SLIDES OF TALKS Akira Masuoka Boston2013

Hopf Algebraic Techniques Applied to

Super Algebraic Groups

Akira Masuoka (U. Tsukuba)

Boston, April 7, 2013

0. What is a space?

Functorial viewpoint: A space (over k) is a functor defined on the category (ComAlg_k) of com. algebras.

Geometric viewpoint: It is a ringed space (X , O_X).

Want to show that the functorial (≑ Hopf-algebraic) viewpoint is effective for algebraic groups, and even more so for super algebraic groups.

Work over a field k (unless otherwise stated).

1. What is an algebraic group?

An affine group (scheme) over k is a representable functor G : (ComAlg_k) → (Group).

It is uniquely represented by a com. Hopf algebra A. Thus, (AffineGroup_k) ≃ (ComHopfAlg_k)^op,

G 7→ O(G ), Sp(A) ^7→ A.

Here, Sp(A) : R 7→ Alg_k(A, R), a group under ∗.

By an algebraic group, we mean an affine group G = Sp(A) s.t. A is finitely generated. Thus,

(AlgGroup_k) ≃ (AffineHopfAlg_k)^op

Hopf-algebraic study of algebraic groups

G. Hochschild char k = 0 using Lie algebras

M. Takeuchi characteristic-free using hyperalgebras

A hyperalgebra is a synonym of an irreducible cocom. Hopf alg. The hyperalgebra hy(G ) of an algebraic group G = Sp(A) is defined by

hy(G ) = ^∪

n>0

(A/(A⁺)ⁿ)^∗ inA^∗ (A⁺= Ker ε).

It coincides with U(Lie(G )) if char k = 0.

2. Invitation to the super (= Z

-graded) world

In what follows assume char k ̸= 2. A super vector space is a vector space

V = V0⊕ V1

graded by Z2= {0, 1}. The super vector spaces form a tensor category (SuperVecSpace_k) w.r.t the tensor product

V ⊗ W = ^⊕

i+j=0

(Vi⊗ Wj) ⊕ ^⊕

i+j=1

(Vi⊗ Wj),

and the unit object k (= k0). It is symmetric w.r.t. the super symmetry c_{V ,W} : V ⊗ W −→ W ⊗ V given by^≃

cV ,W(v ⊗ w ) = (−1)^{|v | |w |}w ⊗ v =

{−w ⊗ v if |v | = |w | = 1 w ⊗ v otherwise.

Algebras, Lie algebras, Hopf algebras, . . . in (VecSpace_{k, ⊗, k,} twist)

are generalized, as purely even objects, by

super algebras, super Lie algebras, super Hopf algebras, . . . in (SuperVecSpace_{k, ⊗, k, c}V ,W).

Example. V a vector space.

∧(V ) = ^⊕

neven

∧ⁿ(V ) ⊕ ^⊕

n odd

∧ⁿ(V )

This is a super-com. (xy = (−1)^{|x| |y |}yx), (super-)cocom. (super) Hopf algebra in which every v ∈ V is supposed to be odd

primitive. If dim V < ∞, this is self-dual.

3. Super groups from the functorial viewpoint

A super affine group over k is a representable functor G : (SuperComAlg_k) → (Group).

It is uniquely represented by a super-com. Hopf algebra A. (SuperAffineGroup_k) ≃ (SuperComHopfAlg_k)^op,

G 7→ O(G ), SSp(A) ^7→ A.

Here, SSp(A) : R 7→ Z2^-GrAlg_k^{(A, R).}

By a super algebraic group, we mean a super affine group G = SSp(A) s.t. A is finitely generated. Thus,

(SuperAlgGroup_k) ≃ (SuperAffineHopfAlg_k)^op

Example. V = V₀⊕ V₁, m = dim V₀, n = dim V₁. GLV = GL(m|n) : R 7→ Z2^-GrAutR(V ⊗ R) O(GLV) = k[xij, ykℓ, det(X )⁻¹, det(Y )⁻¹] ⊗ ∧(pi ℓ, qkj)

(X P

Q Y )

=^{( xij pi ℓ} q_kj y_kℓ

)

, 1 ≤ i, j ≤ m, 1 ≤ k, ℓ ≤ n.

∆^{(X P} Q Y

)

=^{(X P} Q Y

)

⊗^{(X P} Q Y

)

, ε^{(X P} Q Y

)

=^{( I O}

O I

) ,

S(X ) = (X − PY⁻¹Q)⁻¹, S(Y ) = (Y − QX⁻¹P)⁻¹, S(P) = −X⁻¹PS(Y ), S(Q) = −Y⁻¹QS(X ).

Characteristic-free study of super affine groups seems relatively new. Main sources of this talk are:

[MZ] A. M., Alexadr Zubkov, Quotient sheaves of algebraic supergroups are superschemes, J. Algebra 348 (2011).

[M] A. M., Harish-Chandra pairs for algebraic affine supergroup schemes over an arbitrary field, Transform. Groups .

[MS] A. M., Taiki Shibata, Algebraic supergroups over a PID, in preparation.

[MP] A. M., Craig Pastro, Integrals for algebraic supergroups, in preparation.

4. The quotient G ˜/H

Thm [MZ]. G = SSp(A) a super algebraic group, H = SSp(A/I ) a closed super subgroup. The k-sheaf G ˜/H obtained as the sheafification of the ‘naive’ functor

R 7→ G (R)/H(R), the cosets

is a Noether super scheme, i.e., has a finite covering of open sub-functors represented by Noether super com. algebras.

Note. A functor X : (SuperComAlg_k) → (Set) is called a k-sheaf (under the fppf topology), if

(i) X (^∏n_ı=1R_i) ≃^∏n_ı=1X (R_i),

(ii) for R → S in (SuperComAlg_k) s.t. S is faithfully flat and finitely presented over R,

1 → X (R) → X (S) ⇒ X (S ⊗_RS) is exact.

Given G = SSp(A) as above, the restriction of the domain gives an algebraic group,

G := G |_(ComAlg

k^{) : (ComAlgk}) → (Group), which is represented by

A := A/(A₁).

Thm [MZ]. H < G super algebraic groups as above. There are associated algebraic groups, H < G . Then

G ˜/H is affine ⇔ G ˜/H is affine.

▶ These equivalent conditions hold if H ◁ G .

▶ In particular, G ˜/H is affine if H is reductive; a positive answer to J. Brundan’s question.

5. A key result—Tensor product decomposition

G = SSp(A) a super algebraic group. A := A/(A₁) = O(G |_(ComAlg

k⁾), an affine Hopf algebra.

W^A:= A₁/A⁺₀A₁= T_ε^∗(G )₁, the odd part of the cotangent space of G at ε; this is a f.d. vector space.

A is regarded as a left A-comodule algebra along A → A.

Thm(A.M.) ∃a counit-preserv. left A-comod. super algebra isom. A−→ A ⊗ ∧(W^{≃ A})

The proof uses Cleftness Theorem due to Doi and Takeuchi. Question. Can A recover from A, W^A plus something else? Answer. Yes, by Harish-Chandra pairs!

6. Harish-Chandra pairs—Hopf-algebraic formulation

A Harish-Chandra pair is a pair (C , W ) of an affine Hopf algebra C and a f.d. right C -comodule W which is accompanied with a bracket [ , ] : W^∗× W^∗ → P(C^◦) satisfying some conditions. One can prove:

A 7→ (A, W^A) gives a category equivalence

(SuperAffineHopfAlg_k) ≈ (Harish-ChandraPair_k).

A more Hopf-algebraic and useful reformulation (with the restriction “connected” added) is the following:

Thm [MS]. G 7→ (hy(G ), G ) gives a category equivalence from

▶ the connected super algebraic groups G to

▶ the pairs (H, F ) of super hyperalgebras H and connected algebraic groups F s.t.

(i) the largest ordinary Hopf subalgebra H in H equals hy(F ), (ii) the vec. space P(H)1of odd primitives in H is finite-dim’nal, (iii) the adjoint action of H on P(H)1arises (uniquely) from an

F -module structure.

Note. If char k = 0, the second category is identified with

▶ the pairs (L, F ) of f.d. super Lie algebras L and connected algebraic groups F s.t.

(i^′) L0= Lie(F ),

(iii^′) the adjoint action of L0on L1 arises (uniquely) from an F -module structure.

Let’s observe G 7→ (hy(G ), G ). Set A = O(G ), H = hy(G ). Then O(G ) = A := A/(A₁). We have natural embeddings,

A ֒→ H^◦, A ֒→ H^◦.

Recall H ⊂ H is the largest ordinary Hopf subalgebra. We have cano : H^◦→ H^◦, which is shown to be surjective.

Prop [MS]. We have a com. diagram of super algebras with counit,

H^◦ H^◦⊗ ∧(W^A)

A A ⊗ ∧(W^A)

H^◦-comodule isom.

A-comodule isom.

≃ _//

≃ _//

OO OO

⟲

It follows that A is characterized in the left H^◦-comodule H^◦ as the largest A-subcomodule.

Assume G is a connected reductive group with a split maximal torus T . Then the characterization above proves:

Thm [MS]. There is a natural category isomorphism between

▶ the super G -modules and

▶ those locally finite super hy(G )-modules whose restricted hy(T )-module structures arise (uniquely) from T -module structures.

Remark [MS]. The last two theorems, with some minor modification, hold true when k is a PID. The first theorem then gives an alternative construction of the Chevalley supergroups over Z,due to Fioresi and Gavarini, in a generalized situation.

7. Application—characterizations of 3 classes

Class 1. Simply connected super algebraic groups

Def. A connected super algebraic group G is said to be simply connected, if an epimorphism η : F ↠ G of connected super algebraic groups with Ker η purely even, finite etale is necessarily isomorphic.

Thm in char zero (cf. Hochschild 1970). k = k, char k = 0. G 7→ Lie(G ) gives a bijection, modulo isom., from

▶ the simply connected super algebraic groups G to

▶ the f.d. super Lie algebras L s.t. (i) RadL0 is nilpotent,

(ii) adⁿ(L0)(L1) = 0 for some n > 0.

Thm in positive char (cf. Takeuchi 1975). _{k perfect,} char k > 2.

G 7→ hy(G ), H 7→ Sp(H^◦) give a category equivalence between

▶ the simply connected super algebraic groups G and

▶ the super hyperalgebras H s.t.

(i) the super Lie algebra P(H) of all primitives in H is finite-dim’nal,

(ii) the abelianization H^ab of the largest ordinary Hopf subalgebra H of H is finite-dim’nal,

(iii) H is proper, i.e., ^∩

dim H/I <∞

I = 0.

Class 2. Unipotent super algebraic groups

Def. A super algebraic group G is said to be unipotent, if simple super G -modules are exhausted by the purely even or odd trivial super G -module k, or equivalently if O(G ) is irreducible, i.e., Corad(O(G )) = k.

Thm [M] (A. Zubkov). G is unipotent if and only if G = G |_(ComAlgk) is unipotent.

Class 3. Linearly reductive super algebraic groups

Def. A super algebraic group G is said to be linearly reductive, if every super G -module is semisimple, or equivalently if O(G ) is cosemisimple, i.e., Corad(O(G )) = O(G ).

Linearly reductive super algebraic groups are rather restricted.

Linearly reductive super algebraic groups are rather restricted.

Thm (R. Weissauer 2009). k = k, char k = 0.

Linearly reductive super algebraic groups are exhausted by (a lin. reductive alg. group) ⋉^∏

r

Spo(1, 2r )^nr,

where Spo(1, 2r ) denotes the orthosymplectic super group.

Thm [M]. char k > 2.

A linearly reductive super algebraic group G is necessarily purely even, i.e., is an ordinary algebraic group. Hence by Nagata’s Theorem, O(G ) ⊗ k is spanned by group-likes.

8. Existence of Integrals

An integral for a super algebraic group G = SSp(A) is a non-zero, left or right A-comodule map ϕ : A → k.

▶ Recall A is said to be co-Frobenius if such a ϕ exists.

▶ ϕ is said to be total if ϕ(1) = 1.

G is linearly reductive ⇔ G has a total integral.

▶ As was seen, the last equiv. condition are rarely satisfied. Compare with: B. Sullivan: If an algebraic group has an integral, it is necessarily linearly reductive, either if char k = 0 or if the algebraic group is connected.

Fact. For a super algebraic group G , TFAE: (1) G has an integral;

(2) Every injective super G -module is projective;

(3) The injective hull of every finite-dim’nal super G -module is finite-dim’nal.

Thm [MP]. A super algebraic group G has an integral if and only if the associated algebraic group G := G |_(ComAlgk) has an integral.

As consequences, we have:

▶ Suppose G is connected reductive. (i) If char k = 0, then G has an integral.

(ii) If char k > 2, then G does not have any integral unless G is a torus.

The conclusions hold true for the Chevalley super groups.

▶ We have many examples of co-Frobenius and

non-cosemisimple super Hopf algebras, O(G ), in char. zero; their bosonizations by Z2 are co-Frobenius and

non-cosemisimple Hopf algebras.