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Enveloping Algebras of Poisson Thomas Lamkin

Advisor: Dr. Jason Gaddis Department of , Miami University

The Big Idea Lie-Rinehart superalgebras and the PBW theorem PBW theorem for Poisson superalgebras ev In order to prove the PBW theorem for Poisson superalgebras, we first prove a new version of the Let ΩA denote the even Kähler superdifferentials over A. For any Poisson A, the We construct and study the universal enveloping algebra of Poisson superalgebras, general- ev ev PBW theorem for Lie-Rinehart superalgebras. In this section alone we over a commutative pair (A, ΩA ) is a Lie-Rinehart superalgebra if ΩA is given the bracket izing the idea and several theorems from Poisson algebras. ring S rather than a field k. [adf, bdg] = (−1)|b||f|abd{f, g} + a{f, b}dg − (−1)|af||bg|b{g, a}df ev and the anchor map is ρ(df) = {f, ·}. There is a unique isomorphism Λ: U(A) → V (A, ΩA ) Background Definition between the two enveloping algebras, so the following PBW theorem is obtained. Poisson algebras arise in various fields of study including Hamiltonian and algebraic A Lie-Rinehart superalgebra is a pair (A, L), where A is a supercommutative superalgebra, geometry. We wish to study Poisson superalgebras, a generalization which has enjoyed increased L is a as well as an A-, together with a Lie superalgebra PBW Theorem attention due to the development of theories. and A-supermodule morphism ρ : L → Der(A), where Der(A) is the Lie superalgebra of superderivations of A, such that for x, y ∈ L, a ∈ A: Let A be a Poisson superalgebra, and consider the filtration defined on U(A) by Λ. Then there is an A-superalgebra isomorphism [x, ay] = (−1)|a||x|a[x, y] + ρ(x)(a)y. Definition ev ∼ SA(ΩA ) = gr U(A). A superalgebra is a Z2- algebra R = R0 ⊕ R1. An element x is even if x ∈ R0 and odd if x ∈ R1, and we write |x| for the parity of x; such even or odd elements are As with Poisson superalgebras, one can define the universal enveloping algebra of a Lie-Rinehart called homogeneous. A superalgebra is supercommutative if xy = (−1)|x||y|yx for any superalgebra. homogeneous x, y.A Poisson superalgebra is a supercommutative superalgebra R with Further Results bracket {·, ·} satsifying the following for all x, y, z ∈ R: Definition • If R is a Poisson Hopf superalgebra, then the enveloping algebra U(R) is also a Hopf 0 = {x, y} + (−1)|x||y|{y, x} superalgebra 0 = (−1)|x||z|{x, {y, z}} + (−1)|x||y|{y, {z, x}} + (−1)|y||z|{z, {x, y}} Let (A, L) be a Lie-Rinehart superalgebra (A, L). A triple (U, f, g) satisfies property R • If A = R[x] is a Poisson-Ore extension of a Poisson superalgebra R, then the enveloping |x||y| (with respect to (A, L)) if U is an algebra, f : R → U is an algebra homomorphism, and algebra U(A) can be expressed as an iterated Ore extension 0 = {x, yz} − (−1) y{x, z} − {x, y}z. g : L → U is a such that U(A) = U(R)[mx; σ1, η1][hx; σ2, η2] The first two relations make (R, {·, ·}) into a Lie superalgebra. g([x, y]) = g(x)g(y) − (−1)|x||y|g(y)g(x) for appropriate σi, ηi. f(x(a)) = g(x)f(a) − (−1)|a||x|f(a)g(x) We extend the idea of the universal enveloping algebra, first defined and constructed by Oh in g(ax) = f(a)g(x). Acknowledgements [Oh99], to Poisson superalgebras, and extend several well-known theorems to the super case. The universal enveloping algebra of a (A, L) is a triple (V (A, L), α, β) that is universal with respect to property R. Thomas Lamkin was funded for this project through the Undergraduate Summer Scholars pro- Definition gram at Miami University. There is a filtration on V (A, L) as follows: for p ≥ 0, let V denote the left A-subsupermodule Let R be a Poisson superalgebra. A triple (U, α, β) satisfies property P (with respect to R) p References of V (A, L) generated by products of at most p elements of L, and let V = 0. We denote the if U is an algebra, α : R → U is an algebra homomorphism, and β : R → U is a linear map −1 associated graded superalgebra with respect to this filtration by gr(V (A, L)). We remark the such that following PBW theorem is independent of the result in [Rin63]. [LWZ15] Jiafeng Lü, Xingting Wang, and Guangbin Zhuang. β({x, y}) = β(x)β(y) − (−1)|x||y|β(y)β(x) Universal enveloping algebras of Poisson Hopf algebras. |x||y| J. Algebra, 426:92–136, 2015. α({x, y}) = β(x)α(y) − (−1) α(y)β(x) PBW Theorem β(xy) = α(x)β(y) + (−1)|x||y|α(y)β(x). [Oh99] Sei-Qwon Oh. Poisson enveloping algebras. If A and L are free S-, then the canonical epimorphism S (L) → gr(V (A, L)) The universal enveloping algebra of R is a triple (U(R), α, β) that is universal with respect A Comm. Algebra, 27(5):2181–2186, 1999. to property P. is an isomorphism. [Rin63] George S. Rinehart. Differential forms on general commutative algebras. Trans. Amer. Math. Soc., 108:195–222, 1963.