Hopf Algebraic Techniques Applied to
Super Algebraic Groups
Akira Masuoka (U. Tsukuba)
Halifax, June 6, 2013
0. Once upon a time
Around 1992, Moss Sweedler wrote for me a kind letter which contains a list suggesting applications of Hopf algebras, such as Hopf algebras and combinatorics
Hopf algebras and physics Hopf algebras and geometry Hopf algebras and probability. . .
So far, my list contains two,
Hopf algebras and Galois theory of differential/difference equations Hopf algebras and super algebraic groups
Let me talk about the second subject.
We 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 : (ComAlgk) → (Grop).
It is uniquely represented by a com. Hopf algebra A. Thus, (AffineGroupk) ≃ (ComHopfAlgk)op,
G 7→ O(G ), Sp(A) 7→ A.
Here, Sp(A) : R 7→ Algk(A, R), a group under ∗.
By an algebraic group, we mean an affine group G = Sp(A) s.t. A is finitely generated. Thus,
(AlgGroupk) ≃ (AffineHopfAlgk)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
2-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 (SuperVecSpacek) 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 cV ,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 (VecSpacek, ⊗, k, twist)
are generalized, as purely even objects, by
super algebras, super Lie algebras, super Hopf algebras, . . . in (SuperVecSpacek, ⊗, k, cV ,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 functorial viewpoint
A super affine group over k is a representable functor G : (SuperComAlgk) → (Group).
It is uniquely represented by a super-com. Hopf algebra A. (SuperAffineGroupk) ≃ (SuperComHopfAlgk)op,
G 7→ O(G ), SSp(A) 7→ A.
Here, SSp(A) : R 7→ SuperAlgk(A, R).
By a super algebraic group, we mean a super affine group G = SSp(A) s.t. A is finitely generated. Thus,
(SuperAlgGroupk) ≃ (SuperAffineHopfAlgk)op
To each super affine (algebraic) group G = SSp(A), there is associated an affine (algebraic) group,
G := G |(ComAlgk) : (ComAlgk) → (Group), which is represented by
A := A/(A1). Thus
G = Sp(A).
Example. V = V0⊕ V1, m = dim V0, n = dim V1. GLsupV = GL(m|n) : R 7→ SuperAutR(V ⊗ R) O(GLsupV ) = k[xij, ykℓ, det(X )−1, det(Y )−1] ⊗ ∧(pi ℓ, qkj)
(X P
Q Y )
=( xij pi ℓ qkj ykℓ
)
, 1 ≤ i, j ≤ m, 1 ≤ k, ℓ ≤ n.
∆(X P Q Y
)
=(X P Q Y
)
⊗(XQ YP )
, 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 ).
The associated algebraic group of GLsupV is GLV0× GLV1.
Characteristic-free study of super affine groups seems relatively new. Main sources of this talk are:
[M1] A. M., Quotient sheaves of algebraic supergroups are superschemes, J. Pure Appl. Algebra 202 (2005).
[M2] A. M., Harish-Chandra pairs for algebraic affine supergroup schemes over an arbitrary field, Transform. Groups 17 (2012). [MS] A. M., Taiki Shibata, Algebraic supergroups over a PID, on arXiv.
[MPS] A. M., Craig Pastro, Taiki Shibata, Integrals for algebraic supergroups, in preparation.
4. Main results
G = SSp(A) a super algebraic group. A := A/(A1) = O(G |(ComAlg
k)), an affine Hopf algebra.
WA:= A1/A+0A1= Tε∗(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 [M1]. ∃a counit-preserv. left A-comod. super algebra isom. A−→ A ⊗ ∧(W≃ A).
Question. Can A recover from A, WA plus something else? Answer. Yes, byHarish-Chandra pairs, though we are going to present the result so that those pairs are not quoted explicitly.
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◦⊗ ∧(WA)
A A ⊗ ∧(WA)
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 super groups over Z,due to Fireosi and Gavarini, in a generalized situation.
5. Application 1—Cosemisimplicity
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. Thm (R. Weissauer 2009). k = k, char k = 0.
Linearly reductive super algebraic group 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.
6. Application 2—Integrals or co-Frobeinusness
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 in most cases including the case char k = 0.
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 [MPS]. For a super algebraic group G , TFAE: (i) G has an integral;
(ii) The associated algebraic group G := G |(ComAlgk) has an
integral.
Sullivan tells us that if char k = 0, Condition (ii) is equivalent to (iii) G is linearly reductive, or O(G ) is cosemisimple,
and if char k > 2, Condition (ii) is equivalent to
(iv) (G0k)red is a torus, or (O(G ) ⊗ k)/√0 is a group algebra. Conclusion. In characteristic zero, we have many examples of co-Frobenius and non-cosemisimple super Hopf algebras, O(G ), which includes those for the Chevalley super groups G ; their bosonizations by Z2 are co-Frobenius and non-cosemisimple Hopf algebras.
