Let α be a polynomial Poisson bivector on a finite-dimensional vector space V over C. Then Kontsevich [K97] gives a formula for a quantization f ⋆ g of the algebra S(V )∗. We give a construction of an algebra with the PBW property defined from α by generators and relations. Namely, we define an algebra as the quotient of the free tensor algebra T (V ∗) by relations xi ⊗ xj − xj ⊗ xi = Rij(~) wh...

We discuss relations of linear Nambu-Poisson structures to Filippov algebras and define a Filippov algebroid – a generalization of a Lie algebroid. We also prove results describing multiplicative Nambu-Poisson structures on Lie groups. In particular, it is shown that simple Lie groups do not admit multiplicative Nambu-Poisson structures of order > 2.

Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds for any complex affine Poisson algebra; that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is sho...

The main purpose of the paper is to study hyperkähler structures from the viewpoint of symplectic geometry. We introduce a notion of hypersymplectic structures which encompasses that of hyperkähler structures. Motivated by the work of Kronheimer on (co)adjoint orbits of semi-simple Lie algebras [10] [11], we define hyper-Lie Poisson structures associated with a compact semi-simple Lie algebra a...

In this paper we study the class of laterally complete commutative unital regular algebras A over arbitrary fields. We introduce a notion passport Γ(X) for faithful Amodules X, which consist uniquely defined partition unity in Boolean algebra all idempotents and set pairwise different cardinal numbers. prove that A-modules X Y are isomorphic if only = Γ(Y). Further Banach case C∞(Q) or C∞(Q)+i ...

The theory of the last multipliers as solutions of the Liouville’s transport equation, previously developed for vector fields, is extended here to general multivectors. Characterizations in terms of Witten and Marsden differentials are reobtained as well as the algebraic structure of the set of multivectors with a common last multiplier, namely Gerstenhaber algebra. Applications to Poisson bive...

We investigate how the matrix representation of SU(N) algebra approaches that of the Poisson algebra in the large N limit. In the adjoint representation, the (N2 − 1)× (N2 − 1) matrices of the SU(N) generators go to those of the Poisson algebra in the large N limit. However, it is not the case for the N × N matrices in the fundamental representation.

We introduce a dual notion of the Poisson algebra, called transposed by exchanging roles two binary operations in Leibniz rule defining algebra. The algebra shares common properties and arises naturally from Novikov-Poisson taking commutator Lie Novikov Consequently, classic construction commutative with commuting derivations similarly applies to More broadly, captures algebraic structures when...

We describe geometric non-commutative formal groups in terms of a geometric commutative formal group with a Poisson structure on its splay algebra. We describe certain natural properties of such Poisson structures and show that any such Poisson structure gives rise to a non-commutative formal group. We describe geometric non-commutative formal groups in terms of a geometric commutative formal g...