scholarly journals Admissible Poisson bialgebras

2021 ◽  
Author(s):  
Jinting Liang ◽  
Jiefeng Liu ◽  
Chengming Bai
Keyword(s):  
Special Class ◽  
Symmetric Solution ◽  
Poisson Algebra ◽  
Baxter Equation ◽  
Poisson Algebras ◽  
Proper Subclass ◽  
Unexpected Consequence ◽  
Manin Triples ◽  
Direct Correspondence

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 [Formula: see text]-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.

scholarly journals Anti-flexible bialgebras

2021 ◽  
pp. 2250212
Author(s):  
Mafoya Landry Dassoundo ◽  
Chengming Bai ◽  
Mahouton Norbert Hounkonnou
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Symmetric Solution ◽  
Baxter Equation ◽  
Manin Triple ◽  
Unexpected Consequence ◽  
Special Case

We establish a bialgebra theory for anti-flexible algebras in this paper. We introduce the notion of an anti-flexible bialgebra which is equivalent to a Manin triple of anti-flexible algebras. The study of a special case of anti-flexible bialgebras leads to the introduction of anti-flexible Yang–Baxter equation in an anti-flexible algebra which is an analogue of the classical Yang–Baxter equation in a Lie algebra or the associative Yang–Baxter equation in an associative algebra. It is unexpected consequence that both the anti-flexible Yang–Baxter equation and the associative Yang–Baxter equation have the same form. A skew-symmetric solution of anti-flexible Yang–Baxter equation gives an anti-flexible bialgebra. Finally the notions of an [Formula: see text]-operator of an anti-flexible algebra and a pre-anti-flexible algebra are introduced to construct skew-symmetric solutions of anti-flexible Yang–Baxter equation.

scholarly journals Quantization of a Poisson algebra and polynomials associated to links

1990 ◽  
Vol 120 ◽  
pp. 113-127 ◽  
Author(s):  
Tetsuya Ozawa
Keyword(s):  
Riemann Surface ◽  
Lie Algebra ◽  
Poisson Algebra ◽  
Homotopy Classes ◽  
Poisson Algebras

A formal quantization of Poisson algebras was discussed by several authors (see for instance Drinfel’d [D]). A formal Lie algebra generated by homotopy classes of loops on a Riemann surface ∑ was obtained by W. Goldman in [G], and its Poisson algebra was quantized, in the sense of Drinfel’d, by Turaev in [T].

scholarly journals LEFT-SYMMETRIC BIALGEBRAS AND AN ANALOGUE OF THE CLASSICAL YANG–BAXTER EQUATION

2008 ◽  
Vol 10 (02) ◽  
pp. 221-260 ◽  
Author(s):  
CHENGMING BAI
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symmetric Solution ◽  
Baxter Equation ◽  
Symmetric Part ◽  
Lie Bialgebra ◽  
Lie Bialgebras ◽  
Poisson Lie Groups ◽  
Lagrangian Subalgebras

We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakähler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call "S-equation", which is an analogue of the classical Yang–Baxter equation. In a certain sense, the S-equation associated to a left-symmetric algebra reveals the left-symmetry of the products. We show that a symmetric solution of the S-equation gives a parakähler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called "[Formula: see text]-operator", whereas a skew-symmetric solution of the classical Yang–Baxter equation corresponds to the skew-symmetric part of an [Formula: see text]-operator. Thus a method to construct symmetric solutions of the S-equation (hence parakähler Lie algebras) from [Formula: see text]-operators is provided. Moreover, by comparing left-symmetric bialgebras and Lie bialgebras, we observe that there is a clear analogue between them and, in particular, parakähler Lie groups correspond to Poisson–Lie groups in this sense.

scholarly journals Universal enveloping Hom-algebras of regular Hom-Poisson algebras

2022 ◽  
Vol 7 (4) ◽  
pp. 5712-5727
Author(s):  
Xianguo Hu ◽  
Keyword(s):  
Poisson Algebra ◽  
Poisson Algebras

<abstract><p>In this paper, we introduce universal enveloping Hom-algebras of Hom-Poisson algebras. Some properties of universal enveloping Hom-algebras of regular Hom-Poisson algebras are discussed. Furthermore, in the involutive case, it is proved that the category of involutive Hom-Poisson modules over an involutive Hom-Poisson algebra $ A $ is equivalent to the category of involutive Hom-associative modules over its universal enveloping Hom-algebra $ U_{eh}(A) $.</p></abstract>

scholarly journals On (co)homology of Frobenius Poisson algebras

2014 ◽  
Vol 14 (2) ◽  
pp. 371-386 ◽  
Author(s):  
Can Zhu ◽  
Fred Van Oystaeyen ◽  
Yinhuo Zhang
Keyword(s):  
Bilinear Form ◽  
Hochschild Cohomology ◽  
Cohomology Ring ◽  
Poisson Algebra ◽  
Hochschild Homology ◽  
Poisson Algebras ◽  
Poisson Homology ◽  
Poisson Cohomology ◽  
Frobenius Algebras

AbstractIn this paper, we study Poisson (co)homology of a Frobenius Poisson algebra. More precisely, we show that there exists a duality between Poisson homology and Poisson cohomology of Frobenius Poisson algebras, similar to that between Hochschild homology and Hochschild cohomology of Frobenius algebras. Then we use the non-degenerate bilinear form on a unimodular Frobenius Poisson algebra to construct a Batalin-Vilkovisky structure on the Poisson cohomology ring making it into a Batalin-Vilkovisky algebra.

scholarly journals Mutation-periodic quivers, integrable maps and associated Poisson algebras

2011 ◽  
Vol 369 (1939) ◽  
pp. 1264-1279 ◽  
Author(s):  
Allan P. Fordy
Keyword(s):  
Poisson Bracket ◽  
Canonical Transformation ◽  
Hamiltonian Structure ◽  
Poisson Algebra ◽  
Complete Integrability ◽  
Canonical Coordinates ◽  
Poisson Algebras ◽  
Invariant Functions ◽  
Liouville’S Equation

We consider a class of map, recently derived in the context of cluster mutation. In this paper, we start with a brief review of the quiver context, but then move onto a discussion of a related Poisson bracket, along with the Poisson algebra of a special family of functions associated with these maps. A bi-Hamiltonian structure is derived and used to construct a sequence of Poisson-commuting functions and hence show complete integrability. Canonical coordinates are derived, with the map now being a canonical transformation with a sequence of commuting invariant functions. Compatibility of a pair of these functions gives rise to Liouville’s equation and the map plays the role of a Bäcklund transformation.

scholarly journals Left-right noncommutative Poisson algebras

2014 ◽  
Vol 12 (1) ◽  
Author(s):  
José Casas ◽  
Tamar Datuashvili ◽  
Manuel Ladra
Keyword(s):  
Poisson Algebra ◽  
Cohomological Dimension ◽  
Poisson Algebras ◽  
Special Cases ◽  
Crossed Modules ◽  
Noncommutative Analogue

AbstractThe notions of left-right noncommutative Poisson algebra (NPlr-algebra) and left-right algebra with bracket AWBlr are introduced. These algebras are special cases of NLP-algebras and algebras with bracket AWB, respectively, studied earlier. An NPlr-algebra is a noncommutative analogue of the classical Poisson algebra. Properties of these new algebras are studied. In the categories AWBlr and NPlr-algebras the notions of actions, representations, centers, actors and crossed modules are described as special cases of the corresponding wellknown notions in categories of groups with operations. The cohomologies of NPlr-algebras and AWBlr (resp. of NPr-algebras and AWBr) are defined and the relations between them and the Hochschild, Quillen and Leibniz cohomologies are detected. The cases P is a free AWBr, the Hochschild or/and Leibniz cohomological dimension of P is ≤ n are considered separately, exhibiting interesting possibilities of representations of the new cohomologies by the well-known ones and relations between the corresponding cohomological dimensions.

scholarly journals The generalized Weyl Poisson algebras and their Poisson simplicity criterion

2019 ◽  
Vol 110 (1) ◽  
pp. 105-119
Author(s):  
V. V. Bavula
Keyword(s):  
Large Class ◽  
Poisson Structure ◽  
Poisson Algebra ◽  
Explicit Description ◽  
Poisson Algebras ◽  
Weyl Algebras ◽  
Generalized Weyl Algebras

Abstract A new large class of Poisson algebras, the class of generalized Weyl Poisson algebras, is introduced. It can be seen as Poisson algebra analogue of generalized Weyl algebras or as giving a Poisson structure to (certain) generalized Weyl algebras. A Poisson simplicity criterion is given for generalized Weyl Poisson algebras, and an explicit description of the Poisson centre is obtained. Many examples are considered (e.g. the classical polynomial Poisson algebra in 2n variables is a generalized Weyl Poisson algebra).

scholarly journals On extension of classical Baer results to Poisson algebras

2021 ◽  
Vol 31 (1) ◽  
pp. 84-108
Author(s):  
L. A. Kurdachenko ◽  
◽  
A. A. Pypka ◽  
I. Ya. Subbotin ◽  
◽  
...  
Keyword(s):  
Poisson Algebra ◽  
Finite Codimension ◽  
Poisson Algebras ◽  
Abelian Algebra ◽  
Specific Field ◽  
Zero Divisors ◽  
Finite Dimensional

In this paper we prove that if P is a Poisson algebra and the n-th hypercenter (center) of P has a finite codimension, then P includes a finite-dimensional ideal K such that P/K is nilpotent (abelian). As a corollary, we show that if the nth hypercenter of a Poisson algebra P (over some specific field) has a finite codimension and P does not contain zero divisors, then P is an abelian algebra.

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