar group

For the Poincar group (fundamental group) of a topological space, see Fundamental group.

The Poincar group, named after Henri Poincar,[1] is the group of Minkowski spacetime isometries.[2][3] It is aten-generator non-abelian Lie group of fundamental importance in physics.

6.1 OverviewA Minkowski spacetime isometry has the property that the interval between events is left invariant. For example,if everything was postponed by two hours including two events and the path you took to go from one to the other,then the time interval between the events recorded by a stop-watch you carried with you would be the same. Or ifeverything was shifted ve miles to the west, or turned 60 degrees to the right, you would also see no change in theinterval. It turns out that the proper length of an object is also unaected by such a shift. A time or space reversal (areection) is also an isometry of this group.InMinkowski space (i.e. ignoring the eects of gravity), there are ten degrees of freedom of the isometries, whichmaybe thought of as translation through time or space (four degrees, one per dimension); reection through a plane (threedegrees, the freedom in orientation of this plane); or a "boost" in any of the three spatial directions (three degrees).Composition of transformations is the operator of the Poincar group, with proper rotations being produced as thecomposition of an even number of reections.In classical physics, the Galilean group is a comparable ten-parameter group that acts on absolute time and space.Instead of boosts, it features shear mappings to relate co-moving frames of reference.

6.2 DetailsThe Poincar group is the group of Minkowski spacetime isometries. It is a ten-dimensional noncompact Lie group.The abelian group of translations is a normal subgroup, while the Lorentz group is also a subgroup, the stabilizer ofthe origin. The Poincar group itself is the minimal subgroup of the ane group which includes all translations andLorentz transformations. More precisely, it is a semidirect product of the translations and the Lorentz group,

R1;3 o SO(1; 3) :Another way of putting this is that the Poincar group is a group extension of the Lorentz group by a vector representationof it; it is sometimes dubbed, informally, as the "inhomogeneous Lorentz group". In turn, it can also be obtained as agroup contraction of the de Sitter group SO(4,1) ~ Sp(2,2), as the de Sitter radius goes to innity.Its positive energy unitary irreducible representations are indexed by mass (nonnegative number) and spin (integer orhalf integer) and are associated with particles in quantum mechanics (see Wigners classication).In accordance with the Erlangen program, the geometry of Minkowski space is dened by the Poincar group:Minkowski space is considered as a homogeneous space for the group.

19

20 CHAPTER 6. POINCAR GROUP

The Poincar algebra is the Lie algebra of the Poincar group. It is a Lie algebra extension of the Lie algebraof the Lorentz group. More specically, the proper (det=1), orthochronous (001) part of the Lorentz sub-group (its identity component), SO+(1, 3), is connected to the identity and is thus provided by the exponentiationexp(iaP) exp(iM/2) of this Lie algebra. In component form, the Poincar algebra is given by the commuta-tion relations:[4][5]

where P is the generator of translations, M is the generator of Lorentz transformations, and is the Minkowski metric(see Sign convention).The bottom commutation relation is the (homogeneous) Lorentz group, consisting of rotations, Ji = imnMmn/2,and boosts, Ki =Mi0. In this notation, the entire Poincar algebra is expressible in noncovariant (but more practical)language as

[Jm; Pn] = imnkPk ;

[Ji; P0] = 0 ;

[Ki; Pk] = iikP0 ;

[Ki; P0] = iPi ;[Jm; Jn] = imnkJk ;

[Jm;Kn] = imnkKk ;

[Km;Kn] = imnkJk ;where the bottom line commutator of two boosts is often referred to as a Wigner rotation. Note the importantsimplication [Jm+i Km , Jni Kn] = 0, which permits reduction of the Lorentz subalgebra to su(2)su(2) andecient treatment of its associated representations.The Casimir invariants of this algebra are PP and W W where W is the PauliLubanski pseudovector; theyserve as labels for the representations of the group.The Poincar group is the full symmetry group of any relativistic eld theory. As a result, all elementary particles fallin representations of this group. These are usually specied by the four-momentum squared of each particle (i.e. itsmass squared) and the intrinsic quantum numbers JPC, where J is the spin quantum number, P is the parity and C isthe charge-conjugation quantum number. In practice, charge conjugation and parity are violated by many quantumeld theories; where this occurs, P and C are forfeited. Since CPT symmetry is invariant in quantum eld theory, atime-reversal quantum number may be constructed from of those given.As a topological space, the group has four connected components: the component of the identity; the time reversedcomponent; the spatial inversion component; and the component which is both time-reversed and spatially inverted.

6.3 Poincar symmetryPoincar symmetry is the full symmetry of special relativity. It includes:

translations (displacements) in time and space (P), forming the abelian Lie group of translations on space-time; rotations in space, forming the non-Abelian Lie group of three-dimensional rotations (J); boosts, transformations connecting two uniformly moving bodies (K).

The last two symmetries, J and K, together make the Lorentz group (see also Lorentz invariance); the semi-directproduct of the translations group and the Lorentz group then produce the Poincar group. Objects which are invariantunder this group are then said to possess Poincar invariance or relativistic invariance.

6.4. SEE ALSO 21

6.4 See also Euclidean group Representation theory of the Poincar group Wigners classication Symmetry in quantum mechanics Center of mass (relativistic) PauliLubanski pseudovector Particle physics and representation theory

6.5 Notes[1] Poincar, Henri, "Sur la dynamique de llectron",Rendiconti del Circolomatematico di Palermo 21: 129176, doi:10.1007/bf03013466

(Wikisource translation: On the Dynamics of the Electron).

[2] Minkowski, Hermann, "Die Grundgleichungen fr die elektromagnetischen Vorgnge in bewegten Krpern", Nachrichtenvon der Gesellschaft der Wissenschaften zu Gttingen, Mathematisch-Physikalische Klasse: 53111 (Wikisource translation:The Fundamental Equations for Electromagnetic Processes in Moving Bodies).

[3] Minkowski, Hermann, "Raum und Zeit", Physikalische Zeitschrift 10: 7588

[4] N.N. Bogolubov (1989). General Principles of Quantum Field Theory (2nd ed.). Springer. p. 272. ISBN 0-7923-0540-X.

[5] T. Ohlsson (2011). Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum FieldTheory. Cambridge University Press. p. 10. ISBN 1-13950-4320.

6.6 References Wu-Ki Tung (1985). Group Theory in Physics. World Scientic Publishing. ISBN 9971-966-57-3. Weinberg, Steven (1995). The Quantum Theory of Fields 1. Cambridge: Cambridge University press. ISBN978-0-521-55001-7.

L.H. Ryder (1996). Quantum Field Theory (2nd ed.). Cambridge University Press. p. 62. ISBN 0-52147-8146.

Chapter 7

PoissonLie group

In mathematics, a PoissonLie group is a Poisson manifold that is also a Lie group, with the group multiplicationbeing compatible with the Poisson algebra structure on the manifold. The algebra of a PoissonLie group is a Liebialgebra.

7.1 DenitionA PoissonLie group is a Lie group G equipped with a Poisson bracket for which the group multiplication :G G ! G with (g1; g2) = g1g2 is a Poisson map, where the manifold GG has been given the structure of aproduct Poisson manifold.Explicitly, the following identity must hold for a PoissonLie group:

ff1; f2g(gg0) = ff1 Lg; f2 Lgg(g0) + ff1 Rg0 ; f2 Rg0g(g)where f1 and f2 are real-valued, smooth functions on the Lie group, while g and g' are elements of the Lie group.Here, Lg denotes left-multiplication and Rg denotes right-multiplication.If P denotes the corresponding Poisson bivector on G, the condition above can be equivalently stated as

P(gg0) = Lg(P(g0)) +Rg0(P(g))Note that for Poisson-Lie group always ff; gg(e) = 0 , or equivalently P(e) = 0 . This means that non-trivialPoisson-Lie structure is never symplectic, not even of constant rank.

7.2 HomomorphismsA PoissonLie group homomorphism : G ! H is dened to be both a Lie group homomorphism and a Poissonmap. Although this is the obvious denition, neither left translations nor right translations are Poisson maps. Also,the inversion map : G! G taking (g) = g1 is not a Poisson map either, although it is an anti-Poisson map:

ff1 ; f2 g = ff1; f2g for any two smooth functions f1; f2 on G.

7.3 References Doebner, H.-D.; Hennig, J.-D., eds. (1989). Quantum groups. Proceedings of the 8th International Workshopon Mathematical Physics, Arnold Sommerfeld Institute, Claausthal, FRG. Berlin: Springer-Verlag. ISBN 3-

22

7.3. REFERENCES 23

540-53503-9.

Chari, Vyjayanthi; Pressley, Andrew (1994). A Guide to Quantum Groups. Cambridge: Cambridge UniversityPress. ISBN 0-521-55884-0.

Chapter 8

Polar decomposition

In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linearoperator is a factorization analogous to the polar form of a nonzero complex number z as z = rei where r is theabsolute value of z (a positive real number), and ei is an element of the circle group.

8.1 Matrix polar decompositionThe polar decomposition of a square complex matrix A is a matrix decomposition of the form

A = UP

where U is a unitary matrix and P is a positive-semidenite Hermitian matrix. Intuitively, the polar decompositionseparates A into a component that stretches the space along a set of orthogonal axes, represented by P, and a rotation(with possible reection) represented by U. The decomposition of the complex conjugate of A is given by A = U P.This decomposition always exists; and so long as A is invertible, it is unique, with P positive-denite. Note that

detA = detU detP = rei

gives the corresponding polar decomposition of the determinant of A, since detP = r = j detAj and detU = ei .The matrix P is always unique, even if A is singular, and given by

P =pAA

where A* denotes the conjugate transpose of A. This expression is meaningful since a positive-semidenite Hermitianmatrix has a unique positive-semidenite square root. If A is invertible, then the matrix U is given by

U = AP1:

In terms of the singular value decomposition of A, A = W V*, one has

P = V V

U = WV

conrming that P is positive-denite and U is unitary. Thus, the existence of the SVD is equivalent to the existenceof polar decomposition.

24

8.2. BOUNDED OPERATORS ON HILBERT SPACE 25

One can also decompose A in the form

A = P 0U

Here U is the same as before and P is given by

P 0 = UPU1 =pAA = WW :

This is known as the left polar decomposition, whereas the previous decomposition is known as the right polar de-composition. Left polar decomposition is also known as reverse polar decomposition.The matrix A is normal if and only if P = P. Then U = U, and it is possible to diagonalise U with a unitarysimilarity matrix S that commutes with , giving S U S* = 1, where is a diagonal unitary matrix of phases ei.Putting Q = V S*, one can then re-write the polar decomposition as

A = (QQ)(QQ);

so A then thus also has a spectral decomposition

A = QQ

with complex eigenvalues such that * = 2 and a unitary matrix of complex eigenvectors Q.

8.2 Bounded operators on Hilbert spaceThe polar decomposition of any bounded linear operator A between complex Hilbert spaces is a canonical factor-ization as the product of a partial isometry and a non-negative operator.The polar decomposition for matrices generalizes as follows: if A is a bounded linear operator then there is a uniquefactorization of A as a product A = UP where U is a partial isometry, P is a non-negative self-adjoint operator andthe initial space of U is the closure of the range of P.The operator U must be weakened to a partial isometry, rather than unitary, because of the following issues. If A isthe one-sided shift on l2(N), then |A| = {A*A} = I. So if A = U |A|, U must be A, which is not unitary.The existence of a polar decomposition is a consequence of Douglas lemma:

Lemma If A, B are bounded operators on a Hilbert space H, and A*A B*B, then there exists a con-traction C such that A = CB. Furthermore, C is unique if Ker(B*) Ker(C).

The operator C can be dened by C(Bh) := Ah for all h in H, extended by continuity to the closure of Ran(B), and byzero on the orthogonal complement to all of H. The lemma then follows since A*A B*B implies Ker(A) Ker(B).In particular. If A*A = B*B, then C is a partial isometry, which is unique if Ker(B*) Ker(C). In general, for anybounded operator A,

AA = (AA)12 (AA)

12 ;

where (A*A) is the unique positive square root of A*A given by the usual functional calculus. So by the lemma, wehave

A = U(AA)12

26 CHAPTER 8. POLAR DECOMPOSITION

for some partial isometry U, which is unique if Ker(A*) Ker(U). Take P to be (A*A) and one obtains the polardecomposition A = UP. Notice that an analogous argument can be used to show A = P'U' , where P' is positive andU' a partial isometry.WhenH is nite-dimensional,U can be extended to a unitary operator; this is not true in general (see example above).Alternatively, the polar decomposition can be shown using the operator version of singular value decomposition.By property of the continuous functional calculus, |A| is in the C*-algebra generated by A. A similar but weakerstatement holds for the partial isometry: U is in the von Neumann algebra generated by A. If A is invertible, the polarpart U will be in the C*-algebra as well.

8.3 Unbounded operatorsIf A is a closed, densely dened unbounded operator between complex Hilbert spaces then it still has a (unique) polardecomposition

A = U jAjwhere |A| is a (possibly unbounded) non-negative self adjoint operator with the same domain as A, and U is a partialisometry vanishing on the orthogonal complement of the range Ran(|A|).The proof uses the same lemma as above, which goes through for unbounded operators in general. If Dom(A*A)= Dom(B*B) and A*Ah = B*Bh for all h Dom(A*A), then there exists a partial isometry U such that A = UB.U is unique if Ran(B) Ker(U). The operator A being closed and densely dened ensures that the operator A*Ais self-adjoint (with dense domain) and therefore allows one to dene (A*A). Applying the lemma gives polardecomposition.If an unbounded operator A is aliated to a von Neumann algebra M, and A = UP is its polar decomposition, thenU is inM and so is the spectral projection of P, 1B(P), for any Borel set B in [0, ).

8.4 Quaternion polar decompositionThe polar decomposition of quaternions H depends on the sphere fxi + yj + zk 2 H : x2 + y2 + z2 = 1g ofsquare roots of minus one. Given any r on this sphere, and an angle < a , the versor ear = cos(a) + r sin(a)is on the 3-sphere of H. For a = 0 and a = , the versor is 1 or 1 regardless of which r is selected. The norm t of aquaternion q is the Euclidean distance from the origin to q. When a quaternion is not just a real number, then thereis a unique polar decomposition q = tear:

8.5 Alternative planar decompositionsIn the Cartesian plane, alternative planar ring decompositions arise as follows:

If x 0, z = x ( 1 + (y/x) ) is a polar decomposition of a dual number z = x + y , where 2 = 0, i.e. isnilpotent. In this polar decomposition, the unit circle has been replaced by the line x = 1, the polar angle bythe slope y/x, and the radius x is negative in the left half-plane.

If x2 y2, then the unit hyperbola x2 y2 = 1 and its conjugate x2 y2 = 1 can be used to form a polardecomposition based on the branch of the unit hyperbola through (1,0). This branch is parametrized by thehyperbolic angle a and is written

cosh(a) + j sinh(a) = exp(aj) = eaj

where j 2 = +1 and the arithmetic[1] of split-complex numbers is used. The branch through (1,0) istraced by ea j . Since the operation of multiplying by j reects a point across the line y = x, the second

8.6. NUMERICAL DETERMINATION OF THE MATRIX POLAR DECOMPOSITION 27

hyperbola has branches traced by jea j or jea j . Therefore a point in one of the quadrants has a polardecomposition in one of the forms:

reaj ;reaj ; rjeaj ;rjeaj ; r > 0

The set { 1, 1, j, j } has products that make it isomorphic to the Klein four-group. Evidently polardecomposition in this case involves an element from that group.

8.6 Numerical determination of the matrix polar decompositionTo compute an approximation of the polar decomposition A=UP, usually the unitary factor U is approximated.[2][3]The iteration is based on Herons method for the square root of 1 and computes, starting from U0 = A , the sequence

Uk+1 =12

Uk + (U

k )1 , k=0,1,2,...

The combination of inversion and Hermite conjugation is chosen so that in the singular value decomposition, theunitary factors remain the same and the iteration reduces to Herons method on the singular values.This basic iteration may be rened to speed up the process:

Every step or in regular intervals, the range of the singular values of Uk is estimated and then the matrix isrescaled to kUk to center the singular values around 1. The scaling factor k is computed using matrix normsof the matrix and its inverse. Examples of such scale estimates are:

k =4

skU1k k1 kU1k k1kUkk1 kUkk1

using the row-sum and column-sum matrix norms or

k =

skU1k kFkUkkF

using the Frobenius norm. Including the scale factor, the iteration is nowUk+1 =

12

kUk +

1

k(Uk )

1, k=0,1,2,...

The QR decomposition can be used in a preparation step to reduce a singular matrix A to a smaller regularmatrix, and inside every step to speed up the computation of the inverse.

Heron' method for computing roots of x21 = 0 can be replaced by higher order methods, for instance basedon Halleys method of third order, resulting in

Uk+1 = Uk (I + 3UkUk)

1(3 I + UkUk) , k=0,1,2,...

This iteration can again be combined with rescaling. This particular formula has the benet that it alsoapplicable to singular or rectangular matrices A.

8.7 See also Cartan decomposition Algebraic polar decomposition

28 CHAPTER 8. POLAR DECOMPOSITION

8.8 References[1] Sobczyk, G.(1995) Hyperbolic Number Plane, College Mathematics Journal 26:26880

[2] Higham, Nicholas J. (1986). Computing the polar decomposition with applications. SIAM J. Sci. Stat. Comput. (Philadel-phia, PA, USA: Society for Industrial and Applied Mathematics) 7 (4): 11601174. doi:10.1137/0907079. ISSN 0196-5204.

[3] Byers, Ralph; Hongguo Xu (2008). A New Scaling for Newtons Iteration for the Polar Decomposition and its BackwardStability. SIAM J. Matrix Anal. Appl. (Philadelphia, PA, USA: Society for Industrial and Applied Mathematics) 30 (2):822843. doi:10.1137/070699895. ISSN 0895-4798.

8.9 Literature Conway, J.B.: A Course in Functional Analysis. Graduate Texts in Mathematics. New York: Springer 1990 Douglas, R.G.: On Majorization, Factorization, and Range Inclusion of Operators on Hilbert Space. Proc.Amer. Math. Soc. 17, 413-415 (1966)

Helgason, Sigurdur (1978), Dierential geometry, Lie groups, and symmetric spaces, Academic Press, ISBN0-8218-2848-7

8.10 External links Polar Decomposition on www.continuummechanics.org

Chapter 9

Pre-Lie algebra

In mathematics, a pre-Lie algebra is an algebraic structure on a vector space, that describes some properties ofobjects such as rooted trees and vector elds on ane space.The notion of pre-Lie algebra has been introduced by Murray Gerstenhaber in his work on deformations of algebras.Pre-Lie algebras have been considered under some other names, among which one can cite left-symmetric algebras,right-symmetric algebras or Vinberg algebras.

9.1 Denition

A pre-Lie algebra (V; /) is a vector space V with a bilinear map / : V V ! V , satisfying the relation (x / y) /z x / (y / z) = (x / z) / y x / (z / y):This identity can be seen as the invariance of the associator (x; y; z) = (x / y) / z x / (y / z) under the exchangeof the two variables y and z .Every associative algebra is hence also a pre-Lie algebra, as the associator vanishes identically.

9.2 Examples

Vector elds on the ane space

If we denote by f(x)@x the vector eld x 7! f(x) , and if we dene / as f(x) / g(x) = f 0(x)g(x) , we can see thatthe operator / is exactly the application of the g(x)@x eld to f(x)@x eld. (g(x)@x)(f(x)@x) = g(x)@xf(x)@x =g(x)f 0(x)@x

If we study the dierence between (x/ y) / z and x/ (y / z) , we have (x/ y) / zx/ (y / z) = (x0y)0zx0y0z =x0y0zx00yz z0y0z = x00yz which is symmetric on y and z.

Rooted trees

Let T be the vector space spanned by all rooted trees.One can introduce a bilinear productx on T as follows. Let 1 and 2 be two rooted trees.1 x 2 =

Ps2Vertices(1) 1 s 2

where 1 s 2 is the rooted tree obtained by adding to the disjoint union of 1 and 2 an edge going from the vertexs of 1 to the root vertex of 2 .Then (T;x) is a free pre-Lie algebra on one generator.

29

30 CHAPTER 9. PRE-LIE ALGEBRA

9.3 References Chapoton, F.; Livernet, M. (2001), Pre-Lie algebras and the rooted trees operad, International MathematicsResearch Notices 8 (8): 395408, doi:10.1155/S1073792801000198, MR 1827084.

Szczesny,M. (2010), Pre-Lie algebras and incidence categories of colored rooted trees 1007, p. 4784, arXiv:1007.4784,Bibcode:2010arXiv1007.4784S.

Chapter 10

Principal homogeneous space

For the term torsor in algebraic geometry, see torsor (algebraic geometry).

In mathematics, a principal homogeneous space,[1] or torsor, for a group G is a homogeneous space X for G inwhich the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group G isa non-empty set X on which G acts freely and transitively, meaning that for any x, y in X there exists a unique g inG such that xg = y where denotes the (right) action of G on X. An analogous denition holds in other categorieswhere, for example,

G is a topological group, X is a topological space and the action is continuous, G is a Lie group, X is a smooth manifold and the action is smooth, G is an algebraic group, X is an algebraic variety and the action is regular.

If G is nonabelian then one must distinguish between left and right torsors according to whether the action is on theleft or right. In this article, we will use right actions. To state the denition more explicitly, X is a G-torsor if X isnonempty and is equipped with a map (in the appropriate category) X G X such that

x1 = xx(gh) = (xg)h

for all x X and all g,h G and such that the map X G X X given by

(x; g) 7! (x; x g)is an isomorphism (of sets, or topological spaces or ..., as appropriate). Note that this means that X and G areisomorphic. However and this is the essential point, there is no preferred 'identity' point in X. That is, X looksexactly like G except that which point is the identity has been forgotten. This concept is often used in mathematicsas a way of passing to a more intrinsic point of view, under the heading 'throw away the origin'.Since X is not a group we cannot multiply elements; we can, however, take their quotient. That is, there is a mapX X G that sends (x,y) to the unique element g = x \ y G such that y = xg.The composition of this operation with the right group action, however, yields a ternary operation X (X X) X G X that serves as an ane generalization of group multiplication and is sucient to both characterize a principalhomogeneous space algebraically, and intrinsically characterize the group it is associated with. If x/y z is the resultof this operation, then the following identities

x/y y = x = y/y xv/w (x/y z) = (v/w x)/y z

31

32 CHAPTER 10. PRINCIPAL HOMOGENEOUS SPACE

will suce to dene a principal homogeneous space, while the additional property

x/y z = z/y x

identies those spaces that are associated with abelian groups. The group may be dened as formal quotients xnysubject to the equivalence relation

(x/w y)nz = yn(w/x z)

with the group product, identity and inverse dened, respectively, by

(wny) (xnz) = yn(w/x z) = (x/w y)nz

e = xnx(xny)1 = ynx;and the group action by

x (ynz) = x/y z:

10.1 ExamplesEvery groupG can itself be thought of as a left or rightG-torsor under the natural action of left or right multiplication.Another example is the ane space concept: the idea of the ane space A underlying a vector space V can be saidsuccinctly by saying that A is a principal homogeneous space for V acting as the additive group of translations.The ags of any regular polytope form a torsor for its symmetry group.Given a vector space V we can take G to be the general linear group GL(V), and X to be the set of all (ordered)bases of V. Then G acts on X in the way that it acts on vectors of V ; and it acts transitively since any basis can betransformed via G to any other. What is more, a linear transformation xing each vector of a basis will x all v inV, hence being the neutral element of the general linear group GL(V) : so that X is indeed a principal homogeneousspace. One way to follow basis-dependence in a linear algebra argument is to track variables x in X. Similarly, thespace of orthonormal bases (the Stiefel manifold Vn(Rn) of n-frames) is a principal homogeneous space for theorthogonal group.In category theory, if two objects X and Y are isomorphic, then the isomorphisms between them, Iso(X,Y), forma torsor for the automorphism group of X, Aut(X), and likewise for Aut(Y); a choice of isomorphism between theobjects gives an isomorphism between these groups and identies the torsor with these two groups, and giving thetorsor a group structure (as it is a base point).

10.2 ApplicationsThe principal homogeneous space concept is a special case of that of principal bundle: it means a principal bundle withbase a single point. In other words the local theory of principal bundles is that of a family of principal homogeneousspaces depending on some parameters in the base. The 'origin' can be supplied by a section of the bundlesuchsections are usually assumed to exist locally on the basethe bundle being locally trivial, so that the local structureis that of a cartesian product. But sections will often not exist globally. For example a dierential manifold M has aprincipal bundle of frames associated to its tangent bundle. A global section will exist (by denition) only whenM isparallelizable, which implies strong topological restrictions.In number theory there is a (supercially dierent) reason to consider principal homogeneous spaces, for elliptic curvesE dened over a eld K (and more general abelian varieties). Once this was understood various other examples were

10.3. OTHER USAGE 33

collected under the heading, for other algebraic groups: quadratic forms for orthogonal groups, and SeveriBrauervarieties for projective linear groups being two.The reason of the interest for Diophantine equations, in the elliptic curve case, is that K may not be algebraicallyclosed. There can exist curves C that have no point dened over K, and which become isomorphic over a larger eldto E, which by denition has a point over K to serve as identity element for its addition law. That is, for this casewe should distinguish C that have genus 1, from elliptic curves E that have a K-point (or, in other words, provide aDiophantine equation that has a solution in K). The curves C turn out to be torsors over E, and form a set carrying arich structure in the case that K is a number eld (the theory of the Selmer group). In fact a typical plane cubic curveC over Q has no particular reason to have a rational point; the standard Weierstrass model always does, namely thepoint at innity, but you need a point over K to put C into that form over K.This theory has been developed with great attention to local analysis, leading to the denition of the Tate-Shafarevichgroup. In general the approach of taking the torsor theory, easy over an algebraically closed eld, and trying to getback 'down' to a smaller eld is an aspect of descent. It leads at once to questions of Galois cohomology, since thetorsors represent classes in group cohomology H1.

10.3 Other usageThe concept of a principal homogeneous space can also be globalized as follows. LetX be a space (a scheme/manifold/topologicalspace etc.), and let G be a group over X, i.e., a group object in the category of spaces over X. In this case, a (right,say) G-torsor E on X is a space E (of the same type) over X with a (right) G action such that the morphism

E X G! E X Egiven by

(x; g) 7! (x; xg)is an isomorphism in the appropriate category, and such that E is locally trivial on X, in that E X acquires a sectionlocally on X. Isomorphism classes of torsors in this sense correspond to classes in the cohomology group H1(X,G).When we are in the smooth manifold category, then a G-torsor (for G a Lie group) is then precisely a principalG-bundle as dened above.Example: if G is a compact Lie group (say), then EG is a G-torsor over the classifying space BG .

10.4 See also Homogeneous space Heap (mathematics)

10.5 Notes[1] S. Lang and J. Tate (1958). Principal Homogeneous Space Over Abelian Varieties. American Journal of Mathematics

80 (3): 659684. doi:10.2307/2372778.

10.6 Further reading Garibaldi, Skip; Merkurjev, Alexander; Serre, Jean-Pierre (2003). Cohomological invariants in Galois coho-mology. University Lecture Series 28. Providence, RI: American Mathematical Society. ISBN 0-8218-3287-5. Zbl 1159.12311.

Skorobogatov, A. (2001). Torsors and rational points. Cambridge Tracts in Mathematics 144. Cambridge:Cambridge University Press. ISBN 0-521-80237-7. Zbl 0972.14015.

34 CHAPTER 10. PRINCIPAL HOMOGENEOUS SPACE

10.7 External links Torsors made easy by John Baez

Chapter 11

Projective linear group

Projective group redirects here. For other uses, see Projective group (disambiguation).In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the

Relation between the projective special linear group PSL and the projective general linear group PGL.

projective general linear group or PGL) is the induced action of the general linear group of a vector space V onthe associated projective space P(V). Explicitly, the projective linear group is the quotient group

PGL(V) = GL(V)/Z(V)

where GL(V) is the general linear group of V and Z(V) is the subgroup of all nonzero scalar transformations of V;these are quotiented out because they act trivially on the projective space and they form the kernel of the action, andthe notation Z reects that the scalar transformations form the center of the general linear group.The projective special linear group, PSL, is dened analogously, as the induced action of the special linear groupon the associated projective space. Explicitly:

PSL(V) = SL(V)/SZ(V)

where SL(V) is the special linear group over V and SZ(V) is the subgroup of scalar transformations with unitdeterminant. Here SZ is the center of SL, and is naturally identied with the group of nth roots of unity in K(where n is the dimension of V and K is the base eld).

35

36 CHAPTER 11. PROJECTIVE LINEAR GROUP

PGL and PSL are some of the fundamental groups of study, part of the so-called classical groups, and an elementof PGL is called projective linear transformation, projective transformation or homography. If V is the n-dimensional vector space over a eld F, namely V = Fn, the alternate notations PGL(n, F) an