Graded Poisson Algebras

Abstract

1. Definitions 1.1. Graded vector spaces. By a Z-graded vector space (or simply, graded vector space) we mean a direct sum A = ⊕ i∈Z A i of vector spaces over a field k of characteristic zero. The A i are called the components of A of degree i and the degree of a homogeneous element a ∈ A is denoted by |a|. We also denote by A[n] the graded vector space with degree shifted by n, namely,The tensor product of two graded vector spaces A and B is again a graded vector space whose degree r component is given byThe symmetric and exterior algebra of a graded vector space A are defined respectively as S(A) = T (A)/I S and (A) = T (A)/I ∧ , where T (A) = ⊕ n≥0 A ⊗n is the tensor algebra of A and I S (resp. I ∧ ) is the two-sided ideal generated by elements of the form a ⊗ b − (−1) |a| |b| b ⊗ a (resp. a ⊗ b + (−1) |a| |b| b ⊗ a), with a and b homogeneous elements of A. The images of A ⊗n in S(A) and (A) are denoted by S n (A) and n (A) respectively. Notice that there is a canonical decalage isomorphism1.2. Graded algebras and graded Lie algebras. We say that A is a graded algebra (of degree zero) if A is a graded vector space endowed with a degree zero bilinear associative product · : A⊗A → A. A graded algebra is graded commutative if the product satisfies the conditionfor any two homogeneous elements a, b ∈ A of degree |a| and |b| respectively. A graded Lie algebra of degree n is a graded vector space A endowed with a graded Lie bracket on A[n]. Such a bracket can be seen as a degree −n Lie bracket on A, i.e., as bilinear operation {·, ·} : A ⊗ A → A[−n] satisfying graded antisymmetry and graded Jacobi relations: {a, b} = −(−1) (|a|+n)(|b|+n) {b, a} {a, {b, c}} = {{a, b}, c} + (−1) (|a|+n)(|b|+n) {a{b, c}}

Secondary Products in Supersymmetric Field Theory

“…18 One succinct way to define the Ω-background, at least formally, comes from considering our 3d N = 4 theory as a 1d N = 4 theory. 19 From the 1d point of view, the U(1) symmetry generated by X is just a global symmetry, and the Ω-background can be viewed as a complex twisted-mass deformation associated to this global symmetry. In particular, even after the deformation, we still have a conventional 1d N = 4 theory.…”

The structures on the universal enveloping algebras of differential graded Poisson Hopf algebras

“…for any homogeneous elements a, b, c ∈ A, then A is called a graded Poisson algebra [3]. If in addition, there is a k-linear homogeneous map d : A → A of degree 1 such that d 2 = 0 and…”

“…Finally, it only remains to show that dS = S d. Note that dS (a) =dS (a (1) ε(a (2) )) = d(S (a (1) )ε(a (2) )) =d(S (a (1) )a (2) S (a (3) )) = dS (a (1) )a (2) S (a (3) ) + (−1) |a (1) | S (a (1) )d(a (2) S (a (3) )) =dS (a (1) )a (2) S (a (3) ) + (−1) |a (1) | S (a (1) )(d(a (2) )S (a (3) ) + (−1) |a (2) | a (2) dS (a (3) )) =dS (a) + (−1) |a (1) | S (a (1) )d(a (2) )S (a (3) ) + (−1) |a (1) a (2) | dS (a), we get dS (a) = −(−1) |a (2) | S (a (1) )d(a (2) )S (a (3) ).…”

DG Poisson algebra and its universal enveloping algebra

“…In this section, examples of DG Poisson algebras are provided arising from Lie theory, differential geometry, homological algebra and deformation theory. See more examples in [5,9]. 5.1.…”

“…The Poisson bracket was originally introduced by French Mathematician Siméon Denis Poisson in search for integrals of motion in Hamiltonian mechanics. For Poisson algebras, many important generalizations have been obtained in both commutative and noncommutative settings recently: Poisson orders [1], graded Poisson algebras [6,9], noncommutative Leibniz-Poisson algebras [7], left-right noncommutative Poisson algebras [8], Poisson PI algebras [19], double Poisson algebras [26], noncommutative Poisson algebras [27], Novikov-Poisson algebras [28] and Quiver Poisson algebras [30]. One of the interesting aspects of Poisson algebras is the notion of Poisson universal enveloping algebra, which was first introduced by Oh in order to describe the category of Poisson modules [22].…”