Jacobi and Poisson algebras

Abstract

Abstract. Jacobi/Poisson algebras are algebraic counterparts of Jacobi/Poisson manifolds. We introduce representations of a Jacobi algebra A and Frobenius Jacobi algebras as symmetric objects in the category. A characterization theorem for Frobenius Jacobi algebras is given in terms of integrals on Jacobi algebras. For a vector space V a non-abelian cohomological type object J H 2 (V, A) is constructed: it classifies all Jacobi algebras containing A as a subalgebra of codimension equal to dim(V ). Representations of A are used in order to give the decomposition of J H 2 (V, A) as a coproduct over all Jacobi A-module structures on V . The bicrossed product P ⊲⊳ Q of two Poisson algebras recently introduced by Ni and Bai appears as a special case of our construction. A new type of deformations of a given Poisson algebra Q is introduced and a cohomological type object HA 2 P, Q | (⊳, ⊲, ↼, ⇀) is explicitly constructed as a classifying set for the bicrossed descent problem for extensions of Poisson algebras. Several examples and applications are provided.

Hochschild products and global non-abelian cohomology for algebras. Applications

“…The formula for the comultiplication of any such coalgebra (C * ⋆ k n ) * can be written down effectively by using (41) and (3). This observation shows that the GE-problem applied for finite dimensional algebras and finite dimensional vector spaces gives the answer at the level of finite dimensional coalgebras to what we have called the extending structures problem studied in [2,4,5] for Jacobi, Lie and respectively associative algebras.…”

“…We will prove now the assertion from step (2). Assume that ϕ : A (λ,Λ,ϑ) → A (λ ′ ,u ′ ) is an algebra map.…”

“…Thus, the results obtained in the previous section can be applied for the classification problem of supersolvable coalgebras of a given dimension. 2 Using the above facts, Corollary 2.18 classifies in fact all 3-dimensional supersolvable coalgebras: the isomorphism classes are the duals A * of the algebras listed in the statement. For example, the dual coalgebra associated to the noncommutative algebra k < x, y | x 2 = 1, y 2 = 0, xy = −yx = y > is the non-cocommutative supersolvable coalgebra having {f 1 , f 2 , f 3 } as a basis and the comultiplication and the counit given by:…”

“…The difficulty of the problem consists in explicitly computing H 2 nab (A, B): lacking an efficient cohomology tool as in the abelian case [35,38], it needs to be computed 'case by case' using different computational and combinatorial approaches. This paper deals, at the level of associative algebras, with a generalization of the extension problem, called the global extension (GE) problem, introduced recently in [3,33] for Poisson/Leibniz algebras as a categorical dual of the extending structures problem [2,4,5]. The GE-problem can be formulated for any category C using a simple idea: in the classical extension problem we drop the hypothesis 'B is a fixed object in C' and replace it by a weaker one, namely 'B has a fixed dimension'.…”

“…The equivalence classes of n(n+1) 2 + 1 -dimensional algebras that have an algebra projection on T n (k) are the families of algebras having {f, e ij | i, j = 1, 2, · · · , n, i ≤ j} as a basis over k and the multiplication given below (we only write down the non-zero products): Supersolvable coalgebras. We recall that a coalgebra C = (C, ∆, ε) is a vector space C equipped with a comultiplication ∆ : C → C ⊗ C and a counit ǫ : C → k such that (∆ ⊗ Id) • ∆ = (Id ⊗ ∆) • ∆ and (Id ⊗ ε) • ∆ = (ε ⊗ Id) • ∆ = Id, where ⊗ = ⊗ k and Id is the identity map on C. We use the Σ-notation for comultiplication: ∆(c) = c (1) ⊗ c (2) , for all c ∈ C (summation understood). The base field k, with the obvious structures, is the final object in the category of coalgebras.…”