Seminar Sophus Lie 2 (1992) 3–9

Lie Superalgebras and Lie Supergroups, II

Helmut Boseck

6. The Hopf Dual.

Let H = H¯0 ⊕ H¯1 denote an affine Hopf superalgebra, i.e. a ℤ_{2}-graded commutative, finitely generated Hopf algebra. The set G(H) = hom_{alg}(H; K) of algebra homomorphisms mapping the superalgebra H onto the base field is a group with respect to convolution, called the structure group of H.

The convolution is defined by γ_{1} ∗ γ_{2} = (γ_{1} ⊗ γ_{2})Δ. It holds γ_{1} ∗ γ_{2} ∈ G(H) if γ_{1}, γ_{2} ∈ G(H), and γ ∗ ε = γ = ε ∗ γ, and γ ∗ (γσ) = ε = (γσ) ∗ γ.

By an easy calculation we have f.i. γ∗ε = (γ⊗ε)Δ = γ(id_{H}⊗ε)Δ = γid_{H} = γ, and (γσ) ∗ γ = ((γσ) ⊗ γ)Δ = (γ ⊗ γ)(σ ⊗ id_{H})Δ = γμ(σ ⊗ id_{H})Δ = γιε = ε.

Let H_{∘} = H/(H¯1) denote the quotient of H by the homogeneous ideal generated by H¯1, which is a coideal too: Δ(H) ⊆ H¯1 ⊗ H¯0 + H¯0 ⊗ H¯1, ε(H¯1) = 0, and σ(H¯1) ⊆ H¯1.

H_{∘} is an affine Hopf algebra and there is a canonical isomorphism between the structure groups of H and H_{∘}

G(H) ≅ G(H_{∘}).

Let Λ denote a Grassmann algebra and let G(H; Λ) = hom_{alg}(H; Λ) denote the set of algebra homomorphisms mapping H into Λ, then G(H; Λ) is a group with respect to convolution. The equation π_{Λ}(γ_{Λ}) = p_{0}γ_{Λ} with γ_{Λ} ∈ G(H; Λ) defines an epimorphism π_{Λ} mapping G(H; Λ) onto G(H). Moreover, by the inclusion K ⊆ Λ we have a canonical embedding ι_{Λ} of G(H) into G(H; Λ). The diagram

G(H) ⟶^{ι_{Λ}} G(H; Λ) ⟶^{π_{Λ}} G(H)

implies a semidirect product structure of G(H; Λ)

G(H; Λ) ≅ G(H) ⊙ K(H; Λ).

The normal subgroup K(H; Λ) = ker π_{Λ} is a unipotent group. In the representation by block matrices from Mat(m, n; Λ) the group K(H; Λ) consists of matrices of the form

( \begin{matrix}E_{m} & B_{1}(Λ) \\ B_{0}(Λ) & E_{n} \end{matrix} ) .

A differentiation λ : H ⟶ K of H is a linear mapping satisfying the equation

λ(XY) = λ(X)ε(Y) + ε(X)λ(Y), X, Y ∈ H.

Let λ_{1} and λ_{2} denote homogeneous differentiations of H. Then

[λ_{1}, λ_{2}] = λ_{1} ∗ λ_{2} − (−1)^{|λ_{1}||λ_{2}|}λ_{2} ∗ λ_{1}

is a homogeneous differentiation of H.

Let L¯0(H) and L¯1(H) denote the linear space of even and odd differentiations of H respectively.

The linear space L(H) = L¯0(H) ⊕ L¯1(H) is a Lie superalgebra with respect to the brackets defined above.

L(H ) is called the Liesuperalgebra of the Hopf superalgebra H.

Let λ_{Λ} : H ⟶ Λ denote an even linear mapping satisfying the equation λ_{Λ}(XY) = λ_{Λ}(X)ε(Y)+ε(X)λ_{Λ}(Y), X, Y ∈ H. We shall call it a Λ-differentiation of H. Then it holds

The linear space L(H; Λ) of Λ- differentiations of H is a Lie algebra with respect to the brackets [λ_{1Λ}, λ_{2Λ}] = λ_{1Λ} ∗ λ_{2Λ} − λ_{2Λ} ∗ λ_{1Λ}. The Lie algebra L(H; Λ) is isomorphic to the Grassmann-hull of the Lie superalgebra L(H).

The algebraic dual H′ = H′ ; ¯0 ⊕ H′ ; ¯1 of the Hopf superalgebra H is an associative superalgebra with unit. The product of H′ is the convolution, i.e. the restriction of the dual mapping Δ′ : (H ⊗ H)′ ⟶ H′ of the coproduct Δ to the subspace H′ ⊗ H′ of (H ⊗ H)′.

Let H^{∘} denote the subalgebra of H′ consisting of those linear functionals φ ∈ H′ which vanish on a homogeneous ideal of finite codimension in H. Then it is known, that the restriction of the dual mapping μ′ of the product map

μ of H to H^{∘} maps H^{∘} into H^{∘} ⊗ H^{∘} and hence defines a coproduct Δ^{∘} in H^{∘}. Moreover, the restriction of the dual map σ′ of the antipode σ of H to H^{∘} is an antipode σ^{∘} of H^{∘}. At last, the counit of H^{∘} is the evaluation of the linear functionals at the unit of H.

H^{∘} = H∘ ; ¯0 ⊕ H∘ ; ¯1 is a cocommutative Hopf superalgebra. It is called the Hopfdual of H.

The structure group G(H) consists of the group-like elements of H^{∘} : γ ∈ G(H) iff Δ^{∘}(γ) = γ ⊗ γ.

The structure group of H is the group of units in H^{∘}.

The Lie superalgebra L(H) consists of the primitive or Liealgebra-like elements of H^{∘} : λ ∈ L(H) iff Δ^{∘}(λ) = λ ⊗ ε + ε ⊗ λ.

It is easy to verify the following equations :

< Δ^{∘}(γ), X ⊗ Y > = < γ, XY >
= < γ, X >< γ, Y > = < γ ⊗ γ, X ⊗ Y >,
and < Δ^{∘}(λ), X ⊗ Y > = < λ, XY >
= < λ, X >< ε, Y > + < ε, X >< λ, Y >
= < λ ⊗ ε + ε ⊗ λ, X ⊗ Y >, X, Y ∈ H.

Structure Theorem of cocommutative Hopf superalgebras . (Sweedler,Kostant) Let H denote a cocommutative Hopf superalgebra, G its group of group-like elements and L its Lie superalgebra of primitive elements. Let K(G) denote the group algebra of G and U(L) the enveloping superalgebra of L. Then H is a smashed product of K(G) and U(L).

As a corollary from this theorem we mention

The Hopf dual H^{∘} of a Hopf superalgebra H is generated by its structure group G(H) and its Lie superalgebra L(H).

7. Affine Algebraic Supergroups.

Let G denote a group. A representative function on G is a K-valued function f on G with the property that span{f_{g}; g ∈ G} ( or span{_{g}f; g ∈ G} ) is finite dimensional. It is f_{g}(g′) = f(g′g) and _{g}f(g′) = f(g^{−1}g′).

Proposition.(Hochschild) The representative functions of G make up an affine Hopf algebra R_{∘}(G). The coproduct Δ_{∘} is defined by the equation f̃(g_{1}, g_{2}) = f(g_{1}g_{2}) using the isomorphism of R_{∘}(G × G) with R_{∘}(G) ⊗ R_{∘}(G). The counit ε_{∘} is the evaluation at the identity : ε_{∘}(f) = f(e) and the antipode σ_{∘} is defined by σ_{∘}(f)(g) = f(g^{−1}).

Definition.(Hochschild) The structure of an affine algebraic group is a pair (G, P_{∘}) consisting of a group G and a sub Hopf algebra P_{∘} of R_{∘}(G) satisfying the following properties

i.) P_{∘} separates the points of G ;

ii.) every algebra homomorphism γ_{∘} : P_{∘} ⟶ K is the evaluation at a group element.

The algebra P_{∘} is called the algebra of polynomial functions on G.

The properties i.) and ii.) imply a canonical isomorphism G ≅ G(P_{∘}) and L¯0 = L(P_{∘}) is the Lie algebra of the affine algebraic group (G, P_{∘}).

Proposition.(Hochschild) Let H_{∘} denote an affine Hopf algebra, then (G(H_{∘}), H_{∘}) is an affine algebraic group structure.

Definition. The structure of an affine algebraic supergroup is a pair (G, P) consisting of a group G and an affine Hopf superalgebra P satisfying the following property:.

There is a sub Hopf algebra P_{∘} of R_{∘}(G) and a finite dimensional vector space W such that it holds

i.) P ≅ P_{∘} ⊗ Λ(W) is an isomorphism of associative superalgebras; ii.) the canonical projection p : P ⟶ P_{∘} is a morphism of supercoalgebras

compatible with the antipodes; iii.) (G, P_{∘}) is the structure of an affine algebraic group. Property ii.) is equivalent to the equations : (p ⊗ p)Δ = Δ_{∘}p , ε =

ε_{∘}p, and pσ = σ_{∘}p. Property iii.) implies canonical isomorphisms G(P) ≅ G(P_{∘}) ≅ G. Every algebra homomorphism γ : P ⟶ K is of the following type: projection by p and evaluation at a group element.

P = P¯0⊕P¯1 is called the superalgebra of polynomial functions in commuting and anticommuting variables on G .

L(P) = L = L¯0 ⊕ L¯1 is called the Lie superalgebra of the affine algebraic supergroup (G, P) .

Evidently holds: dim L¯0 = dim(G, P_{∘}) = degree of transcendency of Q(P_{∘})/K; Q(P_{∘}) denotes the quotient field of P_{∘} provided the algebraic group structure (G, P_{∘}) is irreducible, i.e. P_{∘} is an integral domain; and dim L¯1 = dim W.

Proposition. Let H = H¯0 ⊕ H¯1 denote an affine Hopf superalgebra, then (G(H), H) is the structure of an affine algebraic supergroup.

If H ≅ H_{∘} ⊗ Λ(W), then it holds G(H) ≅ G(H_{∘}) canonically and (G(H), H_{∘}) is the structure of an affine algebraic group, the underlying algebraic group of the algebraic supergroup (G(H), H).

Assume L = L¯0 ⊕ L¯1 to be a finite dimensional Lie superalgebra. Denote by U(L) its enveloping algebra, U(L) is a cocommutative Hopf superalgebra: Δ(λ) = λ ⊗ ε + ε ⊗ λ ; λ ∈ L , ε denotes the unit element of U(L). Its Hopf dual U^{∘}(L) is a commutative Hopf superalgebra.

It holds the following isomorphism of associative, commutative superalgebras:

U^{∘}(L) ≅ U^{∘}(L¯0) ⊗ Λ(L′ ; ¯1).

Proposition. The following statements are equivalent

(i) U^{∘}(L) is an affine Hopf superalgebra;

(ii) [L¯0, L¯0] = L¯0; (iii) L(U^{∘}(L)) ≅ L. If one of the statements (i) - (iii) holds, then G̃ = G(U^{∘}(L)) ≅ G(U^{∘}(L¯0))

is the connectend and simply connected algebraic Lie group associated to L¯0 and (G̃, U^{∘}(L)) is the structure of an affine algebraic supergroup.

Corollary. Let L denote a finite dimensional Lie superalgebra with semisimple even part L¯0, and let G̃ denote the connected, simply connected, semisimple Lie group corresponding to L¯0, then (G̃, U^{∘}(L¯0)) is the structure of an affine algebraic supergroup.

8. Representations.

Let us start with a motivation.

Let L = L¯0 ⊕ L¯1 denote a Lie superalgebra, and let V = V¯0 ⊕ V¯1 denote a finite dimensional linear superspace. Assume that ρ : L ⟶ L_{L}(V) is a representation of L and U(ρ) : U(L) ⟶ L(V) its lift to the enveloping algebra U(L). Put δ(v)(u) = (−1)^{|u||v|}U(ρ)(u)v, with u ∈ U(L), v ∈ V and homogeneous. δ(v) is a V -valued representative function of L : δ(v) ∈ U^{∘}(L) ⊗ V . The map δ : V ⟶ U^{∘}(L) makes V an U^{∘}(L)-left supercomodule.

Every representation of the Lie superalgebra L defines an U^{∘}(L)-left supercomodule structure on the representation superspace V .

Definition. Let H = H¯0 ⊕ H¯1 denote a Hopf superalgebra. A H-left supercomodule is a linear superspace V = V¯0 ⊕ V¯1 endowed with an

even linear mapping δ : V ⟶ H ⊗ V satisfying the following equations:

(Δ ⊗ id_{V})δ = (id_{H} ⊗ δ)δ ; (ε ⊗ id_{V})δ = id_{V} .

Example 1 ^{⋄}. Let H = H¯0⊕H¯1 denote a Hopf superalgebra. Put V = H and δ = Δ.

H is a H-left (-right) supercomodul.

Example 2 ^{⋄}. Assume H = SP(m, n) ≅ K[X, X′]/(detX − detX′) ⊗ Λ(Y, Y′)

X = ( \begin{matrix}X & Y′ \\ Y & X′ \end{matrix} ) ; L = 𝔰𝔩(m, n).

Take V = K^{m,n} = K^{m,0}⊕K^{0,n}, and let e_{1}, ..., e_{m} and f_{1}, ..., f_{n} denote the canonical basis of K^{m,0} and K^{0,n} respectively. Write

E = ( e_{1} ; ⋮ ; e_{m} ; f_{1} ; ⋮ ; f_{n} ) and δ(E) = X ⊗ E.

In more detail

δ(e_{μ}) = ∑_{j=1}^{m} X_{μj} ⊗ e_{j} + ∑_{k=1}^{n} Y′_{μk} ⊗ f_{k},
δ(f_{ν}) = ∑_{j=1}^{m} Y_{νj} ⊗ e_{j} + ∑_{k=1}^{n} X′_{νk} ⊗ f_{k}.

It holds (Δ⊗id_{V})δ(E) = (Δ⊗id_{V})(X ⊗E) = ΔX ⊗E = (X ⊗X)⊗E = X ⊗ (X ⊗E) = (id_{H} ⊗δ)δ(E); (ε⊗id_{V})δ(E) = (ε⊗id_{V})(X ⊗E) = ε(X)⊗E = EE . K^{m,n} is a SP(m, n)-left supercomodule .

Let H = H¯0 ⊕ H¯1 denote a Hopf superalgebra. A block matrix X ∈ Mat(m, n; H) is called multiplicative, if it holds

Δ(X) = X ⊗ X .

Proposition.(Manin) The H-left supercomodules (K^{m,n}, δ) are in oneto-one correspondence to the multiplicative block matrices X ∈ Mat(m, n; H).

Given δ : K^{m,n} ⟶ H ⊗ K^{m,n} we may compute X from

δ(E) = X ⊗ E,

and given a multiplicative X ∈ Mat(m, n; H) , δ is defined by the same equation.

A linear mapping f : K^{m,n} ⟶ K^{m′,n′} is a morphism of the supercomodules (K^{m,n}, δ) and (K^{m′,n′}, δ′), if there is a block matrix F interchanging the corresponding multiplicative block matrices X and X′ : FX′ = X F and f(E) = FE′.

Let H = H¯0 ⊕ H¯1 denote a Hopf superalgebra and let (V, δ) denote a H-left supercomodule ( not necessarily finite dimensional). If φ ∈ H′ = H′ ; ¯0 ⊕ H′ ; ¯1 then the equation

ρ′(φ) = (φ ⊗ id_{V})δ

defines a representation ρ′ of the associative superalgebra H′ over V .

By suitable restrictions of ρ′ we get representations of the superalgebra H^{∘}, of the structure group G(H), and of the Lie superalgebra L(H):

ρ^{∘}(φ) = (φ ⊗ id_{V})δ , φ ∈ H^{∘};
ρ_{G}(γ) = (γ ⊗ id_{V})δ , γ ∈ G(H);
ρ_{L}(λ) = (λ ⊗ id_{V})δ , λ ∈ L(H).

Let (G, P) denote an affine algebraic supergroup and assume V = P as P -left supercomodule (P, Δ). The corresponding representation ρ_{L} of the Lie superalgebra L = L(P) is called the left regular representation of L.

Proposition. The left regular representation of the Lie superalgebra L = L(P) of the affine algebraic supergroup (G, P) is an isomorphism of L onto the Lie superalgebra Der^{(r)}P of right invariant derivations on P : L(P) ≅ Der^{(r)}P.

Let us start with an affine algebraic supergroup (G, P). If (K^{m,n}, δ) is a P - left supercomodule , and X denotes the corresponding multiplicative block matrix, then we may get matrix representations of the group G and its Lie superalgebra L = L(P) as well, in the following way:

ρ_{G}(γ) = γ(X) = ( \begin{matrix}γ(X) & 0 \\ 0 & γ(X′) \end{matrix} );
ρ_{L}(λ) = Λ(X) = ( \begin{matrix}λ(X) & λ(Y′) \\ λ(Y) & Λ(X′) \end{matrix} ) .

We may also get representations of the group G_{Λ} = G(P; Λ) and of the Lie algebra L_{Λ} = L(P; Λ) by Λ matrices :

ρ_{G}(γ_{Λ}) = γ_{Λ}(X) = ( \begin{matrix}γ_{Λ}(X) & γ_{Λ}(Y′) \\ γ_{Λ}(Y) & γ_{Λ}(X′) \end{matrix} );
ρ_{L}(λ_{Λ}) = λ_{Λ}(X) = ( \begin{matrix}λ_{Λ}(X) & λ_{Λ}(Y′) \\ λ_{Λ}(Y) & λ_{Λ}(X′) \end{matrix} ) .

References

[1]  Abe, E., “Hopf algebras,” Cambridge University Press 1977.

[2] Berezin, F. A., “Introduction to algebra and analysis with anticommuting variables,” (Russian) Moscow University 1983.

[3] Boseck H., On representative functions of Lie superalgebras, Math.Nachr. 123 (1985), 61–72; Correction to my paper: ‘On representative functions...’, Math. Nachr. 130 (1987), 137–138.

[4]  —, Affine Lie supergroups, Math. Nachr. 143 (1987), 303–327.
[5]  —, Classical Lie supergroups, Math. Nachr. 148 (1990), 81–115.

[6] —, Lie superalgebras and Lie supergroups, I Seminar Sophus Lie (Heldermann Verlag Berlin), 2 (1992), 109–122.

[7] Corwin, L., Y. Ne’eman, and S. Sternberg, Graded Lie algebras in Mathematics and Physics (Bose-Fermi-symmetry), Rev. of Mod. Physics 47 (1975), 57–63.

[8] Heynemann, R. G., and M. E. Sweedler Affine Hopf algebras I , J. of Algebra 13 (1969), 192–241.

[9] Hochschild, G., Algebraic groups and Hopfalgebras, Illinois J. Math. 14 (1970), 52–65.

[10] —, “Introduction to Affine Algebraic Groups,” Holden Day Inc., San Francisco etc., 1971.

[11] —, “Basic Theory of Algebraic Groups and Lie Algebras,” Graduate Texts in Math. 75, Springer Verlag Berlin etc., 1981.

[12] Kac, V. G., Lie superalgebras, Adv. in Math. 26 (1977), 8–96.

[13] Kostant, B., Graded manifolds, graded Lie theory, and prequantization, Lect. Notes in Math. 570 (1977), 177–306.

[14] Manin, Y., “Quantum Groups and Noncommutative Geometry,” CRM Université de Montreal 1988.

[15] Milnor J., and J. Moore, On the structure of Hopf algebras, Ann. of Math. 81 (1965), 211–264.

[16] Scheunert M., “The Theory of Lie Superalgebras,” Lect. Notes in Math. 716, Springer Verlag, Berlin etc., 1979.

[17] Sweedler M. E. ”Hopf Algebras”, Benjamin Inc., New York, 1969.

Fachrichtungen Mathematik/Informatik Ernst Moritz Arndt Universität Greifswald Friedrich Ludwig Jahn-Straße 15a O-2200 Greifswald, Germany

Received March 31, 1992