Poisson Algebra Research Articles

The second cohomology spaces of 𝒦(1) with coefficients in the superspace of weighted densities and deformations of the superspace of symbols on S 1|1

Abstract We explicitly describe the second cohomology of the Lie superalgebra 𝒦 ⁢ ( 1 ) {\mathcal{K}(1)} of contact vector fields on the supercircle S 1 | 1 {S^{1|1}} with coefficients in the spaces of weighted densities. We deduce the second cohomology of 𝒦 ⁢ ( 1 ) {\mathcal{K}(1)} with coefficients in the Poisson algebra of pseudodifferential symbols on S 1 | 1 {S^{1|1}} . We study formal deformations of the standard embedding of 𝒦 ⁢ ( 1 ) {\mathcal{K}(1)} into the Poisson superalgebra of pseudodifferential symbols.

The coalgebra symmetry and the superintegrable discrete-time systems

In this paper, we classify all the variational discrete-time systems in quasi-standard form in N degrees of freedom admitting coalgebra symmetry with respect to the generic realisation of the Lie–Poisson algebra . This approach naturally yields several quasi-maximally and maximally superintegrable discrete-time systems, both known and new. We conjecture that this exhausts the (super)integrable cases associated with this algebraic construction.

Finite generation of Lie algebras of Poisson brackets

Let L be a finite dimensional Lie algebra. Let U(L) and Pol(L) be the universal enveloping algebra of L and the related Poisson algebra. We find conditions for finite generation of Lie algebras [U(L),U(L)] and [Pol(L), Pol(L)].

Transposed Poisson structures on Witt type algebras

We describe 12-derivations, and hence transposed Poisson algebra structures, on Witt type Lie algebras V(f), where f:Γ→C is non-trivial and f(0)=0. More precisely, if |f(Γ)|≥4, then all the transposed Poisson algebra structures on V(f) are mutations of the group algebra structure (V(f),⋅) on V(f). If |f(Γ)|=3, then we obtain the direct sum of 3 subspaces of V(f), corresponding to cosets of Γ0 in Γ, with multiplications being different mutations of ⋅. The case |f(Γ)|=2 is more complicated, but also deals with certain mutations of ⋅. As a consequence, new Lie algebras that admit non-trivial Hom-Lie algebra structures are found.

Local observables in SUq(2) lattice gauge theory

We consider a deformation of 3D lattice gauge theory in the canonical picture, first classically, based on the Heisenberg double of $\mathrm{SU}(2)$, then at the quantum level. We show that classical spinors can be used to define a fundamental set of local observables. They are invariant quantities that live on the vertices of the lattice and are labeled by pairs of incident edges. Any function on the classical phase space, e.g., Wilson loops, can be rewritten in terms of these observables. At the quantum level, we show that spinors become spinor operators. The quantization of the local observables then requires the use of the quantum $\mathcal{R}$ matrix, which we prove to be equivalent to a specific parallel transport around the vertex. We provide the algebra of the local observables, as a Poisson algebra classically, then as a $q$ deformation of ${\mathfrak{so}}^{*}(2n)$ at the quantum level. This formalism can be relevant to any theory relying on lattice gauge theory techniques such as topological models, loop quantum gravity or of course lattice gauge theory itself.

A POISSON ALGEBRA STRUCTURE OVER THE EXTERIOR ALGEBRA OF A QUADRATIC SPACE

We construct a Poisson algebra structure of degree $-2$ over the exterior algebra of a quadratic space. Here we do not use Clifford algebra as in [4].

Symmetry-induced decoherence-free subspaces

Preservation of coherence is a fundamental yet subtle phenomenon in open systems. We uncover its relation to symmetries respected by the system Hamiltonian and its coupling to the environment. We discriminate between local and global classes of decoherence-free subspaces for many-body systems through the introduction of "ghost variables". The latter are orthogonal to the symmetry and the coupling to the environment does not depend on them. Constructing them is facilitated in classical phase space and can be transferred to quantum mechanics through the equivalent role that Poisson and Lie algebras play for symmetries in classical and quantum mechanics, respectively. Examples are given for an interacting spin system.

The algebraic and geometric classification of transposed Poisson algebras

The algebraic and geometric classification of all complex 3-dimensional transposed Poisson algebras is obtained. Also we discuss special 3-dimensional transposed Poisson algebras.

Geometry of Tangent Poisson–Lie Groups

Let G be a Poisson–Lie group equipped with a left invariant contravariant pseudo-Riemannian metric. There are many ways to lift the Poisson structure on G to the tangent bundle TG of G. In this paper, we induce a left invariant contravariant pseudo-Riemannian metric on the tangent bundle TG, and we express in different cases the contravariant Levi-Civita connection and curvature of TG in terms of the contravariant Levi-Civita connection and the curvature of G. We prove that the space of differential forms Ω*(G) on G is a differential graded Poisson algebra if, and only if, Ω*(TG) is a differential graded Poisson algebra. Moreover, we show that G is a pseudo-Riemannian Poisson–Lie group if, and only if, the Sanchez de Alvarez tangent Poisson–Lie group TG is also a pseudo-Riemannian Poisson–Lie group. Finally, some examples of pseudo-Riemannian tangent Poisson–Lie groups are given.

Циклические фробениусовы алгебры

In this paper, we introduce the notion of cyclic Frobenius algebras (CF-algebras). Canonical structures of CF-algebras exist on associative and Poisson algebras. It turns out that the modern theory of integrable systems yields non-trivial examples of CF-algebras. In the theory of the KdV hierarchy there is a structure of CF-algebra which leads to explicit expressions for the first integrals of the N-th Novikov hierarchy, suitable for classical, non-Abelian and quantum cases.

Center of Poisson and Skein algebras associated to loops on surfaces

Center of Poisson and Skein algebras associated to loops on surfaces

On the Kantor product, II

We describe the Kantor square (and Kantor product) of multiplications, extending the classification proposed in [J. Algebra Appl. 2017, 16 (9), 1750167]. Besides, we explicitly describe the Kantor square of some low dimensional algebras and give constructive methods for obtaining new transposed Poisson algebras and Poisson-Novikov algebras; and for classifying Poisson structures and commutative post-Lie structures on a given algebra.

Scalar Field Cosmology from a Modified Poisson Algebra

We investigate the phase space of a scalar field theory obtained by minisuperspace deformation. We consider quintessence or phantom scalar fields in the action that arises from minisuperspace deformation on the Einstein–Hilbert action. We use a modified Poisson algebra where Poisson brackets are the α-deformed ones and are related to the Moyal–Weyl star product. We discuss early- and late-time attractors and reconstruct the cosmological evolution. We show that the model can have the ΛCDM model as a future attractor if we initially consider a massless scalar field without a cosmological constant term.

Hom–Jordan–Malcev–Poisson algebras

UDC 512.5 We provide and study a Hom-type generalization of Jordan–Malcev–Poisson algebras called Hom–Jordan–Malcev–Poisson algebras. We show that they are closed under twisting by suitable self-maps and give a characterization of admissible Hom–Jordan–Malcev–Poisson algebras. In addition, we introduce the notion of pseudo-Euclidian Hom–Jordan–Malcev–Poisson algebras and describe its T * -extension. Finally, we generalize the notion of Lie–Jordan–Poisson triple system to the Hom setting and establish its relationships with Hom–Jordan–Malcev–Poisson algebras.

Scale for codimension growth of Poisson PI-algebras

A. Regev proved that the codimension growth of an associative PI-algebra is at most exponential. The author established a scale for the codimension growth of Lie PI-algebras, which includes a series of functions between exponential and factorial functions. We prove that the same scale stratifies the ordinary codimension growth of Poisson PI-algebras satisfying Lie identical relations. As a byproduct, we obtain a new bound on the codimension growth of Lie PI-algebras in terms of complexity functions. We also study very fast codimension growth of some Poisson algebras without Lie identities. We essentially use techniques of exponential generating functions and growth of respective fast growing entire functions.

The algebraic classification and degenerations of nilpotent Poisson algebras

We generalize the Skjelbred–Sund method, used to classify nilpotent low-dimensional Lie algebras, in order to classify Poisson algebras with non-trivial annihilator. We develop this method with the purpose of classifying nilpotent Poisson algebras, obtaining from it the algebraic classification of the nilpotent Poisson algebras up to dimension four. Additionally, we obtain the geometric classification of the variety of nilpotent Poisson algebras up to dimension four, by adapting some notions and results used for varieties of algebras with a single multiplication.

Star product approach for loop quantum cosmology

Guided by recent developments toward the implementation of the deformation quantization program within the loop quantum cosmology (LQC) formalism, in this paper we address the introduction of both the integral and differential representation of the star product for LQC. To this end, we consider the Weyl quantization map for cylindrical functions defined on the Bohr compactification of the reals. The integral representation contains all of the common properties that characterize a star product which, in the case under study here, stands for a deformation of the usual pointwise product of cylindrical functions. Our construction also admits a direct comparison with the integral representation of the Moyal product which may be reproduced from our formulation by judiciously substituting the appropriate characters that identify such representation. Further, we introduce a suitable star commutator that correctly reproduces both the quantum representation of the holonomy-flux algebra for LQC and, in the proper limit, the holonomy-flux classical Poisson algebra emerging in the cosmological setup. Finally, we propose a natural way to obtain the quantum dynamical evolution in LQC in terms of this star commutator for cylindrical functions as well as a differential representation of the star product using discrete finite differences. We expect that our findings may contribute to a better understanding of certain issues arising within the LQC program, in particular, those related to the semiclassical limit and the dynamical evolution of quantum states.

Dimensioned Algebra: Mathematics with Physical Quantities

A rigorous mathematical theory of dimensional analysis, systematically accounting for the use of physical quantities in science and engineering, perhaps surprisingly, was not developed until relatively recently. We claim that this has shaped current mathematical models of theoretical physics, which generally lack any explicit reference to units of measurement, and we propose a novel mathematical framework to alleviate this. Our proposal is a generalization of the usual categories of algebraic structures used to formulate physical theories (groups, rings, vector spaces...), herein dubbed dimensioned, that can naturally articulate the structure of physical dimension. Our goal in the present work is not so much to define an algebraic theory of physical quantities—this has already been done—but to define a theory of algebra informed by how physical quantities are used in practice. We conclude by studying the dimensioned analogue of Poisson algebras in some detail due to their relevance in Jacobi geometry and classical mechanics.

Li filtrations of SUSY vertex algebras

Any vertex algebra has a canonical decreasing filtration, called the Li filtration, whose associated graded space has a natural structure of a vertex Poisson algebra. In this note, we introduce an analogous filtration for any SUSY vertex algebra, which was introduced by Heluani and Kac as a superfield formalism for supersymmetric vertex algebras. We prove that the associated graded superspace of our filtration has a structure of a SUSY vertex Poisson algebra. We also introduce and discuss related notions, such as Zhu’s \(C_2\)-Poisson superalgebras, associated superschemes, and singular supports, for SUSY vertex algebras.

Double Multiplicative Poisson Vertex Algebras

Abstract We develop the theory of double multiplicative Poisson vertex algebras. These structures, defined at the level of associative algebras, are shown to be such that they induce a classical structure of multiplicative Poisson vertex algebra on the corresponding representation spaces. Moreover, we prove that they are in one-to-one correspondence with local lattice double Poisson algebras, a new important class among Van den Bergh’s double Poisson algebras. We derive several classification results, and we exhibit their relation to non-abelian integrable differential-difference equations. A rigorous definition of double multiplicative Poisson vertex algebras in the non-local and rational cases is also provided.