classification of connected hopf algebras up to prime-cube ... · as-regular algebras....

Classification of connected Hopf algebras up toprime-cube dimension

Xingting Wang

University of California, San Diego

2015 AMS Special Session on Hopf Algebras and Tensor Categories, San Antonio

January 12, 2015

Motivations (long-term goal)

We want to understand the behavior of finite quantum groups(finite-dimensional Hopf algebras) in positive characteristic.

• Representation theory: irreducible representations, gaugeequivalence, quiver algebras and Auslander-Reiten quivers.

• Cohomology theory: cohomology rings of finite-dimensionalHopf algebras.

• Noncommutative invariant theory: quantum group actions onAS-regular algebras.

• Yetter-Drinfeld category and Nichols algebra.

Motivations (short-term goal)

We study the behavior of unipotent quantum groups (connectedHopf algebras) by classifying all such Hopf algebras of dimension p,p2 and p3 over an algebraically closed field of characteristic p > 0.

Outlines

Part I: Basic definitions and notations.

Part II: Classification list.

Part III: Primitive control deformation (PCD).

Part IV: Future projects.

Part I

Throughout, let k be a base field, algebraically closed ofcharacteristic p > 0. Use the standard notation(

H,m, u,∆, ε,S)

to denote a Hopf algebra. Regarding the coalgebra structure of(H,∆, ε), we have

• the coradical of H is the sum of all simple subcoalgebras,which is denoted by H0;

• the primitive space P(H) = x ∈ H|∆(x) = x ⊗ 1 + 1⊗ x;• P(H) is a restricted Lie algebra, where the Lie bracket is given

by the commutator and the restricted map is given by thep-th power in H.

Part I

We say H is connected if its coradical H0 is one-dimensional. Factsabout finite-dimensional connected Hopf algebras:

• They only appear in positive characteristic.

• They all have dimension pn for some integer n > 0.

• They can be constructed from p-groups, finite unipotentgroup schemes and finite restricted Lie algebras.

• They are in one-to-one correspondence with finite-dimensionallocal Hopf algebras by Cartier duality.

Part II

We provide new examples of finite unipotent quantum groups byclassifying all such algebras of dimension p, p2 and p3 over k. Alist of isomorphism classes is obtained with explicit generators andrelations.

• dimension p, there are two isomorphism classes.

• dimension p2, there are eight isomorphism classes.

• dimension p3.

♥ dimP(H) = 1, there are four isomorphism classes and oneinfinite parametric family.

♠ dimP(H) = 2 and nonabelian, there are three isomorphismclasses.

♣ (suppose p > 2). dimP(H) = 2 and abelian, there are thirtythree isomorphism classes, one finite parametric family andeight infinite parametric families.

♦ dimP(H) = 3, there are fifteen isomorphism classes and onefinite parametric family.

Part II

This classification involves:

• groups of order p, p2 and p3 (Holder 1893 and Burnside1897)

• order p: Cp

• order p2: Cp × Cp and Cp2

• order p3: Abelian Cp3 , Cp2 × Cp and Cp × Cp × Cp; NonabelianD4, Q8 for p = 2 and (Cp × Cp) o Cp and Cp2 o Cp for p > 2

• finite unipotent group schemes of dimension p, p2 and p3

• restricted Lie algebras of dimension 1, 2 and 3 (simple Liealgebras over an algebraically closed field of characteristicp ≥ 5 is recently classified by Premet, Strade and others)

Part II

Infinite family in ♥: let λ ∈ k, and A(λ) the k-algebra generatedby elements x , y , z , subject to the following relations

[x , y ] = 0, [x , z ] = 0, [y , z ] = x , xp = 0, yp = 0, zp + xp−1y = λx .

Take the expression ω(t) =∑

1≤i≤p−1

(pi

)/pt i ⊗ tp−i . Therefore

A(λ) becomes a connected Hopf algebra via

∆(x) = x ⊗ 1 + 1⊗ x , ∆(y) = y ⊗ 1 + 1⊗ y + ω(x),

∆(z) = z ⊗ 1 + 1⊗ z + ω(x)(y ⊗ 1 + 1⊗ y)p−1 + ω(y),

ε(x) = ε(y) = ε(z) = 0, S(x) = −x ,S(y) = −y ,S(z) = −z .

When p > 2, A(λ) ∼= A(λ′)⇔ λ = γλ′ for some γ ∈ p2+p−1√

1, or

the isomorphism classes of A(λ) are parametrized by k/ p2+p−1√

1.

Part III

Techniques used in classification:

• Dimension p: choose 0 6= x ∈ P(H). Since xp ∈ P(H), wehave xp = λx for some λ ∈ k. By rescaling, we can takeλ = 0 or 1.

• Dimension p2: restricted Lie algebra theory and Hochschildcohomology of coalgebras (cobar construction).

• Dimension p3: ♣ Primitive control deformations (PCDs) ofrestricted universal enveloping algebras.

Part III: A concrete example of PCD

Consider the trivial extension T of restricted Lie algebras:

T : 0 // g // g⊕ h // h // 0

satisfying

• dim g = 2, dim h = 1. Fix basis x , y for g and z for h.

• By Strade and Farnsteiner’s terminology, suppose g and h aretori such that [x , y ] = 0 and xp = x , yp = y , zp = z .

The restricted universal enveloping algebra of T is

u(g⊕ h) =k⟨x , y , z

⟩([x , y ], [y , z ], [x , z ], xp − x , yp − y , zp − z

) .


Any PCD of the extension T is given by[u(g⊕ h), χ,Θ

]:= k[x , y , z ]/J

where the relation J is generated by

xp − x , yp − y , zp − z + Θ.

and

∆(x) = x ⊗ 1 + 1⊗ x , ∆(y) = y ⊗ 1 + 1⊗ y ,

∆(z) = z ⊗ 1 + 1⊗ z + χ.


Let B = u(g). The parametric space of A := [u(g⊕ h), χ,Θ] is

P = Z 2(ΩB)× B+ 3 (χ,Θ) ,

where Z 2(ΩB)

is the set of all cocycles in the cobar constructionon B:

k // B+ d1// B+ ⊗ B+ d2

// B+ ⊗ B+ ⊗ B+ // · · · .

Now all PCDs of the extension T correspond to the subset of Psatisfying

(1) grA = k[X ,Y ,Z ]/(X p,Y p,Zp) with respect to the coradicalfiltration.

(2) P(A) = g.

Part III: A concrete example of PCDIn order to classify all PCDs of the extension T , we construct asubset quotient H2(T ) of P by imposing the equivalence relationof extensions

1 // u(g) // A // u(h) // 1 .

We show that there is a bijection

H2(T ) oo // S = Fp × Fp × Fp \ (0, 0, 0)

In details, any point P = (a, b, c) ∈ S can be represented in theparametric space P by

χP = ax ⊗ y +

p−1∑i=1

(p

i

)/p (bx + cy)i ⊗ (bx + cy)p−i ,

ΘP = 0.

Part III: A concrete example of PCDs

Moreover, we define an automorphism group

Aut(T ) = Aut(g)× Aut(h) = GL(2,Fp)× GL(1,Fp).

Choose any φ = M × γ ∈ GL(2,Fp)× GL(1,Fp). There is anembedding Aut(T ) → GL(3,Fp) via

φ = M × γ →(γ det(M) 0

0 γM

).

Then φ acts on any point P = (a, b, c) ∈ S as follows:

φ [a, (b, c)] = [γ(detM)a, γ(b, c)M].

We show that Aut(T )-orbits in S are in 1-1 correspondence withisomorphism classes of all PCDs of the extension T .

Part III: A concrete example of PCDs

Consider the group action on S = Fp × Fp × Fp \ (0, 0, 0) by thefollowing two normal subgroups.

• Aut(h) = F×p acts on S by multiplication

φ(a, b, c) = (γa, γb, γc).

• Aut(g) = GL(2,Fp) acts on S via the embedding

M →(

det(M) 00 M

).

Since S/F×p = P2,

Aut(T )-orbits in S oo // Aut(g)-orbits in P2

via the previous embedding GL(2,Fp) → PGL(3,Fp).

Part III: A concrete example of PCDThe Aut(g)-orbits in P2 contain three points

A = [1 : 0 : 0], B = [1 : 1 : 0], C = [0 : 1 : 0]

The corresponding PCDs are

k[x , y , z ]

(xp − x , yp − y , zp − z)

where x , y are primitive elements and

A : ∆(z) = z ⊗ 1 + 1⊗ z + x ⊗ y

B : ∆(z) = z ⊗ 1 + 1⊗ z + x ⊗ y +

p−1∑i=1

(p

i

)/p x i ⊗ xp−i

C : ∆(z) = z ⊗ 1 + 1⊗ z +

p−1∑i=1

(p

i

)/p x i ⊗ xp−i


By Masuoka’s result, A, B, C correspond to the dual of the groupalgebra kG , whose Frattini group is Cp. There are only three ofthem

A : ∆(z) = z ⊗ 1 + 1⊗ z + x ⊗ y ↔ (Cp × Cp) o Cp

B : ∆(z) = z ⊗ 1 + 1⊗ z + x ⊗ y +

p−1∑i=1

(p

i

)/p x i ⊗ xp−i ↔ Cp2 o Cp

C : ∆(z) = z ⊗ 1 + 1⊗ z +

p−1∑i=1

(p

i

)/p x i ⊗ xp−i ↔ Cp2 × Cp

Part III: Classification of ♣

♣: Suppose p > 2, dimP(H) = 2 and abelian. There are thirtythree isomorphism classes, one finite parametric family and eightinfinite parametric families.

All Hopf algebras in ♣ come from PCDs of u(go h) satisfying

• dim g = 2 and g is abelian.

• dim h = 1.

• The primitive space of the deformation is isomorphic to g.


We first classify all possible extensions

T : 0 // g // go h // h // 0

with data T = (g, h, ρ). There are sixteen isomorphism classes ofsuch extensions.

Classification of g :

A : xp = 0, yp = 0, B : xp = 0, yp = 0,

C : xp = y , yp = 0, D : xp = x , yp = x .

Classification of h : N : zp = 0, S : zp = z .


Type g h Algebraic representation

(T1) A N ρz = 0

(T2) A N ρz(x) = y , ρz(y) = 0

(T3) A S ρz = 0

(T4) A S ρz(x) = x , ρz(y) = λy for λ ∈ Fp and λ 6= −1

(T5) B N ρz = 0

(T6) B N ρz(x) = 0, ρz(y) = x

(T7) B S ρz = 0

(T8) B S ρz(x) = 0, ρz(y) = y

(T9) B S ρz(x) = 0, ρz(y) = x + y


Type g h Algebraic representation

(T10) C N ρz = 0

(T11) C N ρz(x) = y , ρz(y) = 0

(T12) C S ρz = 0

(T13) C S ρz(x) = x , ρz(y) = 0

(T14) C S ρz(x) = x + y , ρz(y) = 0

(T15) D N ρz = 0

(T16) D S ρz = 0


Type Aut(T )-orbits in H2(T )

(T1) eight points

(T2) six points and k/±1, k

(T3) NONE

(T4) one point for each −1 6=λ ∈ Fp, totally p+12 points

(T5) four points and k/p−1

2√

1

(T6) one point and k/p2−1

2√

1

(T7) one point

(T8) three points

(T9) k/(F×p )2 and k


Type Aut(T )-orbits in H2(T )

(T10) four points and k/ p2−p−1√

1

(T11) one point and k/ p2−p+1√

1

(T12) NONE

(T13) one point

(T14) one point

(T15) NONE

(T16) three

Part IV: Future projects

Suppose H is a finite-dimensional connected Hopf algebra.

• Only as algebras, when is H isomorphic to some restricteduniversal enveloping algebra u(l)? (Anti-example from ♠)?

• Is H always basic or H∗ pointed?

• What does H•(H, k) := Ext•H(k, k) look like?

• For a fixed dimension pn, can we parametrize all isoclasses ofH, apart from a finite number of them, by a finite number of1-parameter families (tameness)?

References

X. Wang,

Connected Hopf algebras of dimension p2,J. Algebra 391 (2013), 93-113.

X. Wang,

Local Criteria for cocommutative Hopf algebras,Comm. Algebra 42 (2014) no.12, 5180-5191.

X. Wang4,

Another proof of Masuoka’s Theorem for semisimple irreducible Hopf algebras,preprint, arXiv:1212.0622.

X. Wang3,

Isomorphism classes of finite dimensional Hopf algebras in positive characteristic.preprint, arXiv:1403.3130.

L. Wang and X. Wang

Classification of pointed Hopf algebras of dimension p2 over any algebraically closed field,Algebr. Represent. Theory 17 (2014), no.4, 1267-1276.

V. Nguyen, L. Wang and X. Wang,

Classification of Connected Hopf algebras of dimension p3 I,J. Algebra, 424 (2015), 473-505.

V. Nguyen, L. Wang and X. Wang,

Classification of Connected Hopf algebras of dimension p3 II,in preparation.

classification of connected hopf algebras up to prime-cube ... · as-regular algebras....

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