a freudenthal-tits supermagic square - universiteit gent
TRANSCRIPT
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A Freudenthal-Tits Supermagic Square
Alberto Elduque
Universidad de Zaragoza
October 19, 2012Universiteit Gent
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Freudenthal-Tits Magic Square
A supermagic rectangle
A supermagic square
Some conclusions
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Freudenthal-Tits Magic Square
A supermagic rectangle
A supermagic square
Some conclusions
![Page 4: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/4.jpg)
Exceptional Lie algebras
G2, F4, E6, E7, E8
G2 = derO (Cartan 1914)
F4 = derH3(O)(Chevalley-Schafer 1950)
E6 = str0 H3(O)
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Exceptional Lie algebras
G2, F4, E6, E7, E8
G2 = derO (Cartan 1914)
F4 = derH3(O)(Chevalley-Schafer 1950)
E6 = str0 H3(O)
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Exceptional Lie algebras
G2, F4, E6, E7, E8
G2 = derO (Cartan 1914)
F4 = derH3(O)(Chevalley-Schafer 1950)
E6 = str0 H3(O)
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Tits construction (1966)
I C a Hurwitz algebra (unital composition algebra),
I J a central simple Jordan algebra of degree 3,
then
T (C , J) = derC ⊕ (C0 ⊗ J0)⊕ der J
is a Lie algebra (char 6= 2, 3) under a suitable Lie bracket:
[a⊗x , b⊗y ] =1
3tr(xy)Da,b+
([a, b]⊗
(xy−1
3tr(xy)1
))+2t(ab)dx ,y .
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Tits construction (1966)
I C a Hurwitz algebra (unital composition algebra),
I J a central simple Jordan algebra of degree 3,
then
T (C , J) = derC ⊕ (C0 ⊗ J0)⊕ der J
is a Lie algebra (char 6= 2, 3) under a suitable Lie bracket:
[a⊗x , b⊗y ] =1
3tr(xy)Da,b+
([a, b]⊗
(xy−1
3tr(xy)1
))+2t(ab)dx ,y .
![Page 9: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/9.jpg)
Tits construction (1966)
I C a Hurwitz algebra (unital composition algebra),
I J a central simple Jordan algebra of degree 3,
then
T (C , J) = derC ⊕ (C0 ⊗ J0)⊕ der J
is a Lie algebra (char 6= 2, 3) under a suitable Lie bracket:
[a⊗x , b⊗y ] =1
3tr(xy)Da,b+
([a, b]⊗
(xy−1
3tr(xy)1
))+2t(ab)dx ,y .
![Page 10: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/10.jpg)
Freudenthal-Tits Magic Square
T (C , J) H3(k) H3(k × k) H3(Mat2(k)) H3(C (k))
k A1 A2 C3 F4
k × k A2 A2 ⊕ A2 A5 E6
Mat2(k) C3 A5 D6 E7
C (k) F4 E6 E7 E8
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Tits construction rearranged
J = H3(C ′) ' k3 ⊕(⊕2
i=0ιi (C′)),
J0 ' k2 ⊕(⊕2
i=0ιi (C′)),
der J ' tri(C ′)⊕(⊕2
i=0ιi (C′)),
T (C , J) = derC ⊕ (C0 ⊗ J0)⊕ der J
' derC ⊕ (C0 ⊗ k2)⊕(⊕2
i=0C0 ⊗ ιi (C ′))⊕(tri(C ′)⊕ (⊕2
i=0ιi (C′))
)'
(tri(C )⊕ tri(C ′)
)⊕
(⊕2
i=0ιi (C ⊗ C ′))
tri(C ) = {(d0, d1, d2) ∈ so(C )3 : d0(xy) = d2(x)y+xd1(y) ∀x , y ∈ C}
is the triality Lie algebra of C .
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Tits construction rearranged
J = H3(C ′) ' k3 ⊕(⊕2
i=0ιi (C′)),
J0 ' k2 ⊕(⊕2
i=0ιi (C′)),
der J ' tri(C ′)⊕(⊕2
i=0ιi (C′)),
T (C , J) = derC ⊕ (C0 ⊗ J0)⊕ der J
' derC ⊕ (C0 ⊗ k2)⊕(⊕2
i=0C0 ⊗ ιi (C ′))⊕(tri(C ′)⊕ (⊕2
i=0ιi (C′))
)'
(tri(C )⊕ tri(C ′)
)⊕
(⊕2
i=0ιi (C ⊗ C ′))
tri(C ) = {(d0, d1, d2) ∈ so(C )3 : d0(xy) = d2(x)y+xd1(y) ∀x , y ∈ C}
is the triality Lie algebra of C .
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Tits construction rearranged
J = H3(C ′) ' k3 ⊕(⊕2
i=0ιi (C′)),
J0 ' k2 ⊕(⊕2
i=0ιi (C′)),
der J ' tri(C ′)⊕(⊕2
i=0ιi (C′)),
T (C , J) = derC ⊕ (C0 ⊗ J0)⊕ der J
' derC ⊕ (C0 ⊗ k2)⊕(⊕2
i=0C0 ⊗ ιi (C ′))⊕(tri(C ′)⊕ (⊕2
i=0ιi (C′))
)'
(tri(C )⊕ tri(C ′)
)⊕(⊕2
i=0ιi (C ⊗ C ′))
tri(C ) = {(d0, d1, d2) ∈ so(C )3 : d0(xy) = d2(x)y+xd1(y) ∀x , y ∈ C}
is the triality Lie algebra of C .
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Tits construction rearranged
J = H3(C ′) ' k3 ⊕(⊕2
i=0ιi (C′)),
J0 ' k2 ⊕(⊕2
i=0ιi (C′)),
der J ' tri(C ′)⊕(⊕2
i=0ιi (C′)),
T (C , J) = derC ⊕ (C0 ⊗ J0)⊕ der J
' derC ⊕ (C0 ⊗ k2)⊕(⊕2
i=0C0 ⊗ ιi (C ′))⊕(tri(C ′)⊕ (⊕2
i=0ιi (C′))
)'
(tri(C )⊕ tri(C ′)
)⊕(⊕2
i=0ιi (C ⊗ C ′))
tri(C ) = {(d0, d1, d2) ∈ so(C )3 : d0(xy) = d2(x)y+xd1(y) ∀x , y ∈ C}
is the triality Lie algebra of C .
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Tits construction rearranged
The product in
g(C ,C ′) =(tri(C )⊕ tri(C ′)
)⊕
(⊕2
i=0ιi (C ⊗ C ′)),
is given by:
I tri(C )⊕ tri(C ′) is a Lie subalgebra of g(C ,C ′),
I [(d0, d1, d2), ιi (x ⊗ x ′)] = ιi(di (x)⊗ x ′
),
I [(d ′0, d′1, d′2), ιi (x ⊗ x ′)] = ιi
(x ⊗ d ′i (x
′)),
I [ιi (x ⊗ x ′), ιi+1(y ⊗ y ′)] = ιi+2
((x y)⊗ (x ′y ′)
)(indices
modulo 3),
I [ιi (x ⊗ x ′), ιi (y ⊗ y ′)] = q′(x ′, y ′)θi (tx ,y ) + q(x , y)θ′i (t ′x ′,y ′),
wheretx ,y =
(q(x , .)y−q(y , .)x , 1
2q(x , y)1−RxRy ,12q(x , y)1−LxLy
)(Vinberg, Allison-Faulkner, Barton-Sudbery, Landsberg-Manivel)
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Tits construction rearrangedThe product in
g(C ,C ′) =(tri(C )⊕ tri(C ′)
)⊕
(⊕2
i=0ιi (C ⊗ C ′)),
is given by:
I tri(C )⊕ tri(C ′) is a Lie subalgebra of g(C ,C ′),
I [(d0, d1, d2), ιi (x ⊗ x ′)] = ιi(di (x)⊗ x ′
),
I [(d ′0, d′1, d′2), ιi (x ⊗ x ′)] = ιi
(x ⊗ d ′i (x
′)),
I [ιi (x ⊗ x ′), ιi+1(y ⊗ y ′)] = ιi+2
((x y)⊗ (x ′y ′)
)(indices
modulo 3),
I [ιi (x ⊗ x ′), ιi (y ⊗ y ′)] = q′(x ′, y ′)θi (tx ,y ) + q(x , y)θ′i (t ′x ′,y ′),
wheretx ,y =
(q(x , .)y−q(y , .)x , 1
2q(x , y)1−RxRy ,12q(x , y)1−LxLy
)(Vinberg, Allison-Faulkner, Barton-Sudbery, Landsberg-Manivel)
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Tits construction rearrangedThe product in
g(C ,C ′) =(tri(C )⊕ tri(C ′)
)⊕
(⊕2
i=0ιi (C ⊗ C ′)),
is given by:
I tri(C )⊕ tri(C ′) is a Lie subalgebra of g(C ,C ′),
I [(d0, d1, d2), ιi (x ⊗ x ′)] = ιi(di (x)⊗ x ′
),
I [(d ′0, d′1, d′2), ιi (x ⊗ x ′)] = ιi
(x ⊗ d ′i (x
′)),
I [ιi (x ⊗ x ′), ιi+1(y ⊗ y ′)] = ιi+2
((x y)⊗ (x ′y ′)
)(indices
modulo 3),
I [ιi (x ⊗ x ′), ιi (y ⊗ y ′)] = q′(x ′, y ′)θi (tx ,y ) + q(x , y)θ′i (t ′x ′,y ′),
wheretx ,y =
(q(x , .)y−q(y , .)x , 1
2q(x , y)1−RxRy ,12q(x , y)1−LxLy
)(Vinberg, Allison-Faulkner, Barton-Sudbery, Landsberg-Manivel)
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Freudenthal-Tits Magic Square (char 3)
This more symmetric construction is valid too in characteristic 3:
dimC ′
g(C ,C ′) 1 2 4 8
1 A1 A2 C3 F4
2 A2 A2 ⊕ A2 A5 E6
dimC4 C3 A5 D6 E7
8 F4 E6 E7 E8
I A2 denotes a form of pgl3, so [A2, A2] is a form of psl3.
I A5 denotes a form of pgl6, so [A5, A5] is a form of psl6.
I E6 is not simple, but [E6, E6] is a codimension 1 simple ideal.
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Freudenthal-Tits Magic Square (char 3)
This more symmetric construction is valid too in characteristic 3:
dimC ′
g(C ,C ′) 1 2 4 8
1 A1 A2 C3 F4
2 A2 A2 ⊕ A2 A5 E6
dimC4 C3 A5 D6 E7
8 F4 E6 E7 E8
I A2 denotes a form of pgl3, so [A2, A2] is a form of psl3.
I A5 denotes a form of pgl6, so [A5, A5] is a form of psl6.
I E6 is not simple, but [E6, E6] is a codimension 1 simple ideal.
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Freudenthal-Tits Magic Square (char 3)
This more symmetric construction is valid too in characteristic 3:
dimC ′
g(C ,C ′) 1 2 4 8
1 A1 A2 C3 F4
2 A2 A2 ⊕ A2 A5 E6
dimC4 C3 A5 D6 E7
8 F4 E6 E7 E8
I A2 denotes a form of pgl3, so [A2, A2] is a form of psl3.
I A5 denotes a form of pgl6, so [A5, A5] is a form of psl6.
I E6 is not simple, but [E6, E6] is a codimension 1 simple ideal.
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Freudenthal-Tits Magic Square (char 3)
This more symmetric construction is valid too in characteristic 3:
dimC ′
g(C ,C ′) 1 2 4 8
1 A1 A2 C3 F4
2 A2 A2 ⊕ A2 A5 E6
dimC4 C3 A5 D6 E7
8 F4 E6 E7 E8
I A2 denotes a form of pgl3, so [A2, A2] is a form of psl3.
I A5 denotes a form of pgl6, so [A5, A5] is a form of psl6.
I E6 is not simple, but [E6, E6] is a codimension 1 simple ideal.
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Freudenthal-Tits Magic Square
A supermagic rectangle
A supermagic square
Some conclusions
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A look at the rows of Tits construction
T (C , J) = derC ⊕ (C0 ⊗ J0)⊕ der J
(C a composition algebra, J a suitable Jordan algebra.)
First row dimC = 1, T (C , J) = der J.
Second row dimC = 2, T (C , J) ' J0 ⊕ der J ' str0(J).
Third row dimC = 4, so C = Q is a quaternion algebra and
T (C , J) = adQ0 ⊕(Q0 ⊗ J0)⊕ der J
' (Q0 ⊗ J)⊕ der J.
(This is Tits version (1962) of theTits-Kantor-Koecher construction T KK(J))
Up to now, everything works for arbitrary Jordan algebras incharacteristic 6= 2, and even for Jordan superalgebras.
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A look at the rows of Tits construction
T (C , J) = derC ⊕ (C0 ⊗ J0)⊕ der J
(C a composition algebra, J a suitable Jordan algebra.)
First row dimC = 1, T (C , J) = der J.
Second row dimC = 2, T (C , J) ' J0 ⊕ der J ' str0(J).
Third row dimC = 4, so C = Q is a quaternion algebra and
T (C , J) = adQ0 ⊕(Q0 ⊗ J0)⊕ der J
' (Q0 ⊗ J)⊕ der J.
(This is Tits version (1962) of theTits-Kantor-Koecher construction T KK(J))
Up to now, everything works for arbitrary Jordan algebras incharacteristic 6= 2, and even for Jordan superalgebras.
![Page 25: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/25.jpg)
A look at the rows of Tits construction
T (C , J) = derC ⊕ (C0 ⊗ J0)⊕ der J
(C a composition algebra, J a suitable Jordan algebra.)
First row dimC = 1, T (C , J) = der J.
Second row dimC = 2, T (C , J) ' J0 ⊕ der J ' str0(J).
Third row dimC = 4, so C = Q is a quaternion algebra and
T (C , J) = adQ0 ⊕(Q0 ⊗ J0)⊕ der J
' (Q0 ⊗ J)⊕ der J.
(This is Tits version (1962) of theTits-Kantor-Koecher construction T KK(J))
Up to now, everything works for arbitrary Jordan algebras incharacteristic 6= 2, and even for Jordan superalgebras.
![Page 26: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/26.jpg)
A look at the rows of Tits construction
T (C , J) = derC ⊕ (C0 ⊗ J0)⊕ der J
(C a composition algebra, J a suitable Jordan algebra.)
First row dimC = 1, T (C , J) = der J.
Second row dimC = 2, T (C , J) ' J0 ⊕ der J ' str0(J).
Third row dimC = 4, so C = Q is a quaternion algebra and
T (C , J) = adQ0 ⊕(Q0 ⊗ J0)⊕ der J
' (Q0 ⊗ J)⊕ der J.
(This is Tits version (1962) of theTits-Kantor-Koecher construction T KK(J))
Up to now, everything works for arbitrary Jordan algebras incharacteristic 6= 2, and even for Jordan superalgebras.
![Page 27: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/27.jpg)
A look at the rows of Tits construction
T (C , J) = derC ⊕ (C0 ⊗ J0)⊕ der J
(C a composition algebra, J a suitable Jordan algebra.)
First row dimC = 1, T (C , J) = der J.
Second row dimC = 2, T (C , J) ' J0 ⊕ der J ' str0(J).
Third row dimC = 4, so C = Q is a quaternion algebra and
T (C , J) = adQ0 ⊕(Q0 ⊗ J0)⊕ der J
' (Q0 ⊗ J)⊕ der J.
(This is Tits version (1962) of theTits-Kantor-Koecher construction T KK(J))
Up to now, everything works for arbitrary Jordan algebras incharacteristic 6= 2, and even for Jordan superalgebras.
![Page 28: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/28.jpg)
A look at the rows of Tits construction
T (C , J) = derC ⊕ (C0 ⊗ J0)⊕ der J
(C a composition algebra, J a suitable Jordan algebra.)
First row dimC = 1, T (C , J) = der J.
Second row dimC = 2, T (C , J) ' J0 ⊕ der J ' str0(J).
Third row dimC = 4, so C = Q is a quaternion algebra and
T (C , J) = adQ0 ⊕(Q0 ⊗ J0)⊕ der J
' (Q0 ⊗ J)⊕ der J.
(This is Tits version (1962) of theTits-Kantor-Koecher construction T KK(J))
Up to now, everything works for arbitrary Jordan algebras incharacteristic 6= 2, and even for Jordan superalgebras.
![Page 29: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/29.jpg)
A look at the rows of Tits construction
T (C , J) = derC ⊕ (C0 ⊗ J0)⊕ der J
(C a composition algebra, J a suitable Jordan algebra.)
First row dimC = 1, T (C , J) = der J.
Second row dimC = 2, T (C , J) ' J0 ⊕ der J ' str0(J).
Third row dimC = 4, so C = Q is a quaternion algebra and
T (C , J) = adQ0 ⊕(Q0 ⊗ J0)⊕ der J
' (Q0 ⊗ J)⊕ der J.
(This is Tits version (1962) of theTits-Kantor-Koecher construction T KK(J))
Up to now, everything works for arbitrary Jordan algebras incharacteristic 6= 2, and even for Jordan superalgebras.
![Page 30: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/30.jpg)
Fourth row
Fourth row dimC = 8. If the characteristic is 6= 2, 3, thenderC = g2 is simple of type G2, C0 is its smallestnontrivial irreducible module, and
T (C , J) = g2 ⊕ (C0 ⊗ J0)⊕ der J
is a G2-graded Lie algebra. Essentially, all theG2-graded Lie algebras appear in this way[Benkart-Zelmanov 1996].
It makes sense to consider Jordan superalgebras, as long as itsGrassmann envelope satisfies the Cayley-Hamilton equation ofdegree 3.
I T (C , J(V )) ' G (3) (dimV = 2, V = V1).
I T (C ,D2) ' F (4).
I T (C ,K10) in characteristic 5!!This is a new simple modular Lie superalgebra, whose evenpart is so11 and odd part its spin module.
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Fourth row
Fourth row dimC = 8. If the characteristic is 6= 2, 3, thenderC = g2 is simple of type G2, C0 is its smallestnontrivial irreducible module, and
T (C , J) = g2 ⊕ (C0 ⊗ J0)⊕ der J
is a G2-graded Lie algebra. Essentially, all theG2-graded Lie algebras appear in this way[Benkart-Zelmanov 1996].
It makes sense to consider Jordan superalgebras, as long as itsGrassmann envelope satisfies the Cayley-Hamilton equation ofdegree 3.
I T (C , J(V )) ' G (3) (dimV = 2, V = V1).
I T (C ,D2) ' F (4).
I T (C ,K10) in characteristic 5!!This is a new simple modular Lie superalgebra, whose evenpart is so11 and odd part its spin module.
![Page 32: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/32.jpg)
Fourth row
Fourth row dimC = 8. If the characteristic is 6= 2, 3, thenderC = g2 is simple of type G2, C0 is its smallestnontrivial irreducible module, and
T (C , J) = g2 ⊕ (C0 ⊗ J0)⊕ der J
is a G2-graded Lie algebra. Essentially, all theG2-graded Lie algebras appear in this way[Benkart-Zelmanov 1996].
It makes sense to consider Jordan superalgebras, as long as itsGrassmann envelope satisfies the Cayley-Hamilton equation ofdegree 3.
I T (C , J(V )) ' G (3) (dimV = 2, V = V1).
I T (C ,D2) ' F (4).
I T (C ,K10) in characteristic 5!!This is a new simple modular Lie superalgebra, whose evenpart is so11 and odd part its spin module.
![Page 33: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/33.jpg)
Fourth row
Fourth row dimC = 8. If the characteristic is 6= 2, 3, thenderC = g2 is simple of type G2, C0 is its smallestnontrivial irreducible module, and
T (C , J) = g2 ⊕ (C0 ⊗ J0)⊕ der J
is a G2-graded Lie algebra. Essentially, all theG2-graded Lie algebras appear in this way[Benkart-Zelmanov 1996].
It makes sense to consider Jordan superalgebras, as long as itsGrassmann envelope satisfies the Cayley-Hamilton equation ofdegree 3.
I T (C , J(V )) ' G (3) (dimV = 2, V = V1).
I T (C ,D2) ' F (4).
I T (C ,K10) in characteristic 5!!This is a new simple modular Lie superalgebra, whose evenpart is so11 and odd part its spin module.
![Page 34: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/34.jpg)
Fourth row
Fourth row dimC = 8. If the characteristic is 6= 2, 3, thenderC = g2 is simple of type G2, C0 is its smallestnontrivial irreducible module, and
T (C , J) = g2 ⊕ (C0 ⊗ J0)⊕ der J
is a G2-graded Lie algebra. Essentially, all theG2-graded Lie algebras appear in this way[Benkart-Zelmanov 1996].
It makes sense to consider Jordan superalgebras, as long as itsGrassmann envelope satisfies the Cayley-Hamilton equation ofdegree 3.
I T (C , J(V )) ' G (3) (dimV = 2, V = V1).
I T (C ,D2) ' F (4).
I T (C ,K10) in characteristic 5!!This is a new simple modular Lie superalgebra, whose evenpart is so11 and odd part its spin module.
![Page 35: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/35.jpg)
Fourth row
Fourth row dimC = 8. If the characteristic is 6= 2, 3, thenderC = g2 is simple of type G2, C0 is its smallestnontrivial irreducible module, and
T (C , J) = g2 ⊕ (C0 ⊗ J0)⊕ der J
is a G2-graded Lie algebra. Essentially, all theG2-graded Lie algebras appear in this way[Benkart-Zelmanov 1996].
It makes sense to consider Jordan superalgebras, as long as itsGrassmann envelope satisfies the Cayley-Hamilton equation ofdegree 3.
I T (C , J(V )) ' G (3) (dimV = 2, V = V1).
I T (C ,D2) ' F (4).
I T (C ,K10) in characteristic 5!!This is a new simple modular Lie superalgebra, whose evenpart is so11 and odd part its spin module.
![Page 36: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/36.jpg)
A supermagic rectangle
T (C , J) H3(k) H3(k × k) H3(Mat2(k)) H3(C(k)) J(V ) Dt K10
k A1 A2 C3 F4 A1 B(0, 1) B(0, 1)⊕ B(0, 1)
k × k A2 A2 ⊕ A2 A5 E6 B(0, 1) A(1, 0) C(3)
Mat2(k) C3 A5 D6 E7 B(1, 1) D(2, 1; t) F (4)
C(k) F4 E6 E7 E8 G(3) F (4) T (C(k),K10)(t = 2) (char 5)
![Page 37: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/37.jpg)
A supermagic rectangle: the new columns
T (C , J) J(V ) Dt K10
k A1 B(0, 1) B(0, 1)⊕ B(0, 1)
k × k B(0, 1) A(1, 0) C (3)
Mat2(k) B(1, 1) D(2, 1; t) F (4)
C (k) G (3) F (4) T (C (k),K10)(t = 2) (char 5)
![Page 38: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/38.jpg)
Fourth row, characteristic 3
If the characteristic is 3 and dimC = 8, then derC is no longersimple, but contains the simple ideal adC0 (a form of psl3). Itmakes sense to consider:
T (C , J) = adC0 ⊕ (C0 ⊗ J0)⊕ der J
' (C0 ⊗ J)⊕ der J.
T (C , J) becomes a Lie algebra if and only if J is a commutativeand alternative algebra (these conditions imply the Jordanidentity).
The simple commutative alternative algebras are just the fields, sonothing interesting appears here.
![Page 39: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/39.jpg)
Fourth row, characteristic 3
If the characteristic is 3 and dimC = 8, then derC is no longersimple, but contains the simple ideal adC0 (a form of psl3). Itmakes sense to consider:
T (C , J) = adC0 ⊕ (C0 ⊗ J0)⊕ der J
' (C0 ⊗ J)⊕ der J.
T (C , J) becomes a Lie algebra if and only if J is a commutativeand alternative algebra (these conditions imply the Jordanidentity).
The simple commutative alternative algebras are just the fields, sonothing interesting appears here.
![Page 40: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/40.jpg)
Fourth row, characteristic 3
If the characteristic is 3 and dimC = 8, then derC is no longersimple, but contains the simple ideal adC0 (a form of psl3). Itmakes sense to consider:
T (C , J) = adC0 ⊕ (C0 ⊗ J0)⊕ der J
' (C0 ⊗ J)⊕ der J.
T (C , J) becomes a Lie algebra if and only if J is a commutativeand alternative algebra (these conditions imply the Jordanidentity).
The simple commutative alternative algebras are just the fields, sonothing interesting appears here.
![Page 41: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/41.jpg)
Fourth row, characteristic 3
If the characteristic is 3 and dimC = 8, then derC is no longersimple, but contains the simple ideal adC0 (a form of psl3). Itmakes sense to consider:
T (C , J) = adC0 ⊕ (C0 ⊗ J0)⊕ der J
' (C0 ⊗ J)⊕ der J.
T (C , J) becomes a Lie algebra if and only if J is a commutativeand alternative algebra (these conditions imply the Jordanidentity).
The simple commutative alternative algebras are just the fields, sonothing interesting appears here.
![Page 42: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/42.jpg)
Fourth “superrow”, characteristic 3
But there are simple commutative alternative superalgebras[Shestakov 1997] (characteristic 3):
(i) fields,
(ii) J(V ), the Jordan superalgebra of a superform on a twodimensional odd space V ,
(iii) B = B(Γ,D, 0) = Γ⊕ Γu, where
I Γ is a commutative associative algebra,I D ∈ der Γ such that Γ is D-simple,I a(bu) = (ab)u = (au)b, (au)(bu) = aD(b)− D(a)b,∀a, b ∈ Γ.
![Page 43: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/43.jpg)
Fourth “superrow”, characteristic 3
But there are simple commutative alternative superalgebras[Shestakov 1997] (characteristic 3):
(i) fields,
(ii) J(V ), the Jordan superalgebra of a superform on a twodimensional odd space V ,
(iii) B = B(Γ,D, 0) = Γ⊕ Γu, where
I Γ is a commutative associative algebra,I D ∈ der Γ such that Γ is D-simple,I a(bu) = (ab)u = (au)b, (au)(bu) = aD(b)− D(a)b,∀a, b ∈ Γ.
![Page 44: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/44.jpg)
Fourth “superrow”, characteristic 3
But there are simple commutative alternative superalgebras[Shestakov 1997] (characteristic 3):
(i) fields,
(ii) J(V ), the Jordan superalgebra of a superform on a twodimensional odd space V ,
(iii) B = B(Γ,D, 0) = Γ⊕ Γu, where
I Γ is a commutative associative algebra,I D ∈ der Γ such that Γ is D-simple,I a(bu) = (ab)u = (au)b, (au)(bu) = aD(b)− D(a)b,∀a, b ∈ Γ.
![Page 45: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/45.jpg)
Fourth “superrow”, characteristic 3
But there are simple commutative alternative superalgebras[Shestakov 1997] (characteristic 3):
(i) fields,
(ii) J(V ), the Jordan superalgebra of a superform on a twodimensional odd space V ,
(iii) B = B(Γ,D, 0) = Γ⊕ Γu, where
I Γ is a commutative associative algebra,I D ∈ der Γ such that Γ is D-simple,I a(bu) = (ab)u = (au)b, (au)(bu) = aD(b)− D(a)b,∀a, b ∈ Γ.
![Page 46: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/46.jpg)
Fourth “superrow”, characteristic 3
But there are simple commutative alternative superalgebras[Shestakov 1997] (characteristic 3):
(i) fields,
(ii) J(V ), the Jordan superalgebra of a superform on a twodimensional odd space V ,
(iii) B = B(Γ,D, 0) = Γ⊕ Γu, where
I Γ is a commutative associative algebra,I D ∈ der Γ such that Γ is D-simple,I a(bu) = (ab)u = (au)b, (au)(bu) = aD(b)− D(a)b,∀a, b ∈ Γ.
![Page 47: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/47.jpg)
Fourth “superrow”, characteristic 3
Example (Divided powers)
Γ = O(1; n) = span{t(r) : 0 ≤ r ≤ 3n − 1
},
t(r)t(s) =(r+s
r
)t(r+s),
D(t(r)) = t(r−1).
![Page 48: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/48.jpg)
Bouarroudj-Leites superalgebras
Over an algebraically closed field of characteristic 3:
I T(C (k), J(V )
)is a simple Lie superalgebra specific of
characteristic 3 of (super)dimension 10|14,
I T(C (k),O(1, n)⊕O(1, n)u
)= Bj(1; n|7) is a simple Lie
superalgebra of (super)dimension 23 × 3n|23 × 3n.
Both simple Lie superalgebras have been considered in acompletely different way by Bouarroudj and Leites (2006).
![Page 49: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/49.jpg)
Bouarroudj-Leites superalgebras
Over an algebraically closed field of characteristic 3:
I T(C (k), J(V )
)is a simple Lie superalgebra specific of
characteristic 3 of (super)dimension 10|14,
I T(C (k),O(1, n)⊕O(1, n)u
)= Bj(1; n|7) is a simple Lie
superalgebra of (super)dimension 23 × 3n|23 × 3n.
Both simple Lie superalgebras have been considered in acompletely different way by Bouarroudj and Leites (2006).
![Page 50: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/50.jpg)
Bouarroudj-Leites superalgebras
Over an algebraically closed field of characteristic 3:
I T(C (k), J(V )
)is a simple Lie superalgebra specific of
characteristic 3 of (super)dimension 10|14,
I T(C (k),O(1, n)⊕O(1, n)u
)= Bj(1; n|7) is a simple Lie
superalgebra of (super)dimension 23 × 3n|23 × 3n.
Both simple Lie superalgebras have been considered in acompletely different way by Bouarroudj and Leites (2006).
![Page 51: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/51.jpg)
Bouarroudj-Leites superalgebras
Over an algebraically closed field of characteristic 3:
I T(C (k), J(V )
)is a simple Lie superalgebra specific of
characteristic 3 of (super)dimension 10|14,
I T(C (k),O(1, n)⊕O(1, n)u
)= Bj(1; n|7) is a simple Lie
superalgebra of (super)dimension 23 × 3n|23 × 3n.
Both simple Lie superalgebras have been considered in acompletely different way by Bouarroudj and Leites (2006).
![Page 52: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/52.jpg)
Bouarroudj-Leites superalgebras
Over an algebraically closed field of characteristic 3:
I T(C (k), J(V )
)is a simple Lie superalgebra specific of
characteristic 3 of (super)dimension 10|14,
I T(C (k),O(1, n)⊕O(1, n)u
)= Bj(1; n|7) is a simple Lie
superalgebra of (super)dimension 23 × 3n|23 × 3n.
Both simple Lie superalgebras have been considered in acompletely different way by Bouarroudj and Leites (2006).
![Page 53: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/53.jpg)
Bouarroudj-Leites superalgebras
Over an algebraically closed field of characteristic 3:
I T(C (k), J(V )
)is a simple Lie superalgebra specific of
characteristic 3 of (super)dimension 10|14,
I T(C (k),O(1, n)⊕O(1, n)u
)= Bj(1; n|7) is a simple Lie
superalgebra of (super)dimension 23 × 3n|23 × 3n.
Both simple Lie superalgebras have been considered in acompletely different way by Bouarroudj and Leites (2006).
![Page 54: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/54.jpg)
Freudenthal-Tits Magic Square
A supermagic rectangle
A supermagic square
Some conclusions
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Composition superalgebras
Definition
A superalgebra C = C0 ⊕ C1, endowed with a regular quadraticsuperform q = (q0, b), called the norm, is said to be a compositionsuperalgebra in case
q0(x0y0) = q0(x0)q0(y0),
b(x0y , x0z) = q0(x0)b(y , z) = b(yx0, zx0),
b(xy , zt) + (−1)|x ||y |+|x ||z|+|y ||z|b(zy , xt) = (−1)|y ||z|b(x , z)b(y , t),
The unital composition superalgebras are termed Hurwitzsuperalgebras.
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Composition superalgebras
Definition
A superalgebra C = C0 ⊕ C1, endowed with a regular quadraticsuperform q = (q0, b), called the norm, is said to be a compositionsuperalgebra in case
q0(x0y0) = q0(x0)q0(y0),
b(x0y , x0z) = q0(x0)b(y , z) = b(yx0, zx0),
b(xy , zt) + (−1)|x ||y |+|x ||z|+|y ||z|b(zy , xt) = (−1)|y ||z|b(x , z)b(y , t),
The unital composition superalgebras are termed Hurwitzsuperalgebras.
![Page 57: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/57.jpg)
Composition superalgebras: examples (Shestakov)
B(1, 2) = k1⊕ V ,
char k = 3, V a two dim’l vector space with a nonzero alternatingbilinear form 〈.|.〉, with
1x = x1 = x , uv = 〈u|v〉1, q0(1) = 1, b(u, v) = 〈u|v〉,
is a Hurwitz superalgebra. (As a superalgebra, this is just ourprevious J(V ).)
![Page 58: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/58.jpg)
Composition superalgebras: examples (Shestakov)
B(1, 2) = k1⊕ V ,
char k = 3, V a two dim’l vector space with a nonzero alternatingbilinear form 〈.|.〉, with
1x = x1 = x , uv = 〈u|v〉1, q0(1) = 1, b(u, v) = 〈u|v〉,
is a Hurwitz superalgebra. (As a superalgebra, this is just ourprevious J(V ).)
![Page 59: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/59.jpg)
Composition superalgebras: examples (Shestakov)
B(4, 2) = Endk(V )⊕ V ,
k and V as before, Endk(V ) is equipped with the symplecticinvolution f 7→ f , (〈f (u)|v〉 = 〈u|f (v)〉), the multiplication is givenby:
I the usual multiplication (composition of maps) in Endk(V ),
I v · f = f (v) = f · v for any f ∈ Endk(V ) and v ∈ V ,
I u · v = 〈.|u〉v(w 7→ 〈w |u〉v
)∈ Endk(V ) for any u, v ∈ V ,
and with quadratic superform
q0(f ) = det f , b(u, v) = 〈u|v〉,
is a Hurwitz superalgebra.
![Page 60: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/60.jpg)
Composition superalgebras: examples (Shestakov)
B(4, 2) = Endk(V )⊕ V ,
k and V as before, Endk(V ) is equipped with the symplecticinvolution f 7→ f , (〈f (u)|v〉 = 〈u|f (v)〉), the multiplication is givenby:
I the usual multiplication (composition of maps) in Endk(V ),
I v · f = f (v) = f · v for any f ∈ Endk(V ) and v ∈ V ,
I u · v = 〈.|u〉v(w 7→ 〈w |u〉v
)∈ Endk(V ) for any u, v ∈ V ,
and with quadratic superform
q0(f ) = det f , b(u, v) = 〈u|v〉,
is a Hurwitz superalgebra.
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Composition superalgebras: classification
Theorem (E.-Okubo 2002)
Any unital composition superalgebra is either:
I a Hurwitz algebra,
I a Z2-graded Hurwitz algebra in characteristic 2,
I isomorphic to either B(1, 2) or B(4, 2) in characteristic 3.
Hurwitz superalgebras can be plugged into the symmetricconstruction of Freudenthal-Tits Magic Square g(C ,C ′).
![Page 62: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/62.jpg)
Composition superalgebras: classification
Theorem (E.-Okubo 2002)
Any unital composition superalgebra is either:
I a Hurwitz algebra,
I a Z2-graded Hurwitz algebra in characteristic 2,
I isomorphic to either B(1, 2) or B(4, 2) in characteristic 3.
Hurwitz superalgebras can be plugged into the symmetricconstruction of Freudenthal-Tits Magic Square g(C ,C ′).
![Page 63: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/63.jpg)
Composition superalgebras: classification
Theorem (E.-Okubo 2002)
Any unital composition superalgebra is either:
I a Hurwitz algebra,
I a Z2-graded Hurwitz algebra in characteristic 2,
I isomorphic to either B(1, 2) or B(4, 2) in characteristic 3.
Hurwitz superalgebras can be plugged into the symmetricconstruction of Freudenthal-Tits Magic Square g(C ,C ′).
![Page 64: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/64.jpg)
Supermagic Square (char 3, Cunha-E. 2007)
g(C ,C ′) k k × k Mat2(k) C (k) B(1, 2) B(4, 2)
k A1 A2 C3 F4 6|8 21|14
k × k A2 ⊕ A2 A5 E6 11|14 35|20
Mat2(k) D6 E7 24|26 66|32
C (k) E8 55|50 133|56
B(1, 2) 21|16 36|40
B(4, 2) 78|64
Notation: g(n,m) will denote the superalgebra g(C ,C ′), withdimC = n, dimC ′ = m.
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Supermagic Square (char 3, Cunha-E. 2007)
g(C ,C ′) k k × k Mat2(k) C (k) B(1, 2) B(4, 2)
k A1 A2 C3 F4 6|8 21|14
k × k A2 ⊕ A2 A5 E6 11|14 35|20
Mat2(k) D6 E7 24|26 66|32
C (k) E8 55|50 133|56
B(1, 2) 21|16 36|40
B(4, 2) 78|64
Notation: g(n,m) will denote the superalgebra g(C ,C ′), withdimC = n, dimC ′ = m.
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Lie superalgebras in the Supermagic Square
B(1, 2) B(4, 2)
k psl2,2 sp6⊕(14)
k × k(sl2⊕ pgl3
)⊕((2)⊗ psl3
)pgl6⊕(20)
Mat2(k)(sl2⊕ sp6
)⊕((2)⊗ (13)
)so12⊕spin12
C (k)(sl2⊕f4
)⊕((2)⊗ (25)
)e7 ⊕ (56)
B(1, 2) so7⊕2spin7 sp8⊕(40)
B(4, 2) sp8⊕(40) so13⊕spin13
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Lie superalgebras in the Supermagic Square
All these Lie superalgebras are simple, with the exception of g(2, 3)and g(2, 6), both of which contain a codimension one simple ideal.
Only g(1, 3) ' psl2,2 has a counterpart in Kac’s classification incharacteristic 0. The other Lie superalgebras in the SupermagicSquare, or their derived algebras, are new simple Lie superalgebras,specific of characteristic 3.
The simple Lie superalgebra g(2, 3)′ = [g(2, 3), g(2, 3)] isisomorphic to our previous T (C (k), J(V )).
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Lie superalgebras in the Supermagic Square
All these Lie superalgebras are simple, with the exception of g(2, 3)and g(2, 6), both of which contain a codimension one simple ideal.
Only g(1, 3) ' psl2,2 has a counterpart in Kac’s classification incharacteristic 0. The other Lie superalgebras in the SupermagicSquare, or their derived algebras, are new simple Lie superalgebras,specific of characteristic 3.
The simple Lie superalgebra g(2, 3)′ = [g(2, 3), g(2, 3)] isisomorphic to our previous T (C (k), J(V )).
![Page 69: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/69.jpg)
Lie superalgebras in the Supermagic Square
All these Lie superalgebras are simple, with the exception of g(2, 3)and g(2, 6), both of which contain a codimension one simple ideal.
Only g(1, 3) ' psl2,2 has a counterpart in Kac’s classification incharacteristic 0. The other Lie superalgebras in the SupermagicSquare, or their derived algebras, are new simple Lie superalgebras,specific of characteristic 3.
The simple Lie superalgebra g(2, 3)′ = [g(2, 3), g(2, 3)] isisomorphic to our previous T (C (k), J(V )).
![Page 70: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/70.jpg)
Lie superalgebras in the Supermagic Square
All these Lie superalgebras are simple, with the exception of g(2, 3)and g(2, 6), both of which contain a codimension one simple ideal.
Only g(1, 3) ' psl2,2 has a counterpart in Kac’s classification incharacteristic 0. The other Lie superalgebras in the SupermagicSquare, or their derived algebras, are new simple Lie superalgebras,specific of characteristic 3.
The simple Lie superalgebra g(2, 3)′ = [g(2, 3), g(2, 3)] isisomorphic to our previous T (C (k), J(V )).
![Page 71: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/71.jpg)
The Supermagic Square and Jordan algebras
k k × k Mat2(k) C(k)
B(1, 2) psl2,2
(sl2 ⊕ pgl3
)⊕
((2)⊗ psl3
) (sl2 ⊕ sp6
)⊕
((2)⊗ (13)
) (sl2 ⊕f4
)⊕
((2)⊗ (25)
)B(4, 2) sp6 ⊕(14) pgl6 ⊕(20) so12 ⊕spin12 e7 ⊕ (56)
g(3, r) =(sl2⊕ der J
)⊕((2)⊗ J
),
r = 1, 2, 4, 8, J = J0/k1, J = H3(C ).
g(6, r) = (derT )⊕ T ,
r = 1, 2, 4, 8, T =
(k JJ k
), J = H3(C ).
![Page 72: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/72.jpg)
The Supermagic Square and Jordan algebras
k k × k Mat2(k) C(k)
B(1, 2) psl2,2
(sl2 ⊕ pgl3
)⊕
((2)⊗ psl3
) (sl2 ⊕ sp6
)⊕
((2)⊗ (13)
) (sl2 ⊕f4
)⊕
((2)⊗ (25)
)B(4, 2) sp6 ⊕(14) pgl6 ⊕(20) so12 ⊕spin12 e7 ⊕ (56)
g(3, r) =(sl2⊕ der J
)⊕((2)⊗ J
),
r = 1, 2, 4, 8, J = J0/k1, J = H3(C ).
g(6, r) = (derT )⊕ T ,
r = 1, 2, 4, 8, T =
(k JJ k
), J = H3(C ).
![Page 73: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/73.jpg)
The Supermagic Square and Jordan algebras
k k × k Mat2(k) C(k)
B(1, 2) psl2,2
(sl2 ⊕ pgl3
)⊕
((2)⊗ psl3
) (sl2 ⊕ sp6
)⊕
((2)⊗ (13)
) (sl2 ⊕f4
)⊕
((2)⊗ (25)
)B(4, 2) sp6 ⊕(14) pgl6 ⊕(20) so12 ⊕spin12 e7 ⊕ (56)
g(3, r) =(sl2⊕ der J
)⊕((2)⊗ J
),
r = 1, 2, 4, 8, J = J0/k1, J = H3(C ).
g(6, r) = (derT )⊕ T ,
r = 1, 2, 4, 8, T =
(k JJ k
), J = H3(C ).
![Page 74: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/74.jpg)
The Supermagic Square and Jordan algebras
k k × k Mat2(k) C(k)
B(1, 2) psl2,2
(sl2 ⊕ pgl3
)⊕
((2)⊗ psl3
) (sl2 ⊕ sp6
)⊕
((2)⊗ (13)
) (sl2 ⊕f4
)⊕
((2)⊗ (25)
)B(4, 2) sp6 ⊕(14) pgl6 ⊕(20) so12 ⊕spin12 e7 ⊕ (56)
g(3, r) =(sl2⊕ der J
)⊕((2)⊗ J
),
r = 1, 2, 4, 8, J = J0/k1, J = H3(C ).
g(6, r) = (derT )⊕ T ,
r = 1, 2, 4, 8, T =
(k JJ k
), J = H3(C ).
![Page 75: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/75.jpg)
The Supermagic Square and Jordan superalgebras
B(1, 2) B(4, 2)
k der(H3(B(1, 2)
)der
(H3(B(4, 2)
)k × k pstr(H3(B(1, 2)
)pstr
(H3(B(4, 2)
)Mat2(k) T KK
(H3(B(1, 2)
)T KK
(H3(B(4, 2)
)C (k)
B(1, 2) T KK(K9)
B(4, 2)
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Simple modular Lie superalgebras with a Cartan matrix
The finite dimensional modular Lie superalgebras withindecomposable symmetrizable Cartan matrices (or contragredientLie superalgebras) over algebraically closed fields have beenclassified by Bouarroudj, Grozman and Leites (2009), under someextra technical hypotheses.
For characteristic p ≥ 3, apart from the Lie superalgebras obtainedas the analogues of the Lie superalgebras in the classification incharacteristic 0, by reducing the Cartan matrices modulo p, thereare only the following exceptions:
![Page 77: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/77.jpg)
Simple modular Lie superalgebras with a Cartan matrix
The finite dimensional modular Lie superalgebras withindecomposable symmetrizable Cartan matrices (or contragredientLie superalgebras) over algebraically closed fields have beenclassified by Bouarroudj, Grozman and Leites (2009), under someextra technical hypotheses.
For characteristic p ≥ 3, apart from the Lie superalgebras obtainedas the analogues of the Lie superalgebras in the classification incharacteristic 0, by reducing the Cartan matrices modulo p, thereare only the following exceptions:
![Page 78: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/78.jpg)
Simple modular Lie superalgebras with a Cartan matrix
The finite dimensional modular Lie superalgebras withindecomposable symmetrizable Cartan matrices (or contragredientLie superalgebras) over algebraically closed fields have beenclassified by Bouarroudj, Grozman and Leites (2009), under someextra technical hypotheses.
For characteristic p ≥ 3, apart from the Lie superalgebras obtainedas the analogues of the Lie superalgebras in the classification incharacteristic 0, by reducing the Cartan matrices modulo p, thereare only the following exceptions:
![Page 79: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/79.jpg)
Simple modular Lie superalgebras with a Cartan matrix
1. Two exceptions in characteristic 5: br(2; 5) and el(5; 5).(Dimensions 10|12 and 55|32.)
2. The family of exceptions given by the Lie superalgebras in theSupermagic Square in characteristic 3.
3. Another two exceptions in characteristic 3, similar to the onesin characteristic 5: br(2; 3) and el(5; 3). (Dimensions 10|8 and39|32.)
The superalgebra br(2; 3) appeared first related to an eightdimensional symplectic triple system (E. 2006).
The superalgebra el(5; 5) is the Lie superalgebra T (C (k),K10)considered previously.
![Page 80: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/80.jpg)
Simple modular Lie superalgebras with a Cartan matrix
1. Two exceptions in characteristic 5: br(2; 5) and el(5; 5).(Dimensions 10|12 and 55|32.)
2. The family of exceptions given by the Lie superalgebras in theSupermagic Square in characteristic 3.
3. Another two exceptions in characteristic 3, similar to the onesin characteristic 5: br(2; 3) and el(5; 3). (Dimensions 10|8 and39|32.)
The superalgebra br(2; 3) appeared first related to an eightdimensional symplectic triple system (E. 2006).
The superalgebra el(5; 5) is the Lie superalgebra T (C (k),K10)considered previously.
![Page 81: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/81.jpg)
Simple modular Lie superalgebras with a Cartan matrix
1. Two exceptions in characteristic 5: br(2; 5) and el(5; 5).(Dimensions 10|12 and 55|32.)
2. The family of exceptions given by the Lie superalgebras in theSupermagic Square in characteristic 3.
3. Another two exceptions in characteristic 3, similar to the onesin characteristic 5: br(2; 3) and el(5; 3). (Dimensions 10|8 and39|32.)
The superalgebra br(2; 3) appeared first related to an eightdimensional symplectic triple system (E. 2006).
The superalgebra el(5; 5) is the Lie superalgebra T (C (k),K10)considered previously.
![Page 82: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/82.jpg)
Simple modular Lie superalgebras with a Cartan matrix
1. Two exceptions in characteristic 5: br(2; 5) and el(5; 5).(Dimensions 10|12 and 55|32.)
2. The family of exceptions given by the Lie superalgebras in theSupermagic Square in characteristic 3.
3. Another two exceptions in characteristic 3, similar to the onesin characteristic 5: br(2; 3) and el(5; 3). (Dimensions 10|8 and39|32.)
The superalgebra br(2; 3) appeared first related to an eightdimensional symplectic triple system (E. 2006).
The superalgebra el(5; 5) is the Lie superalgebra T (C (k),K10)considered previously.
![Page 83: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/83.jpg)
Simple modular Lie superalgebras with a Cartan matrix
1. Two exceptions in characteristic 5: br(2; 5) and el(5; 5).(Dimensions 10|12 and 55|32.)
2. The family of exceptions given by the Lie superalgebras in theSupermagic Square in characteristic 3.
3. Another two exceptions in characteristic 3, similar to the onesin characteristic 5: br(2; 3) and el(5; 3). (Dimensions 10|8 and39|32.)
The superalgebra br(2; 3) appeared first related to an eightdimensional symplectic triple system (E. 2006).
The superalgebra el(5; 5) is the Lie superalgebra T (C (k),K10)considered previously.
![Page 84: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/84.jpg)
Simple modular Lie superalgebras with a Cartan matrix
1. Two exceptions in characteristic 5: br(2; 5) and el(5; 5).(Dimensions 10|12 and 55|32.)
2. The family of exceptions given by the Lie superalgebras in theSupermagic Square in characteristic 3.
3. Another two exceptions in characteristic 3, similar to the onesin characteristic 5: br(2; 3) and el(5; 3). (Dimensions 10|8 and39|32.)
The superalgebra br(2; 3) appeared first related to an eightdimensional symplectic triple system (E. 2006).
The superalgebra el(5; 5) is the Lie superalgebra T (C (k),K10)considered previously.
![Page 85: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/85.jpg)
Simple modular Lie superalgebras with a Cartan matrix
1. Two exceptions in characteristic 5: br(2; 5) and el(5; 5).(Dimensions 10|12 and 55|32.)
2. The family of exceptions given by the Lie superalgebras in theSupermagic Square in characteristic 3.
3. Another two exceptions in characteristic 3, similar to the onesin characteristic 5: br(2; 3) and el(5; 3). (Dimensions 10|8 and39|32.)
The superalgebra br(2; 3) appeared first related to an eightdimensional symplectic triple system (E. 2006).
The superalgebra el(5; 5) is the Lie superalgebra T (C (k),K10)considered previously.
![Page 86: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/86.jpg)
el(5; 3)
The superalgebra el(5; 3) lives (as a natural maximal subalgebra) inthe Lie superalgebra g(3, 8) of the Supermagic Square as follows:
I el(5; 3)0 = sl2⊕ so9 ≤ sl2⊕f4 = g(3, 8)0,(f4 = der(J), J = H3(C (k))
)I el(5; 3)1 = (2)⊗ (C (k)⊕ C (k)) ≤ (2)⊗ J = g(3, 8)1,(
J = J0/k1 contains three copies of C (k) in the off-diagonal
entries.)
![Page 87: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/87.jpg)
el(5; 3)
The superalgebra el(5; 3) lives (as a natural maximal subalgebra) inthe Lie superalgebra g(3, 8) of the Supermagic Square as follows:
I el(5; 3)0 = sl2⊕ so9 ≤ sl2⊕f4 = g(3, 8)0,(f4 = der(J), J = H3(C (k))
)
I el(5; 3)1 = (2)⊗ (C (k)⊕ C (k)) ≤ (2)⊗ J = g(3, 8)1,(J = J0/k1 contains three copies of C (k) in the off-diagonal
entries.)
![Page 88: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/88.jpg)
el(5; 3)
The superalgebra el(5; 3) lives (as a natural maximal subalgebra) inthe Lie superalgebra g(3, 8) of the Supermagic Square as follows:
I el(5; 3)0 = sl2⊕ so9 ≤ sl2⊕f4 = g(3, 8)0,(f4 = der(J), J = H3(C (k))
)I el(5; 3)1 = (2)⊗ (C (k)⊕ C (k)) ≤ (2)⊗ J = g(3, 8)1,(
J = J0/k1 contains three copies of C (k) in the off-diagonal
entries.)
![Page 89: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/89.jpg)
Freudenthal-Tits Magic Square
A supermagic rectangle
A supermagic square
Some conclusions
![Page 90: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/90.jpg)
Some conclusions
Freudenthal-Tits Magic Square can be extended in several ways toinclude interesting superalgebras:
char. 6= 2, 3: By extending Tits construction with the use of“degree three” Jordan superalgebras.There appear the exceptional Lie superalgebrasD(2, 1; t), G (3) and F (4) in Kac’s classification.
char. 5: In characteristic 5 one can add the new simple Liesuperalgebra (without counterpart in Kac’sclassification) el(5; 5) = T (C (k),K10).
![Page 91: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/91.jpg)
Some conclusions
Freudenthal-Tits Magic Square can be extended in several ways toinclude interesting superalgebras:
char. 6= 2, 3: By extending Tits construction with the use of“degree three” Jordan superalgebras.There appear the exceptional Lie superalgebrasD(2, 1; t), G (3) and F (4) in Kac’s classification.
char. 5: In characteristic 5 one can add the new simple Liesuperalgebra (without counterpart in Kac’sclassification) el(5; 5) = T (C (k),K10).
![Page 92: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/92.jpg)
Some conclusions
Freudenthal-Tits Magic Square can be extended in several ways toinclude interesting superalgebras:
char. 6= 2, 3: By extending Tits construction with the use of“degree three” Jordan superalgebras.There appear the exceptional Lie superalgebrasD(2, 1; t), G (3) and F (4) in Kac’s classification.
char. 5: In characteristic 5 one can add the new simple Liesuperalgebra (without counterpart in Kac’sclassification) el(5; 5) = T (C (k),K10).
![Page 93: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/93.jpg)
Some conclusions
Freudenthal-Tits Magic Square can be extended in several ways toinclude interesting superalgebras:
char. 6= 2, 3: By extending Tits construction with the use of“degree three” Jordan superalgebras.There appear the exceptional Lie superalgebrasD(2, 1; t), G (3) and F (4) in Kac’s classification.
char. 5: In characteristic 5 one can add the new simple Liesuperalgebra (without counterpart in Kac’sclassification) el(5; 5) = T (C (k),K10).
![Page 94: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/94.jpg)
Some conclusions
char. 3: • By using a symmetric construction in terms oftwo Hurwitz algebras, and extending it (only incharacteristic 3) with the use of compositionsuperalgebras.Ten new simple Lie superalgebras are obtained:g(r , 3)′ (r = 2, 4, 8), g(r , 6)′ (r = 1, 2, 4, 8),g(3, 3), g(3, 6) and g(6, 6).
• The new simple Lie superalgebra el(5; 3) appearsas a maximal subalgebra of g(8, 3).
• The new simple Lie superalgebraT (C (k),O(1, n)⊕O(1, n)u) = Bj(1; n|7)
appears by ‘adjusting’ the fourth row of Titsconstruction to characteristic 3.
That’s all. Thanks
![Page 95: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/95.jpg)
Some conclusions
char. 3: • By using a symmetric construction in terms oftwo Hurwitz algebras, and extending it (only incharacteristic 3) with the use of compositionsuperalgebras.Ten new simple Lie superalgebras are obtained:g(r , 3)′ (r = 2, 4, 8), g(r , 6)′ (r = 1, 2, 4, 8),g(3, 3), g(3, 6) and g(6, 6).
• The new simple Lie superalgebra el(5; 3) appearsas a maximal subalgebra of g(8, 3).
• The new simple Lie superalgebraT (C (k),O(1, n)⊕O(1, n)u) = Bj(1; n|7)
appears by ‘adjusting’ the fourth row of Titsconstruction to characteristic 3.
That’s all. Thanks
![Page 96: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/96.jpg)
Some conclusions
char. 3: • By using a symmetric construction in terms oftwo Hurwitz algebras, and extending it (only incharacteristic 3) with the use of compositionsuperalgebras.Ten new simple Lie superalgebras are obtained:g(r , 3)′ (r = 2, 4, 8), g(r , 6)′ (r = 1, 2, 4, 8),g(3, 3), g(3, 6) and g(6, 6).
• The new simple Lie superalgebra el(5; 3) appearsas a maximal subalgebra of g(8, 3).
• The new simple Lie superalgebraT (C (k),O(1, n)⊕O(1, n)u) = Bj(1; n|7)
appears by ‘adjusting’ the fourth row of Titsconstruction to characteristic 3.
That’s all. Thanks
![Page 97: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/97.jpg)
Some conclusions
char. 3: • By using a symmetric construction in terms oftwo Hurwitz algebras, and extending it (only incharacteristic 3) with the use of compositionsuperalgebras.Ten new simple Lie superalgebras are obtained:g(r , 3)′ (r = 2, 4, 8), g(r , 6)′ (r = 1, 2, 4, 8),g(3, 3), g(3, 6) and g(6, 6).
• The new simple Lie superalgebra el(5; 3) appearsas a maximal subalgebra of g(8, 3).
• The new simple Lie superalgebraT (C (k),O(1, n)⊕O(1, n)u) = Bj(1; n|7)
appears by ‘adjusting’ the fourth row of Titsconstruction to characteristic 3.
That’s all. Thanks
![Page 98: A Freudenthal-Tits Supermagic Square - Universiteit Gent](https://reader031.vdocuments.us/reader031/viewer/2022021009/6203b142da24ad121e4c5c4a/html5/thumbnails/98.jpg)
Some conclusions
char. 3: • By using a symmetric construction in terms oftwo Hurwitz algebras, and extending it (only incharacteristic 3) with the use of compositionsuperalgebras.Ten new simple Lie superalgebras are obtained:g(r , 3)′ (r = 2, 4, 8), g(r , 6)′ (r = 1, 2, 4, 8),g(3, 3), g(3, 6) and g(6, 6).
• The new simple Lie superalgebra el(5; 3) appearsas a maximal subalgebra of g(8, 3).
• The new simple Lie superalgebraT (C (k),O(1, n)⊕O(1, n)u) = Bj(1; n|7)
appears by ‘adjusting’ the fourth row of Titsconstruction to characteristic 3.
That’s all. Thanks