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Universal Enveloping Algebras of Poisson Superalgebras

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Universal Enveloping Algebras of Poisson Superalgebras UNIVERSAL ENVELOPING ALGEBRAS OF POISSON SUPERALGEBRAS THOMAS LAMKIN Abstract. In this paper, we define and study the universal enveloping algebra of Poisson superalgebras. In particular, a new PBW theorem for Lie-Rinehart superalgebras is proved, leading to a PBW theorem for Poisson superalgebras; we show the universal enveloping algebra of a Poisson Hopf superalgebra (resp. Poisson-Ore extension) is a Hopf superalgebra (resp. iterated Ore extension); and we determine the universal enveloping algebra for examples such as quadratic polynomial Poisson superalgebras and Poisson symmetric superalgebras. 0. Introduction The notion of a Poisson algebra arises naturally in the study of Hamiltonian mechanics, and has since found use in areas such as Poisson or symplectic geometry, and quantum groups. Due to the development of supersymmetry theories, one has also witnessed the increased use of Poisson superalgebras, along with other super structures such as Lie superalgebras and supermanifolds. One method of studying the representation theory of Poisson algebras is to study its universal enveloping algebra, an idea introduced in [10] by Oh. Since then, universal enveloping algebras have been studied in a variety of contexts, such as for Poisson Hopf algebras [5, 11] and Poisson-Ore extensions [6], and partial PBW theorems have been obtained [3, 5, 13]. We extend several such results to the case of Poisson superalgebras, and obtain a full PBW theorem. Our approach to prove the PBW theorem uses the relation between Lie-Rinehart superalgebras and Poisson superalgebras. The former is a super generalization of Lie-Rinehart algebras which were first given comprehensive treatment by Rinehart [15], and can be viewed as an algebraic analogue of the more geometric notion of Lie algebroids. In his paper, Rinehart defined the universal enveloping algebra of a Lie-Rinehart arXiv:2102.06895v1 [math.RA] 13 Feb 2021 algebra (A, L) and proved that if L was projective as an A-module. In terms of Poisson algebras, this translates to requiring the K¨ahler differentials ΩA to be projective over the Poisson algebra A. While the PBW theorem was known to hold for other Poisson algebras, see e.g. [3], the general problem was still open. We remove this projectivity requirement by showing that if A and L are free (super)modules, the PBW theorem for Lie-Rinehart (super)algebras still holds. In particular, this holds if the base ring is a field, as is the case for Poisson superalgebras. This paper is organized as follows. 2010 Mathematics Subject Classification. 17B63, 17B60, 17B35, 16T05. Key words and phrases. Poisson superalgebra, enveloping algebra, Lie-Rinehart superalgebra, PBW Theorem. 1 In section 1, we recall the basic definitions in supermathematics. In section 2, we define and construct the universal enveloping algebra, as well as proving standard theorems such as the correspondence between Poisson modules over a Poisson superalgebra, and modules over its universal enveloping algebra. Section 3 is dedicated to proving a new PBW theorem for Lie-Rinehart superalgebras: Theorem 1 (Theorem 3.5). The canonical A-superalgebra morphism SA(L) → gr(V (A, L)) is an isomor- phism. In particular, this holds if the base ring of the Lie-Rinehart pair (A, L) is a field. In section 4, we follow [5] and use the results of section 3 to prove the PBW theorem for Poisson superalgebras. In section 5, we study the universal enveloping algebra of Poisson Hopf superalgebras. The following result is a super generalization of [11, Theorem 10]. Theorem 2 (Theorem 5.9). If (A, η, ∇, ε, ∆,S) is a Poisson Hopf superalgebra, then U(A) is a Hopf super- algebra. Section 6 shows that for a quadratic Poisson polynomial algebra, its universal enveloping algebra is a quadratic algebra and further, that its quadratic dual is the universal enveloping algebra of an exterior algebra with appropriate dual bracket. Finally, in section 7, we define Poisson-Ore extensions for Poisson superalgebras and show their universal enveloping algebras are iterated Ore extensions, generalizing [6, Theorem 0.1]. Acknowledgements. The author would like to thank Miami University and the USS program for funding this project. The author would also like to thank Jason Gaddis for suggesting the project idea, as well as for many helpful discussions and advice throughout. 1. Background on Supermathematics Throughout this paper, we work over a field k of characteristic 0, except in section 3 where we work over a commutative ring of characteristic 0. All (super)algebras, (super) tensor products, etc. are assumed to be taken over the base field (or ring) unless specified otherwise. We recall the basic definitions for supermathematics, referring the reader to [7] for further details. Definition 1.1. A superring is a Z2-graded ring R = R0 ⊕ R1 such that RiRj ⊆ Ri+J where the subscripts are taken mod 2. A superring is supercommutative if rs = (−1)|r||s|sr for all homogeneous r, s ∈ R. We shall assume all superrings are supercommutative. Further, any definition which applies to superrings applies to any ring S if we give S the trivial grading S = S ⊕ 0. 2 Definition 1.2. A (left) supermodule over a superring R = R0 ⊕ R1 is a Z2-graded (left) module M = M0 ⊕ M1 such that RiMj ⊆ Mi+j . Elements of M0 are called even, elements of M1 are called odd, and an element which is either even or odd is called homogeneous. We denote parity of a homogeneous element x by |x|. We will say a supermodule M = M0 ⊕M1 is a free supermodule if M has a basis consisting of homogeneous elements. For example, a super vector space is always a free supermodule over its base field. Definition 1.3. A morphism from an R-supermodule M to an R-supermodule N is an R-linear map ϕ : M → N. If ϕ preserves the grading, we say ϕ is an even linear map, whereas if ϕ reserves the grading, we say ϕ is an odd linear map. Definition 1.4. A superalgebra over a superring R = R0 ⊕R1 is a supermodule A = A0 ⊕A1 together with an associative multiplication that admits a unit and satisfies AiAj ⊆ Ai+j . A superalgebra is supercommutative if it is supercommutative as a superring; that is, ab = (−1)|a||b| for all a,b ∈ A. As superalgebras are superrings, we shall assume all superalgebras are supercommutative. Definition 1.5. If A and B are two superalgebras, then the super tensor product of A and B, denoted A ⊗ B, is the superalgebra with grading b (A ⊗ B)0 = (A0 ⊗ B0) ⊕ (A1 ⊗ B1) b (A ⊗ B)1 = (A0 ⊗ B1) ⊕ (A1 ⊗ B0) b with multiplication defined on homogeneous elements by (a ⊗ b)(c ⊗ d) = (−1)|b||c|(ac ⊗ bd). Remark 1.6. In this paper, we shall make a distinction between supermodules (resp. superalgebras) and modules (resp. algebras). The reason for this distinction is most readily seen in the different multiplication for the super tensor product than the normal tensor product. Therefore, we treat the superalgebra A and the underlying algebra A as (slightly) different entities. 2. The Universal Enveloping algebra of a Poisson Superalgebra In this section, we first recall the definition of a Poisson superalgebra and provide some examples. We then define and construct the universal enveloping algebra, prove standard theorems such as the correspondence of module categories, and provide an example computation for the universal enveloping algebra. 3 Definition 2.1. A Poisson superalgebra is a supercommutative superalgebra R with a bracket {·, ·} such that (R, {·, ·}) is a Lie superalgebra, and {·, ·} satisfies the Leibniz rule: {a,bc} = (−1)|a||b|b{a,c} + {a,b}c for all a,b,c ∈ R. That is, {x, ·} is a degree |x| superderivation for all x ∈ R. Example 2.2. Any associative superalgebra becomes a Poisson superalgebra via the supercommutator bracket. Also, any Poisson algebra can be considered a Poisson superalgebra with trivial odd part. Example 2.3. Let Pn = Λ(x1,...,xn,y1,...,yn) be an exterior algebra with grading |xi| = |yj | = 1 for all 1 ≤ i, j ≤ n. Then Pn becomes a Poisson superalgebra via the bracket {xi, xj } =0= {yi,yj }, {xi,yj} = δij where δij is the Kronecker delta. Definition 2.4. Let R be a Poisson superalgebra. A vector space M is a (left) Poisson R-module if there is an algebra homomorphism α : R → End M and a linear map β : R → End M such that β({x, y})= β(x)β(y) − (−1)|x||y|β(y)β(x) α({x, y})= β(x)α(y) − (−1)|x||y|α(y)β(x) β(xy)= α(x)β(y) + (−1)|x||y|α(y)β(x). In order to simplify the definition of Poisson modules, as well as the definition of the universal enveloping algebra, we introduce the following notation. Definition 2.5. Let R be a Poisson superalgebra. We say a triple (U,α,β) satisfies property P (with respect to R) if (1) U is an algebra, (2) α : R → U is an algebra homomorphism, and (3) β : R → U is a linear map such that β({x, y})= β(x)β(y) − (−1)|x||y|β(y)β(x) α({x, y})= β(x)α(y) − (−1)|x||y|α(y)β(x) β(xy)= α(x)β(y) + (−1)|x||y|α(y)β(x). In particular, a vector space M is a Poisson R-module if (End M,α,β) satisfies property P for some α, β. Note that since we do not assume the algebra U in a triple satisfying property P to be Z2-graded. As such, we cannot simplify the definition by saying β is a Lie superalgebra morphism. We are now ready to define and construct the universal enveloping algebra following [10].
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