Hopf Algebraic Techniques Applied to
Super Affine Groups
Pre-canonical isomorphisms and splitting properties
Akira Masuoka (U. Tsukuba)
BIRS, September 9, 2015
1. Pre-canonical isomorphisms = Non-canonical
isomorphisms turning to be canonical after gr
Studying Hopf algebras we sometimes meet such isomorphisms. Example from Sweedler, Page 228. To construct the graded coalgebra gr C =⊕n≥0Cn/Cn−1 from a coalgebra C =∪n≥0Cn filtered by C0⊂ C1 ⊂ C2 ⊂ . . . , we construct a monoidal functor
gr : (Filtered Vector Space) → (Graded Vector Space), gr V =⊕
n≥0
Vn/Vn−1 with an isomorphic monoidal structure
µ = µV ,W : gr V ⊗ gr W −→ gr(V ⊗ W ).≃ We define µ to be the cano. graded linear map induced from
sum : ⊕
i+j=n
Vi⊗ Wj → ∑
i+j=n
Vi⊗ Wj, n≥ 0.
Proof of µ being isomorphisms.
Given V ∈ (Filtered Vector Space), choose a non-canonical filtered linear isomorphism
ϕ : gr V −→ V≃
such that gr ϕ = idgr V. For another such isomorphism ψ : gr W −→ W , the tensor product≃
ϕ ⊗ ψ : gr V ⊗ gr W −→ V ⊗ W≃ in (Filtered Vector Space) induces an isomorphism
gr(ϕ ⊗ ψ) : gr V ⊗ gr W −→ gr(V ⊗ W )≃ in (Graded Vector Space), which is seen to coincide with µ. Q.E.D.
Wish to show this sort of isomorphisms arises for super affine groups or in super algebraic geometry, closely related with splitting properties.
[M1] A. M., Fundamental correspondences in super affine groups and super formal groups, 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). [MZ1] A. M., Alexadr Zubkov, Quotient sheaves of algebraic supergroups are superschemes, J. Algebra 348 (2011).
[MZ2] A. M., Alexadr Zubkov, Solvability and nilpotency for algebraic supergroups, preprint on arXiv.
[MZ3] A. M., Alexadr Zubkov, Regularity and smoothness in algebraic supergeometry, in preparation.
Work over an arbitrary field k, possibly char k = 2.
Let A = A0⊕ A1 be a super algebra. Always assume that it is super-commutativein the sense that
ab= (−1)|a||b|ba, a, b ∈ A0∪ A1; c2= 0, c ∈ A1. Let
(Super Algebra)
denote the category of (super-commutative) superalgebras.
2. Basics on super affine groups
A super affine group is a representable functor G : (Super Algebra) → (Group). It is uniquely represented by a super Hopf algebra A.
(Super Affine Group) ≃ (Super Hopf Algebra)op, G 7→ O(G ), SSp(A) 7→ A.
Here, SSp(A) : R 7→ HomSuperAlg(A, R).
A super algebraic group is a super affine group G = SSp(A) s.t. A is finitely generated. Thus,
(Super Algebraic Group) ≃ (F.g. Super Hopf Algebra)op.
Example. V = V0⊕ V1, m| n = sdim V = dim V0 | dim V1. GLV = GL(m|n) : R 7→ AutR-super-linear(V ⊗ R)
O(GLV) = k[xij, yst, det(X )−1, det(Y )−1] ⊗ ∧(pit, qsj)
(X P
Q Y
)
=( xij pit qsj yst
)
, 1 ≤ i, j ≤ m, 1 ≤ s, t ≤ 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 ).
Correspondence of morphisms
A morphism H → G of super affine groups is called a closed embeddingif O(G ) → O(H) is a surjection. This is a definition. A morphism G → Q of super affine groups is an epimorphism iff O(Q) → O(G ) is an injection. This is NOT a definition, but is based on the following result [M1]: every super Hopf algebra is faithfully flat over every super Hopf subalgebra; proof to be given below.
Waterhouse GTM 66 “epimorphism in a true sense”
= epimorphism of sheaves in fpqc topology
Given a super affine group G = SSp(A), let’s define G = Sp(A) and WA. G is the restricted group functor
G := G |(Algebra): (Algebra) → (Group);
this is an affine group, represented by the ordinary Hopf algebra A:= A/(A1).
WA:= A1/A+0A1= Tε∗(G )1, i.e., the odd part of the cotangent space of G at 1.
3. Splitting of Hopf superalgebras
Let G = SSp(A) ∈ (Super Affine Group). Define IA = (A1) (= A21⊕ A1). Then
gr A =⊕
n≥0
IAn/IAn+1= A ⊕ A1/A31⊕ A21/A41⊕ . . .
is a graded super Hopf algebra with the 0th component A, onto which gr A canonically projects. Since
(gr A)/A+(gr A) ≃
cano ∧ (W A),
we have acanonical isomorphism of graded super Hopf algebras, gr A−→ A≃ >◭∧ (WA) (= smash coproduct)
Thm [M1]. There is a (pre-canonical)isomorphism A−→ A ⊗ ∧(W≃ A)
of left A-comodule super algebras that preserves the counit.
Proof in case char k = 0, A finitely generated
A f.g. Hopf algebra in char. 0 → smooth →equivariantly smooth A0 → A left A-comodule algebra surjection with nilpotent kernel Has a splitting ϕ : A → A0(⊂ A).
By Doi-Takeuchi A⊗coAA−→ A,≃ a⊗ x 7→ ϕ(a) x. Let g = Lie(G ) with G = SSp(A), and choose an embedding
U(g) ←֓ ∧(g1). Dualize and compose to obtain
coAA֒→ A →U(g)∗→ (∧(g
1))∗ = ∧(WA).
This is indeed a pre-canonical isomorphism. Q.E.D.
Proof of faithful flatness. The isom. can be chosen so as compatible with a super Hopf algebra map f : A → B.
A A⊗ ∧(WA)
B B⊗ ∧(WB)
≃ //
≃ //
f
f⊗∧(Wf)
Suppose f is an inclusion A ֒→ B. Then
f : A → B, Wf : WA → WB are both injections, and so
A⊂ B faithfully flat, ∧(WA) ⊂ ∧(WB) free.
∴A⊂ B faithfully flat. Q.E.D.
4. Application of Splitting I – Harish-Chandra pairs
Given G = SSp(A) ∈ (Super Algebraic Group), we have the pair (G , Lie(G )1), where Lie(G )1 = (WA)∗.
Question: Can G recover from the pair? Answer: Yes, roughly speaking.
Def (Koszul). A Harish-Chandra pair is a pair (F , V ) of an algebraic group F and a f.d. right F -module V , which is endowed with an F -equivariant bilinear map [ , ] : V × V → Lie(F ) s.t.
[v , v′] = [v′, v ] and v▹ [v , v ] = 0,
where ▹ represents the action V x Lie(F ) induced from V x F . Thm [M2]. G 7→ (G , Lie(G )1) gives
(Super Algebraic Group) ≈ (Harish-Chandra pair).
Cor ([M2] etc.) G has one of the properties listed below iff G has the same property.
◮ unipotent
◮ simply connected
◮ smooth
◮ connected and solvable
◮ having a non-zero integral
5. Application of Splitting II – the quotient G ˜/H
We have Comparison Theorem [MZ1]: The super schemes defined as a spacial kind of functors (Super Algebra) → (Set) and
the super schemes defined as a special kind of locally super-ringed spaces are equivalent.
Thm [MZ1]. G = SSp(A) a super algebraic group, H= SSp(A/I ) a closed super subgroup. The fppf -sheafification G ˜/H of the functor of cosets
R7→ G (R)/H(R) is a Noetherian super scheme. The quotient G → G ˜/H has nice properties, such as affine and faithfully flat. This answers a question posed by J. Brundan.
Prop [MZ1]. Given G ⊃ H as above, G ˜/H is affine iff G ˜/H is affine. The equiv. conditions are satisfied if G ◃ H.
6. Regular/smooth super algebras
Let A = A0⊕ A1∈ (Super Algebra). Define A= A/(A1) (= A0/A21), the largest ordinary quotient algebra of A.
We have 1-1 correspondences between prime spectrums Spec A= Spec A0 = Spec A,
p0⊕ A1 ↔ p0 ↔ p0/A21 =: p0.
Suppose A is Noetherian, and local with maximal m = m0⊕ A1 with residue field K = A/m (= A0/m0 = A/m0).
Def. Let r | s = sdimK(m/m2). A is regular if
cano : K [X1, . . . , Xr] ⊗K∧(Y1, . . . , Ys) → grm(A) =⊕
n≥0
mn/mn+1
is an isomorphism.
Let A = A0⊕ A1∈ (Super Algebra). Define
IA = (A1) (= A21⊕ A1), A= A/IA. Construct the graded algebra over A
grIA(A) =⊕
n≥0
IAn/IAn+1= A ⊕ A1/A31⊕ A21/A41⊕ . . .
Thm [MZ3]. Suppose that A is Noetherian. TFAE:
(a) A is regular, i.e., the localization Ap0 at every prime/maximal p0 of A0 is regular;
(b) (i) The algebra A is regular, (ii) the f.g. A-module IA/IA2 is projective, and (iii) the canonical graded A-algebra map
κA : ∧A(IA/IA2) → grIA(A) is an isomorphism.
Let A ∈ (Super Algebra). A is smooth over k, if given a surjection B→ C in (Super Algebra) with nilpotent kernel, the induced map Hom(A, B) → Hom(A, C ) is a surjection, or equivalently, if in (Super Algebra), every surjection onto A with nilpotent kernel splits. .
Thm [MZ3]. Suppose that A is Noetherian. TFAE: (a) A is smooth over k;
(b) A is geometrically regular over k, i.e., for any finite field extension ℓ/k, A ⊗ ℓ is regular;
(c) (i) The algebra A is smooth over k, (ii) the f.g. A-module IA/IA2 is projective, and (iii) there exists a(pre-canonical for κA) isomorphism∧A(IA/IA2) ≃ A of superalgebras.
Rem. The last isom. is a weaker variant of A ⊗ ∧(WA) ≃ A for Hopf superalgebras. The latter hold for any (even non-smooth) Hopf superalgebra.
