Narrow Results

We construct an action of the braid group B_N on the twisted quantized enveloping algebra where the elements of B_N act as automorphisms. In the classical limit q → 1, we recover the action of B_N on the polynomial functions on the space of upper triangular matrices with ones on the diagonal. The action preserves the Poisson bracket on the space of polynomials which was introduced by Nelson and Regge in their study of quantum gravity and rediscovered in the mathematical literature. Furthermore, we construct a Poisson bracket on the space of polynomials associated with another twisted quantized enveloping algebra . We use the Casimir elements of both twisted quantized enveloping algebras to reproduce and construct some well-known and new polynomial invariants of the corresponding Poisson algebras.

An admissible Poisson algebra (or briefly, an adm-Poisson algebra) gives an equivalent presentation with only one operation for a Poisson algebra. We establish a bialgebra theory for adm-Poisson algebras independently and systematically, including but beyond the corresponding results on Poisson bialgebras given in [27]. Explicitly, we introduce the notion of adm-Poisson bialgebras which are equivalent to Manin triples of adm-Poisson algebras as well as Poisson bialgebras. The direct correspondence between adm-Poisson bialgebras with one comultiplication and Poisson bialgebras with one cocommutative and one anti-cocommutative comultiplications generalizes and illustrates the polarization–depolarization process in the context of bialgebras. The study of a special class of adm-Poisson bialgebras which include the known coboundary Poisson bialgebras in [27] as a proper subclass in general, illustrating an advantage in terms of the presentation with one operation, leads to the introduction of adm-Poisson Yang–Baxter equation in an adm-Poisson algebra. It is an unexpected consequence that both the adm-Poisson Yang–Baxter equation and the associative Yang–Baxter equation have the same form and thus it motivates and simplifies the involved study from the study of the associative Yang–Baxter equation, which is another advantage in terms of the presentation with one operation. A skew-symmetric solution of adm-Poisson Yang–Baxter equation gives an adm-Poisson bialgebra. Finally, the notions of an ${𝒪}$-operator of an adm-Poisson algebra and a pre-adm-Poisson algebra are introduced to construct skew-symmetric solutions of adm-Poisson Yang–Baxter equation and hence adm-Poisson bialgebras. Note that a pre-adm-Poisson algebra gives an equivalent presentation for a pre-Poisson algebra introduced by Aguiar.

Lie conformal algebras are useful tools for studying vertex operator algebras and their representations. In this paper, we establish close relations between Poisson conformal algebras and representations of Lie conformal algebras. We also calculate explicitly Poisson conformal brackets on the associated graded conformal algebras of universal associative conformal envelopes of the Virasoro conformal algebra and the Neveu–Schwartz conformal superalgebra.

A base of the Malcev Poisson superalgebra related with the free Malcev superalgebra on one odd generator is constructed. As a corollary, a pre-base, that is, a set of elements that spans the free alternative superalgebra on one odd generator is obtained.

Let k be a field of characteristic 0, A a noncommutative Poisson k-algebra, U(A) the ordinary enveloping algebra of A, 𝒞 a quasi-Poisson A-coring that is projective as a left A-module, *𝒞 the left dual ring of 𝒞 (it is a right U(A)-module algebra) and Λ a right quasi-Poisson 𝒞-comodule that is finitely generated as a right U(A)#*𝒞-module. The vector space End^𝒫,𝒞(Λ) of right quasi-Poisson 𝒞-colinear maps from Λ to Λ is a ring. We give necessary and sufficient conditions for projectivity and flatness of a module over End^𝒫,𝒞(Λ). If 𝒞 contains a fixed quasi-Poisson grouplike element, we can replace Λ with A.

We introduce quasi-Poisson cohomology groups for a Poisson algebra, which can be computed by its quasi-Poisson complex. Moreover, there exists a Grothendieck spectral sequence relating quasi-Poisson cohomology to Hochschild cohomology and Lie algebra cohomology.

We define a notion of associative representation for algebras. We prove the existence of faithful associative representations for any alternative, Mal’cev, and Poisson algebra, and prove analogs of Ado-Iwasawa theorem for each of these cases. We construct also an explicit associative representation of the Cayley–Dickson algebra in the matrix algebra $M_8(F).$

We study the relation between Poisson algebras and representations of Lie conformal algebras. We establish a Gröbner–Shirshov basis theory framework for modules over associative conformal algebras and apply this technique to Poisson algebras considered as conformal modules over appropriate associative conformal envelopes of current Lie conformal algebras. As a result, we obtain a setting for the calculation of a Gröbner–Shirshov basis in a Poisson algebra.

We provide a table of multiplication of the Poisson algebra on the minimal set of generators of the invariants of pairs of matrices of degree three.

A Gelfand–Dorfman algebra is called special if it can be embedded into a differential Poisson algebra. We find a new basis of the free Novikov algebra. With its help, we construct the monomial basis of the free special Gelfand–Dorfman algebra.

The purpose of this paper is to study multi-derivations of Poisson algebras. First, we show that the space of multi-derivations of a Lie algebra $𝔤$ with the Nijenhuis–Richardson bracket is a differential graded Lie algebra. Then we introduce the notion of a multi-derivation of a Poisson algebra, and show that the space of multi-derivations of a Poisson algebra with the Nijenhuis–Richardson bracket is also a graded Lie algebra. Finally, we study multi-derivations of three concrete Poisson algebras coming from linear Poisson manifolds, symplectic manifolds and Poisson manifolds with vanishing first cohomology groups. We establish the relationship between multi-derivations of a Lie algebra $𝔤$ and multi-derivations of the linear Poisson manifold ${𝔤}^{\ast }$, and show that a two-derivation of a connected symplectic manifold $(M,\omega )$ is $\lambda \pi$, where $\pi$ is the Poisson bivector field induced by the symplectic structure $\omega$ and $\lambda$ is a real number. We also prove that a two-derivation of a connected Poisson manifold $(M,\pi )$ with trivial Poisson cohomology group $H^1(M,\pi )$ is $f\pi$, where $f$ is a Casimir function.

Quantum hydrodynamics described by Madelung equations is analyzed in the framework of symplectic geometry i.e. in covariant phase space approach to geometric field theory. The pre-symplectic manifolds providing the phase spaces describing the Hamiltonian dynamics of quantum fluid are constructed from the set of all solutions of Madelung equations and their corresponding Lagrangian densities. The Madelung equations under consideration are the Madelung equations associated to Schroedinger equations (in the nonrelativistic case) and Madelung equations associated to Klein–Gordon equations (in the relativistic case). The cases where the coupling with electromagnetic fields is present are also considered here. Our symplectic formulation is different from that of [M. Spera, Moment map and gauge geometric aspects of the Schrodinger and Pauli equations, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1–36] in the choice of fundamental fields or variables. Here we regard density function $\rho$ and phase function $S$ not as canonical pair but as the fundamental fields of the theory. The Hamiltonian vector fields corresponding to an observable are obtained from the Hamiltonian equation generated by the observable. The Poisson bracket of two observables then is determined by the Hamiltonian vector fields associated to each observable. In general, the Poisson bracket of two observables is not unique due to the fact that every observable has more than one corresponding Hamiltonian vector field. It is pointed out that the Poisson bracket has a unique value over a certain subset of the set of all observables defined on the pre-symplectic manifold of the Madelung equation under consideration.

Non-commutative Poisson algebras are the algebras having both an associative algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures on are determined.

This paper surveys the theory of functional identities and its applications. No prior knowledge of the theory is required to follow the paper.

A compatible algebra products of a finite-dimensional semisimple Lie algebra 𝖌 with a Lie bracket [-,-] are studied. If 𝖌 is simple of type A₁ or not of type A_n of n ≧ 2, then the compatible algebra products must be the scalar multiples of the Lie bracket [-,-]. In case that 𝖌 is simple of type A_n of n ≧ 2, such a product is a sum of a scalar multiple of [-,-] and a deformed one of the ordinal associative products on the full (n + 1) × (n + 1) matix algebra. Then we give a alternative proof to the triviality of the compatible associative algebra structures of a semisimple Lie algebra 𝖌.