Frobenius kernels of algebraic supergroups and Steinberg's tensor product theorem

For a split quasireductive supergroup $G$ defined over a field, we study structure and representation of Frobenius kernels $G_r$ of $G$ and we give a necessary and sufficient condition for $G_r$ to be unimodular in terms of the root system of $G$. We also establish Steinberg's tensor product theorem for $G$ under some natural assumptions.

Structure and representation of algebraic group schemes (especially, connected and split reductive groups) over a field have been well studied (see [J, Mi] for example) and provide applications in many areas such as combinatorics or number theory.Over an algebraically closed field k of characteristic zero, the Lie algebra Lie(G) of a connected and split reductive group (scheme) G strongly reflects many properties of G (see [Ho]) and becomes a fundamental tool for studying representations of G.For example, it is known that there exists a category equivalence between the category of left G-modules and the category of locally finite left Lie(G)-T -modules, where T denotes a split maximal torus of G. Here, we say that a Lie(G)-module M is Lie(G)-T -module if the restricted Lie(T )-module structure on M arises from some T -module structure on it.In particular, we can show that for a dominant weight λ, the simple left G-module L(λ) of highest weight λ coincides with the induced representation ind G B (k λ ) of the one-dimensional T -module k λ of weight λ, where B denotes a fixed Borel subgroup of G.The character of L(λ) is explicitly given by Weyl's character formula.
On the other hand, over a field k of positive characteristic, the situation is more complicated, since the simple left G-module L(λ) may be a proper submodule of ind G B (k λ ) in general.In [T1], Takeuchi studied the hyperalgebra hy(G) of G which is a natural refinement of the universal enveloping algebra U(Lie(G)) of Lie(G).Note that, hy(G) is isomorphic to U(Lie(G)) as (cocommutative) Hopf algebras if char(k) = 0.By Hopf-algebraic method, as in the Lie algebra case, he showed hy(G) strongly reflects many properties of G (see [T1, T2, T3]).There also holds a category equivalence between the category of left G-modules and the category of locally finite left hy(G)-T -modules (see [J,Part II,Chapter 1] for example).Over a perfect field k of positive characteristic p, for each positive integer r, the kernel G r of the r-th iterated Frobenius morphism Fr r : G → G, called the r-th Frobenius kernel of G, is a fundamental and powerful tool for studying G.By definition, we have an ascending chain − →r hy(G r ).Moreover, it is known that all Frobenius kernels G r are unimodular, that is, there exists non-zero two-sided integral for G r , see Definition 4.2.Using the categorical equivalence of modules mentioned above, we can show Steinberg's tensor product theorem ([J, Part II, Corollary 3.17]) which states that as a left G-module, the simple left G-module L(λ) decomposes into some tensor products of Fr r -twisted simple left G-modules L(λ r ) [r] such as L(λ) ∼ = L(λ 0 ) ⊗ L(λ 1 ) [1] ⊗ L(λ 2 ) [2] ⊗ • • • ⊗ L(λ m ) [m]   along the "p-adic expansion" λ = λ 0 + pλ 1 + p 2 λ 2 + • • • + p m λ m of λ, where λ r 's are p-restricted weights for G (see Definition 5.18).In particular, the character of L(λ) can be calculated by the product of the character of L(λ r ) [r] .Note that, if we write the character of a G-module M as λ dim(M λ )e λ , then the character of Fr rtwisted G-module M [r] is given by λ dim(M λ )e p r λ .Therefore, the decomposition tells us that to study a simple left G-module, it is enough to consider simple left G-modules with p-restricted weights.
In recent years, supergeometries and superalgebras have attracted much attention.The word "super" is a synonym of "graded by the group Z 2 of order two" (see Section 2.1).The symmetric tensor category of vector spaces is generalized by the category of superspaces (i.e., Z 2 -graded vector spaces) with the familiar tensor product and supersymmetry.The classification of finite dimensional simple Lie superalgebras over an algebraically closed field of characteristic zero was done by Kac [Ka].Since then, many authors have studied the corresponding algebraic supergroup ( [Kost,Kosz,P,Bo,BrKj,Ma1,Z1,CCF,V,FG] for example).Here, an algebraic supergroup is a representable functor from the category of commutative superalgebras to the category of groups; the representing object O( ) is a finitely generated commutative Hopf superalgebra.In this paper, as the super-analogue of the connected and split reductive groups, we study an algebraic supergroup whose "even part" ev is a connected and split reductive group, called a split quasireductive supergroup ([Shi1,Shi2], see also [Se, GZ]), over a field.The class of split quasireductive supergroup has many important algebraic supergroups, for example, the general linear supergroups Ä(m|n), the queer supergroups É(n), the periplectic supergroups È(n), Chevalley supergroups of classical type (including special linear supergroups ËÄ(m|n) and ortho-symplectic supergroups ËÔÇ(m|n)) due to Fioresi and Gavarini [FG], etc.As in the non super-situation, if the base field is of characteristic zero, then representation theory of a split quasireductive supergroup is essentially the same as the Lie superalgebra Lie( ) of .
In this paper, we are interested in modular representation theory of split quasireductive supergroup, that is, the case when the characteristic of the base field k is positive.As in the non super-situation, we can define Frobenius kernels r of a split quasireductive supergroup and these are also powerful tool for studying .
For example, using Frobenius kernels of Ä(m|n), Zubkov and Marko [ZM] provided the linkage principle and described blocks of Ä(m|n).In this paper, we give a necessary and sufficient condition for r to be unimodular in terms of the root system of .Thus, in contrast to the non super-situation, there exists a non unimodular r (see Example 4.17).
Recently, it is shown by several authors that Steinberg's tensor product theorem holds for Ä(m|n) [Ku], É(n) [BrKl] and ËÔÇ(m|n) [SW].See also, [CSW] for (simply connected) Chevalley supergroups of type D(2|1; ζ), G(3) and F (3|1).For a split quasireductive supergroup in general, it has been shown in [MS1] that there exists a category equivalence between the category of left -supermodules and the category of locally finite left hy( )-T -supermodules.Moreover, all simple left -supermodules have been systematically constructed in [Shi1].Therefore, it is natural to ask whether Steinberg's tensor product theorem holds for , in general.To answer this question, one encounters the following two difficulties: (1) not all simple left -supermodules are absolutely simple, that is, there exists a simple left -supermodule which is no longer simple after base change to some field extension of k (see Definition 5.3); (2) the root system of is ill-behaved (see Example 3.4(4) for example) without suitable extra conditions on the "odd part" of .
Note that, in the non super-situation, (1) never happens.
In this paper, we prove that these difficulties (1) and ( 2) can be overcome by attaching appropriate natural conditions.We first show that if the root system △ of does not contain the unit 0 of the character group X(T ) of a fixed split maximal torus of ev , then all simple left -supermodules are absolutely simple (Proposition 5.6).Therefore, to resolve (1), we assume that (1) ′ the base field k is algebraically closed if 0 ∈ △.To resolve (2), we also assume that (2) ′ the root system △ of has a special base (i.e., an existence of even/odd "simple roots" of △), see Definition 5.14 for the detail.We note that typical examples of split quasireductive supergroups (such as Ä(m|n), É(n), È(n) or Chevalley supergroups) satisfy the conditions both (1) ′ and (2) ′ .Under these natural assumptions (1) ′ and (2) ′ , we establish Steinberg's tensor product theorem for (Corollary 5.26); the result includes those by [BrKl, Ku, SW, CSW].
Organization of this paper.This paper is organized as follows: In Section 2, we review some basic definitions and results for Hopf superalgebras and algebraic supergroups defined over a field.The Lie superalgebra Lie( ) and the superhyperalgebra hy( ) of an algebraic supergroup are reviewed in Section 2.3.
In Section 3, we define the notion of split quasireductive supergroups which is the main object of study in this paper.Since the even ev part of a split quasireductive supergroup is a connected and split reductive group (scheme) by definition, we fix a split maximal torus T of ev .Thus, inside of the character group X(T ) of T , we can define the root system △ of with respect to T (Section 3.2) which also has a parity △ = △0 ∪ △1.Over a perfect field, we study structures of Frobenius kernels r of in Section 3.4.In particular, we describe a basis of the Hopf superalgebra O( r ) (see (3.2)) and establish the PBW theorem for the super-hyperalgebra hy( r ) of r (Theorem 3.11).
In Section 4, we discuss the unimodularity of Frobenius kernels of a split quasireductive supergroup over a perfect field.First, we review basic definitions and results for left/right (co)integrals of Hopf superalgebras (Section 4.1).A left (resp.right) integral for an algebraic supergroup is defined to be a left (resp.right) cointegral on the corresponding Hopf superalgebra O( ).We say that is unimodular if there exists a two-sided (i.e., left and right) integral for .In [MSS], it has been shown that has a left (resp.right) integral if and only if its even part ev does.Thus, by Sullivan's result [Su], if the characteristic of the base field is zero, then it follows that algebraic supergroup has a left (or right) integral if and only if is quasireductive.Over a field of characteristic zero, we give a necessary and sufficient condition for a split quasireductive supergroup to be unimodular in terms of its root system △ (Theorem 4.5 and Corollary 4.6).It is known that being unimodular is equivalent to that the distinguished group-like element is trivial (cf [Rad,Chapter 10]).In Section 4.3, we investigate properties of the distinguished group-like element of a finite normal super-subgroup of an algebraic supergroup, in general.In Section 4.4, we study unimodularity of Frobenius kernels r of a split quasireductive supergroup defined over a perfect field.Note that, r always has an integral, since r is finite (i.e., O( r ) is finite-dimensional).Using the result [ZM, Proposition 6.11] by Zubkov and Marko, we get an explicit description of the distinguished group-like element of r , and hence we give a necessary and sufficient condition for all r to be unimodular in terms of △ (Theorem 4.15 and Corollary 4.16).
In Section 5, we establish Steinberg's tensor product theorem for a split quasireductive supergroup under some natural assumptions.In Section 5.1, we review construction of simple -supermodules L(λ) (λ ∈ X(T ) ♭ ) given in [Shi1].In the super-situation, not all of simple -supermodules are absolutely simple (see Example 5.7).We show that if △ does not contain the unit 0 of X(T ), then all simplesupermodules are absolutely simple (Proposition 5.6).In Section 5.2, we construct simple r -supermodules L r (λ) (λ ∈ X(T )) and show that L r (λ) coincides with the r -top of the "highest weight module" M r (λ) of weight λ (Proposition 5.13).In Section 5.3, since the root system △ of is somewhat ill-behaved, we introduce the notion of a special base of △ (see Definition 5.14).We see that typical examples of split quasireductive supergroups have bases of its root systems (Example 5.15).The rest of Section 5.3, we assume that △ has a special base.The set of all p rrestricted weights for is denoted by X r (T ) ♭ , where p is the characteristic of the base field (see Definition 5.18).Then we show that the simple -supermodule L(λ) of highest weight λ ∈ X r (Ì) ♭ coincides with hy( r ) ⇀ L(λ) λ , where L(λ) λ is the λ-weight space of L(λ) (Lemma 5.20).Because of the existence of a non absolutely simple -supermodule, in Section 5.4, we assume that the base field is algebraically closed if 0 ∈ △.This assumption is essentially needed to prove Proposition 5.23 which states L(λ) is isomorphic to L r (λ) as r -supermodules (see Remark 5.24).Using Proposition 5.23, as in the non super-situation, we can establish Steinberg's tensor product theorem for (Theorem 5.25 and Corollary 5.26).
Acknowledgements.The author thanks the anonymous referees for their helpful comments that improved the quality of the manuscript.The author is supported by JSPS KAKENHI Grant Numbers JP19K14517 and JP22K13905.

Preliminaries
Throughout this paper, k denotes a fixed base field of characteristic different from 2. The unadorned ⊗ is the tensor product over k.In this section, we fix notations and collect some known results for Hopf superalgebras and supergroups.
2.1.Hopf superalgebras.Let Z 2 = { 0, 1} be the additive group of order two.The group algebra kZ 2 of Z 2 over k has a unique Hopf algebra structure and a right kZ 2 -comodule is naturally regarded as Z 2 -graded vector space.The category C of right kZ 2 -comodules forms a monoidal category by the tensor product ⊗ over k.Namely, the unit object is k = k⊕0 and (V ⊗W ) ǫ = a,b∈Z2,a+b=ǫ V a ⊗W b (ǫ ∈ Z 2 ) for right kZ 2 -comodules V and W .For a homogeneous element 0 = v ∈ V0 ∪ V1 of V ∈ C, we let |v| denote the degree of v, called the parity of v.We say that V is purely even if V = V0.For simplicity, when we use the symbol |v|, we always assume that v is homogeneous.The following supersymmetry ensures that the category C is symmetric: An object of C is called a superspace.
Example 2.1.For positive integers m and n, the set of all matrices of size m × n whose entries are in k is denoted by Mat m,n (k).Then we can regard Mat m|n (k) := Mat m+n,m+n (k) as a superspace by letting The usual matrix multiplication makes Mat m|n (k) into a superalgebra.For a finite dimensional superspace V , we can identify End k (V ) (resp.End k (V )) with Mat m|n (k)0 (resp.Mat m|n (k)), where m = dim(V0) and n = dim(V1).
Example 2.2.For a vector space V , the exterior algebra H = (V ) of V over k naturally becomes a commutative superalgebra.Moreover, H forms a Hopf superalgebra by letting: As in the non super-situation, if a Hopf superalgebra H is commutative or cocommutative, then the antipode S H : H → H of H satisfies S 2 H = id H .In particular, S H is bijective.
Definition 2.3.Let H be a Hopf superalgebra.A non-zero element g of H is called a group-like elements of H if it satisfies g ∈ H0 and ∆ H (g) = g ⊗ g.The set of all group-like elements of H is denoted by g.l.(H).
2.2.Algebraic supergroups.An affine supergroup scheme (supergroup, for short) over k is a representable functor from the category of commutative superalgebras to the category of groups.By Yoneda lemma, the representing object O( ) of forms a commutative Hopf superalgebra.A supergroup is said to be algebraic (resp.finite) if O( ) is finitely generated as a superalgebra (resp.finitedimensional).
For a supergroup , we define its even part ev as the restricted functor of from the category of commutative algebras to the category of groups.If we set A := O( ), then ev is an (ordinary) affine group scheme represented by the quotient Hopf algebra A := A/(A1), where (A1) is the super-ideal of A generated by the odd part A1 of A. We denote a the image of a ∈ A by the canonical quotient map A ։ A. If is algebraic, then so is ev .An algebraic supergroup is said to be connected if its even part is connected, see [Ma2,Definition 8].
Example 2.4.We list some basic example of algebraic supergroups.In the following, R denotes a commutative superalgebra.
(1) For positive integers m and n, we define the supergroup Ä(m|n), called the general linear supergroup, by , where GL m (resp.Mat m,n (R1)) denotes the general linear group scheme of size m (resp.the set of all m × n matrices whose entries are in R1).It is known that Ä(m|n) is algebraic and its even part Ä(m|n) ev is isomorphic to GL m × GL n , see [BrKj,MZ,Z1] for example.
The even part É(n) ev of É(n) is isomorphic to GL n .In [Br, BrKl], modular representation theory of this supergroup is well-studied.(3) Let (z) denote the exterior superalgebra of a one-dimensional vector space kz (see Example 2.2).The corresponding algebraic supergroup of (z) is denoted by G − a , called the one-dimensional odd unipotent supergroup, see [GZ,MZ2].By definition, we have G − a (R) = R1.Let be a supergroup with representing object O( ).By a left -supermodule we mean a right O( )-supercomodule.A homomorphism of left -supermodules is just a right O( )-supercomodule map.For left -supermodules V and W , we set Hom(V, A non-zero left -supermodule L is said to be simple if L has no non-trivial O( )-super-subcomodule.The parity change Π acts on the set of isomorphism classes of simple left -supermodules Simple( ) as a permutation of order two.We let Simple Π ( ) denote the set of Π-orbits in Simple( ).
(2) As a Lie super-subalgebra of gl(n|n), the Lie superalgebra of the queer supergroup É(n) is isomorphic to This Lie superalgebra q(n) is the so-called queer superalgebra.
For any positive integer n, we regard (O( )/m n ) * as a super-subspace of O( ) * through the dual of the canonical quotient map O( This hy( ) forms a super-subalgebra of O( ) * .We call it the super-hyperalgebra of (it is sometimes called the super-distribution algebra Dist( ) of ).By definition, we see that hy( ) = O( ) * if is finite.Since O( )/m n is finite-dimensional for any positive integer n, one sees that hy( ) has a structure of a cocommutative Hopf superalgebra such that the restriction is connected, then the pairing induces an injection O( ) ֒→ hy( ) * .In particular, the unit element of hy( ) is given by the restriction of the counit Masuoka showed the following ⊗-split type theorem for O( ) and hy( ), see [Ma1,Theorem 4.5] for detail (see also [Ma2,Proposition 22]).
Theorem 2.6.For an algebraic supergroup , there exists a counit (resp.unit) preserving isomorphism In the following, let S : hy( ) → hy( ) denote the antipode of hy( ) for simplicity.Note that, S is the restriction of the dual S * O( ) of the antipode called the super-bracket of hy( ).An element X ∈ hy( ) is said to be primitive if the comultiplication of X is given by X ⊗ 1 + 1 ⊗ X, where 1 denotes the unit element of hy( ).For primitive elements X, Y of hy( ), we have [X, Y ] = XY − (−1) |X||Y | Y X.If we regard Lie( ) as a super-subspace of hy( ), then this shows that Lie( ) coincides with the set of all primitive elements in hy( ).
For a left -supermodule V , we regard V as a left hy( )-supermodule by letting where u ∈ hy( ), v ∈ V .Suppose that V is finite-dimensional.Then the dual superspace V * of V forms a right O( )-supercomodule by using the antipode of O( ).The induced left hy( )-supermodule structure on V * satisfies the following equation. ( where v ∈ V , f ∈ V * and u ∈ hy( ).
and the corresponding Hopf superalgebra O(AE) is isomorphic to the quotient A/I for some Hopf super-ideal I of A. This AE is said to be normal if (as an abstract group) AE(R) is a normal subgroup of (R) for all commutative superalgebra R.

The condition is equivalent to saying that the canonical quotient map
Here, coad A denotes the (left) coadjoint coaction on A given by where S A is the antipode of A. By definition, the even part AE ev of a normal super- subgroup AE of is a normal subgroup of ev .
As the dual notion of (left) coadjoint coaction on A, we define called the (left) adjoint action on hy( ), where S is the antipode of hy( ).Note that, the super-bracket (2.1) can be rewritten as [u, w] = w (u ⊲ w (1) )S(w (2) ).
Let AE be a normal super-subgroup of .By definition, the left coadjoint coaction on O( ) induces a left O( )-supercomodule structure on m AE /m 2 AE , where m AE is the augmentation super-ideal of O(AE).Since m AE /m 2 AE is finite-dimensional, its linear dual Lie(AE) has a left -supermodule structure.Thus by (2.3), we get a left hy( )- We regard Lie(AE) as a super-subspace of hy( ) by the inclusion hy(AE) ⊂ hy( ).Then one sees that the action of hy( ) on Lie(AE) defined above is given by the adjoint action ⊲, see (2.4).In particular, by restricting the action of hy( ) to Lie( ), we see that for all X ∈ Lie( ) and N ∈ Lie(AE).
A Hopf super-subalgebra H of hy( ) is said to be normal (see [Ma1,Theorem is the right adjoint action of hy(AE ev ) on Lie( )1.
Since AE ev is a normal subgroup of ev , the condition (i) is clear by [T1,.By the construction, Lie(AE)1 is ev -stable, and hence the condition (ii) follows.Since Lie(AE) is a Lie super-ideal of Lie( ), the condition (iii) is trivial.Note that, in our case, the value of the counit ε(u) is zero unless u ∈ k1 = {c1 ∈ hy( ) | c ∈ k}.Thus, to show the condition (iv), it is enough to show that X ⊳ u ∈ Lie(AE)1 for all X ∈ Lie( )1 and u ∈ hy(AE ev ).Since hy(AE ev ) is cocommutative, we have S 2 = id and On the other hand, by the construction, Lie( )1 is ev -stable, and hence AE ev - stable.In particular, Lie( )1 is hy(AE ev )-stable under the adjoint action ⊲.Thus, the condition (iv) easily follows from the above formula.] in the variable X with coefficients in k, then it is easy to see that χ(id O( ) )(X) ∈ O( )0 is a group-like element.In this way, we have an isomorphism X( ) ∼ = g.l.O( ) of abstract groups.
For each χ ∈ X( ), we get the one-dimensional left -supermodule k χ so that k χ = k as a purely even superspace and the right O( )-supercomodule structure is given by In other words, g.v = χ(g)v for all commutative superalgebra R and g ∈ (R), v ∈ k χ .If there is no confusion, we sometimes simply denote k χ by χ.In this way, we get a one-to-one correspondence between X( ) ∼ = g.l.(O( )) and the set of all equivalence classes of one-dimensional (simple) left -supermodules under the parity change Π.
3. Split Quasireductive Supergroups 3.1.Split quasireductive supergroups.Recall that, a split and connected reductive Z-group G Z is a connected algebraic group (scheme) over Z having a split maximal torus T Z such that the pair (G Z , T Z ) corresponds to a root datum (cf.[SGA3]).See also [J,  ).An algebraic supergroup Z defined over Z is said to be split quasireductive if its even part of Z is a split and connected reductive group over Z and the odd part of m Z /m 2 Z is finitely generated and free as a Z-module.Here, m Z denotes the augmented ideal of O( Z ).
Note that in [Shi1,Shi2], a split quasireductive supergroup is simply called a quasireductive supergroup.
In the following, we fix a split quasireductive supergroup Z over Z and a split maximal torus T Z of ( Z ) ev .Let (resp.T ) denote the base change of Z (resp.T Z ) to our base field k, that is, O( ) := O( Z ) ⊗ Z k.By definition, is connected.We can identify X(T ) with Z ℓ (ℓ is the rank of ev ) and we often write its group low additively with unit element 0.
Example 3.2.We list some basic examples of split quasireductive supergroups.
st g 00 g 01 g 10 g 11 where t g 00 denotes the matrix transpose of g 00 and I n denotes the identity matrix of size n.One sees that È(n) ev ∼ = GL n .
As we have seen in Section 2.3, for a left -supermodule V , we get a left hy( )-supermodule structure on V .It is easy to see that V is locally finite and has a T -weight decomposition, and hence V becomes a locally finite left hy( )-T -supermodule.Here, we say that a left hy( )-supermodule V is left hy( )-Tsupermodule if the restricted hy(T )-supermodule structure on V arises from some T -supermodule structure on it.In this way, we get a functor from the category of left -supermodules to the category of locally finite left hy( )-T -supermodules.
Theorem 3.3 ([MS1, Theorem 5.8]).The functor discussed above gives an equivalence between the category of left -supermodules and the category of locally finite left hy( )-T -supermodules.
3.2.Root systems.Let g = g0 ⊕ g1 be the Lie superalgebra Lie( ) of .As we have seen in Section 2.4 (for AE = ), the left coadjoint coaction of O( ) induces the adjoint action of on g.Restricting the action to T , the Lie superalgebra g forms a left T -supermodule.Since T is a diagonalizable group scheme, g decomposes into weight superspaces as follows: , where g α denotes the α-weight super-subspace of g.By [J,Part I,7.14], we get where ⊲ is the adjoint action (2.4).Here, we regard X(T ) as a subset of O(T ).Let h := g 0 be the 0-weight super-subspace of g which forms a Lie super-subalgebra of g.Note that, the even part g0 of g coincides with the Lie algebra Lie( ev ).By definition, we see that h0 = Lie(T ).For ǫ ∈ Z 2 , we set We call △ the root system of with respect to T .Note that, △0 is the root system of G with respect to T in the usual sense.Moreover, the quadruple (X(T ), △0, X(T ) ∨ , △ ∨ 0 ) forms a root datum of the pair ( ev , T ), see [Mi, Appendix C].Let λ 1 , . . ., λ ℓ denote a basis of X(T ) ∼ = Z ℓ = ℓ i=1 Zλ i , where ℓ is the rank of ev .Example 3.4.Here we list some examples of root systems.
For each ǫ ∈ Z 2 , we set ℓ ǫ := dim(h ǫ ).Note that, ℓ0 coincides with the rank ℓ of ev .In [Shi1,Theorem 3.11], Poincaré-Birkhoff-Witt (PBW) theorem for hy( ) has been established.It states that we can take a homogeneous basis so that the set of all products of factors of the following type (taken in any fixed total order) forms a basis of hy( ): , 1 ≤ t ≤ ℓ1 and ǫ t , ǫ(γ, j) ∈ {0, 1}.See also Theorem 2.6.Here, we used the symbol of the "divided powers" X (n) α and H (m) i for X α and H i .For more detail, see [Shi1,§3.4].In the following, to simplify the notation, we write Y γ := Y (γ,1) if dim(g γ 1 ) = 1 for γ ∈ △1.One sees that hy( ) is a cocommutative supercoalgebra of Birkhoff-Witt type (for the non super-situation, see [T2,Section 3.3.5]).In particular, if we denote the comultiplication of hy( ) by ∆, then we have for n, m ∈ N ∪ {0} and α ∈ △0.Here, m+n n denotes the binomial coefficient.
3.3.Characters.It is known that ev is generated by the split maximal torus T and the α-root subgroups U α of ev for all α ∈ △0, see [Mi, Theorem 21.11] for example.Since each U α is isomorphic to the one-dimensional additive group (scheme) G a , we see that X(U α ) ∼ = g.l.(O(G a )) is trivial, and hence any character of ev is trivial on U α .In particular, the map X( ev ) → X(T ); χ → χ| T is injective.Remark 3.5.More precisely, it is known (see [J,Part II,1.18]) that gives an isomorphism, where α ∨ ∈ X(T ) ∨ = Hom(G m , T )) denotes the dual root corresponding to α and , denotes the perfect pairing X(T ) × X(T ) ∨ → Z.
Proof.By Lemma 2.9 and Remark 3.5, the claim follows immediately.
3.4.Frobenius kernels.In this subsection, we suppose that k is a perfect field of characteristic p > 2 and fix a positive integer r.Let be an algebraic supergroup over k, in general.For a commutative superalgebra R, we define a commutative superalgebra R (r) so that R (r) = R as a super-ring and the scalar multiplication is given by c.a = c p −r a for all c ∈ k and a ∈ R. We define a supergroup (r) so that (r) (R) := (R (−r) ), and define a morphism Fr r : → (r) of supergroups, called the r-th Frobenius morphism, as follows: The kernel of the morphism Fr r is called the r-th Frobenius kernel of which we denote by r .
Therefore, Lie( r ) = Lie( ) and r is infinitesimal, that is, r is finite and the augmentation super-ideal m r of O( r ) is nilpotent.In particular, r is a finite normal super-subgroup of , and hence hy( r ) = O( r ) * .Let V be a left ev -module.We regard V as a superspace by letting V0 = V and V1 = 0. Using the r-th Frobenius morphism Fr r : → (r) ev , we may consider V as a left -supermodule, which we denote by V [r] , in a natural way.As a right O( )-supercomodule, the structure map of V [r] is given by (1) .
Let M be a left -supermodule M such that r acts trivially on M .Then M naturally forms a left / r -supermodule (for quotient sheaves, see [MZ1]).Since O( / r ) isomorphic to O( ) p r := {a p r ∈ O( ) | a ∈ O( )}, the right O( / r )supercomodule structure map of M can be regarded as M → M ⊗ O( ) p r .Thus, we can define a left ev -supermodule (= right O( ev )-supercomodule) structure on M , which we denote by M [−r] , as follows: (1) .
Example 3.9.Let M be a left -supermodule.For the r -fixed point supersubspace M r of M , we can consider (M r ) [−r] .We naturally regard M as a left r -supermodule via the inclusion r ⊂ .For a finite dimensional leftsupermodule M ′ , we can make r Hom(M ′ , M ) into a left -supermodule by the conjugate action.As a left hy( )-supermodule, the induced action is given by where f ∈ r Hom(M ′ , M ), u ∈ hy( ) and v ∈ M ′ .Here, S denotes the antipode of hy( ).Since M r can be identified with r Hom(k, M ), we can also consider r Hom(M ′ , M ) [−r] .Note that, the "evaluation map" is a morphism of superspaces, since r Hom(M ′ , M ) = r Hom(M ′ , M )0 consists of parity preserving morphisms.Moreover, we get for each u ∈ hy( ), f ∈ r Hom(M ′ , M ) and v ∈ M ′ .This shows that ϕ is actually a -supermodule homomorphism.
In the following, we regard hy( r ) as a Hopf super-subalgebra of hy( ) via the inclusion r ⊂ .The following is a PBW type theorem for the r-th Frobenius kernel r of .
Theorem 3.11.For any total order on the homogeneous basis of g = g0 ⊕ g1, the set of all products of factors of type , 1 ≤ t ≤ ℓ1 and ǫ t , ǫ(γ, j) ∈ {0, 1}), taken in hy( ) with respect to the order, form a basis of hy( r ).
Proof.By Theorem 2.6 for hy( r ) and Proposition 3.8, we have an isomorphism hy( r ) ∼ = hy(( ev ) r ) ⊗ (g1) of (left hy(( ev ) r )-module) supercoalgebras.On the other hand, since ev is split reductive, the set of all products (taken in the fixed order) of factors of type form a basis of hy(( ev ) r ), see [J,Part II,Lemma 3.3].The proof is done.
In particular, it follows that hy( ) is generated by hy( ev ) and hy( r ) as a superalgebra.

Unimodularity of Algebraic Supergroups
In this section, we discuss the unimodularity of Frobenius kernels of split quasireductive supergroups.

(Co)integrals on Hopf superalgebras. Let H be a Hopf superalgebra with unit 1 H and counit
In other words, a left cointegral is an element in the space that is, there exists a non-zero two-sided (i.e, left and right) cointegral on H.
The space of all left (resp.right) integrals in H is denoted by I L H (resp.I R H ). In general, it is known that any finite dimensional Hopf algebra has both nonzero left/right integral.By this fact and the dual result of [MSS,Proposition 3.1], we have dim(  has a left (resp.right) integral if and only if ev does.
Assume for a moment that char(k) = 0. Let F be an algebraic group over k.Then by Sullivan's theorem ( [Su]), F has a left (or right) integral if and only if F is linearly reductive.In particular, in this case, F is automatically unimodular.However, in our super-situation, the existence of an integral does not imply its unimodularity (see Theorem 4.5 below).
By Theorem 4.4 (and Sullivan's theorem again), we note that for a connected and algebraic supergroup defined over a filed of characteristic zero, has a left (or right) integral if and only if is split quasireductive.
Proof.Let ad ′ : g0 → End(g1) be the restriction of the adjoint representation of g.
Then by [MSS,Proposition 3.16], we know that is unimodular if and only if the algebra map χ : U(g0) → k defined by the following is trivial: where the universal enveloping algebra U(g0) of g0.Since hy( ev ) = U(g0) and O( ev ) ⊂ hy( ev ) * (by the connectedness assumption on ev ), we may regard χ with a character of ev .Thus, we see that χ is trivial if and only if the restriction χ | T to the split maximal torus T is trivial by Lemma 3.6.Since the T -weight superspace decomposition of g1 is given as g1 = h1 ⊕ γ∈△1 g γ 1 with h1 = g 0 1 , we can compute for all t ∈ T (R), where R is a commutative algebra.Thus we are done.
Note that, in the above proof, the "order" can be found for such supergroups without assuming that the base field k is of characteristic zero.
Since AE is normal, the left adjoint action Ad of on AE makes B into a Hopf superalgebra object in the category of left A-supermodules.Explicitly, where S A is the antipode of A. Taking the linear dual, B * forms a Hopf superalgebra object in the category of right A-supermodules with the dual supercomodule structure map coad * B : B * → B * ⊗ A of coad B .Since B is finite-dimensional, the space I L B * of left integrals in B * is one dimensional, see Section 4.1.In the following, we take and fix a k-base φ of the space Note that, φ is homogeneous, that is, purely even or odd.
We fix f ∈ B * .Since B * is a Hopf superalgebra object in the category of right A-supermodules, we have where 1 R is the unit element of R. On the other hand, since φ is a left integral in B * , we have By definition, we get ε On the other hand, we calculate the action k ⇀ φ directly.If we identify g.l.(A) with X( ), then χ B ∈ g.l.(B) is identified with the restriction χ| AE ∈ X(AE).Using this, we can rephrase Proposition 4.9 as follows: Theorem 4.10.The restriction χ| AE is trivial if and only if AE is unimodular.In particular, AE is unimodular if χ is trivial.
Remark 4.11.In the non super-situation, Theorem 4.10 tells us that for a connected and split reductive group F , any finite and normal subgroup K of F is unimodular.In particular, all Frobenius kernels of F are unimodular.We give a proof of this fact.The adjoint action Ad : F → Aut(K); f → (k → f kf −1 ) factors through the quotient F/Z(F ), where Z(F ) is the center of F .Thus, the corre- O O Note that, we regard O(F/Z(F )) as a Hopf subalgebra of O(F ) via the canonical quotient F ։ F/Z(F ).Thus, the group-like element χ is in O(F/Z(F )).On the other hand, since F is connected and reductive, the quotient F/Z(F ) coincides with its derived group, see [Mi, Chapter 21] for example.Thus, there is no nontrivial group-like element in O(F/Z(F )), and hence χ must be trivial.Then by Theorem 4.10, K is unimodular.
However, in our super-situation, the proof in Remark 4.11 does not work.One of the reasons is that ( /Z( )) ev = ev /Z( ) ev (by Masuoka and Zubkov [MZ1]) is not isomorphic to ev /Z( ev ), in general.For example, if we take = Ä(m|n), 4.4.Unimodularity of Frobenius kernels.We suppose that the base field k is a perfect field of characteristic p > 2.
In [ZM, Corollary 7.2], it is proved that all Frobenius kernels of the general linear supergroup Ä(m|n) are unimodular.In this subsection, we give a necessary and sufficient condition for Frobenius kernels of a split quasireductive supergroup to be unimodular in terms of the root system of it.Let be a split quasireductive supergroup, and let r be a positive integer.Set g := Lie( ).Since the r-th Frobenius kernel r of is finite and normal, there uniquely exists χ r ∈ g.l.(O( )) ∼ = X( ) such that coad * O( r ) (φ r ) = φ r ⊗ χ r by Lemma 4.8.Here, φ r is a fixed non-zero left integral for r .As a super-analogue of [J,Part I,Proposition 9.7], Zubkov and Marko [ZM] explicitly determined the value of χ r as follows.
Here, the left adjoint action Ad(g) on g is regarded as an element of Mat dim(g0)|dim(g1) (R) with respect to the fixed basis given in (3.2).
Set T r := T ∩ ( ev ) r .Note that, T r is the r-th Frobenius kernel of T .The following is a version of Lemma 3.6: Lemma 4.13.The map X( r ) → X(T r ); χ → χ| Tr is injective.
Proof.For each α ∈ △0, let (U α ) r denote the r-th Frobenius kernel of the α-root subgroup U α of ev .Since U α ∼ = G a , one sees that the corresponding Hopf algebra of (U α ) r is isomorphic to the quotient k[X α ]/(X p r α ) of the polynomial algebra k[X α ].Thus, the character group of (U α ) r is trivial.Since ( ev ) r is generated by (U α ) r and T r , the map X(( ev ) r ) → X(T r ); χ → χ| Tr is injective.Then by Lemma 2.9 and Proposition 3.8, we are done.
Recall that X(T ) ∼ = Z ℓ = ℓ i=1 Zλ i .We shall write down the odd roots by the basis.For each γ ∈ △1, there uniquely exits n Using this notation, we have the following result: Proposition 4.14.The r-th Frobenius kernel r of is unimodular if and only if γ∈△1 dim(g γ 1 )n(γ) i ∈ p r Z for all 1 ≤ i ≤ ℓ.Proof.By Theorem 4.10, we have r is unimodular if and only if the restriction χ r | r is trivial.On the other hand, by Lemma 4.13, the restriction χ r | r is trivial if and only if χ r | Tr is trivial.
Let R be a commutative algebra.By the explicit description of χ r (Proposition 4.12), for each t ∈ T r (R) .
Recall that, the identification X(T ) ∼ = Z ℓ is induced from the fixed isomorphism Theorem 4.15.The following conditions are equivalent: (1) For all positive integer r, the r-th Frobenius kernel r of is unimodular.
Corollary 4.16.Let be one of Ä(m|n), É(n) or a Chevalley supergroup of classical type.For any positive integer r, the r-th Frobenius kernel r of is unimodular.
Proof.As in the proof of Corollary 4.6, for each , we can find a decomposition △1 = △ + 1 ⊔ △ − 1 satisfying the condition (4.1).Thus by Theorem 4.15, we are done.

Steinberg's Tensor Product Theorem
We fix a split quasireductive supergroup with a split maximal torus T of ev .Set g := Lie( ) and h := g 0 as before.In this section, we establish Steinberg's tensor product theorem for under natural assumptions.
5.1.Simple -supermodules.In [Shi1], we defined some special closed supersubgroups of and constructed all simple left -supermodules.In the following, we briefly review the construction.
Proof.For each λ ∈ X(T ), there exists n λ > 0 such that u(λ) ∼ = (k λ ) ⊕n λ as left T -modules.Since X(Ì) is identified with the set of all equivalence classes of onedimensional left Ì-supermodule under the parity change Π, we conclude that X(Ì) is naturally identified with X(T ).This proves the claim.
By counting multiplicity of simple super-submodules inside of V , the claim (1) easily follows.The claim (2) is just a consequence of (1).
In the non super-situation, it is known that all left simple ev -modules are absolutely simple, see [J,Part II,Corollary 2.9] (and [Mi,Section 22.4]).However, the following example shows that this phenomenon is no longer true for the supersituation when 0 ∈ △ (or equivalently, Ì = T ): Example 5.7.Suppose that our base field k satisfies −1 / ∈ (k × ) 2 , that is, k does not contain x such that x 2 = −1.Let be the queer supergroup É(2) over k.
In the following, we fix a positive integer r.As in Section 3.4, the r-th Frobenius kernel + r (resp.r ) of + (resp.) are infinitesimal and normal.
By definition, we get the short exact sequence 0 → p r X(T ) ֒→ X(T ) ։ X(T r ) → 0, where X(T ) → X(T r ) is the restriction map induced from T r ⊂ T .Thus, by Lemma 5.8, for each λ ∈ X(T r ), we can define a left r -supermodule structure on u(λ) in an obvious way.

2. 3 .
Lie superalgebras and super-hyperalgebras.Let be an algebraic supergroup.Set m := Ker(ε O( ) ), called the augmentation super-ideal of O( ), where ε O( ) : O( ) → k is the counit of O( ).Set Lie( ) := (m /m 2 ) * .This naturally forms a Lie superalgebra (see [MS1, Proposition 4.2]), which we call the Lie superalgebra of .Since is algebraic, Lie( ) is finite-dimensional.The even part Lie( )0 of Lie( ) can be identified with the (ordinary) Lie algebra Lie( ev ) of ev .Example 2.5.First, note that, Mat m|n (k) forms a Lie superalgebra with Lie super 2.5.Characters.Let G m := GL 1 denote the one dimensional multiplicative group (scheme).A character of a supergroup is a group homomorphism from to G m .The set of all characters X( ) := Hom( , G m ) of , called the character group of , naturally forms an abstract group.For χ ∈ X( ), we have a group homomorphism χ : (O( )) → G m (O( )), and hence we have a Hopf algebra homomorphism χ(id O( ) ) : O(G m ) → O( ) by the Yoneda lemma.If we realize O(G m ) as the Laurent polynomial algebra k[X ±1 Part II, Chapter 1] and [Mi, §5.2], for example.It is known that O(G Z ) is free as a Z-module and G Z is infinitesimally flat.Definition 3.1 ([Shi1, Definition 3.1]

HL
:= Hom H (H, k), where k is regarded as a trivial left H-supercomodule.The notion of a right cointegral on H and the symbol H R are defined analogously.Using the bosonization technique (see [MZ1, Section 10] for example), we have the following: Proposition 4.1 ([MSS, Corollary 3.2]).Both of dim( H R ) and dim( H L ) are less than or equal to 1, that is, a non-zero left or right cointegral on H is unique up to scalar multiplication if it exists.Moreover, such an element is homogeneous.
3.10]) if H is hy( )-stable under the adjoint action ⊲, that is, u ⊲ h ∈ H for all u ∈ hy( ) and h ∈ H. Proposition 2.8.If AE is a normal super-subgroup of , then hy(AE) ⊂ hy( ) is normal.In particular, hy(AE) is closed under super-bracket of hy( ), that is, [u, x] ∈ hy(AE) for all u ∈ hy( ) and x ∈ hy(AE). .By [Ma2, Proposition 5.5(2)], hy(AE) is normal if and only if the following four conditions are satisfied: [Rad, S H : H → H is the antipode of H.As in the non super-situation (see[Rad, Chapter 10]), one easily sees that the following holds: Proposition 4.3.There uniquely exists α H ∈ g.l.(H * ) such that th = α H , h t for all h ∈ H and t ∈ I L H .The element α H is the so-called distinguished group-like element for H. 4.2.Integrals for supergroups.Let be an algebraic supergroup, in general.We say that has a left (resp.right) integral for if there exists a non-zero left (resp.right) cointegral on O( ).Also, we say that is unimodular if O( ) is unimodular (see Definition 4.2).