scholarly journals Deformations of Courant pairs and Poisson algebras

2020 ◽  
pp. 2150229
Author(s):  
Ashis Mandal ◽  
Satyendra Kumar Mishra
Keyword(s):  
Lie Algebra ◽  
Commutative Algebra ◽  
Poisson Algebra ◽  
Three Dimensional ◽  
Versal Deformation ◽  
Infinitesimal Deformation ◽  
Poisson Algebras ◽  
Cohomology Spaces ◽  
Algebra Base

We study the deformation of Courant pairs with a commutative algebra base. We consider the deformation cohomology bi-complex and describe a universal infinitesimal deformation. In a sequel, we formulate an extension of a given deformation of a Courant pair to another with extended base, which leads to describe the obstruction in extending a given deformation. We also discuss the construction of versal deformation of Courant pairs. As an application, we compute universal infinitesimal deformation of Poisson algebra structures on the three-dimensional complex Heisenberg Lie algebra. We compare the second deformation cohomology spaces of these Poisson algebra structures by considering them in the category of Leibniz pairs and Courant pairs, respectively.

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].

Non-commutative Poisson Algebra Structures on the Lie Algebra $so_n\widetilde{({\Bbb C}_Q)}$

2007 ◽  
Vol 14 (03) ◽  
pp. 521-536 ◽  
Author(s):  
Jie Tong ◽  
Quanqin Jin
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Poisson Algebra ◽  
Algebra Structure ◽  
Poisson Algebras

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 [Formula: see text] are determined.

scholarly journals GROUP PROPERTIES OF A ONE-DIMENSIONAL NONLINEAR POROELASTICITY EQUATION

2019 ◽  
Vol 4 (1) ◽  
pp. 149-155
Author(s):  
Kholmatzhon Imomnazarov ◽  
Ravshanbek Yusupov ◽  
Ilham Iskandarov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Group Classification ◽  
Second Order ◽  
Partial Differential ◽  
One Dimensional ◽  
Preliminary Group

This paper studies a class of partial differential equations of second order , with arbitrary functions and , with the help of the group classification. The main Lie algebra of infinitely infinitesimal symmetries is three-dimensional. We use the method of preliminary group classification for obtaining the classifications of these equations for a one-dimensional extension of the main Lie algebra.

scholarly journals On Theory of logarithmic Poisson Cohomology

2017 ◽  
Vol 9 (4) ◽  
pp. 209
Author(s):  
Joseph Dongho ◽  
Alphonse Mbah ◽  
Shuntah Roland Yotcha
Keyword(s):  
Lie Algebra ◽  
Poisson Structure ◽  
Commutative Algebra ◽  
Linear Representation ◽  
Poisson Cohomology ◽  
Associative Commutative Algebra

We define the notion of logarithmic Poisson structure along a non zero ideal $\cali$ of an associative, commutative algebra $\cal A$ and prove that each logarithmic Poisson structure induce a skew symmetric 2-form and a Lie-Rinehart structure on the $\cal A$-module $\Omega_K(\log \cali)$ of logarithmic K\"{a}hler differential. This Lie-Rinehart structure define a representation of the underline Lie algebra. Applying the machinery of Chevaley-Eilenberg and Palais, we define the notion of logarithmic Poisson cohomology which is a measure obstructions of Linear representation of the underline Lie algebra for which the grown ring act by multiplication.

scholarly journals Three-dimensional higher-order Schrödinger algebras and Lie algebra expansions

2020 ◽  
Vol 2020 (4) ◽  
Author(s):  
Oguzhan Kasikci ◽  
Nese Ozdemir ◽  
Mehmet Ozkan ◽  
Utku Zorba
Keyword(s):  
Lie Algebra ◽  
Three Dimensional ◽  
Higher Order

A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. II. Full Optimal Subalgebraic Systems

1993 ◽  
Vol 48 (4) ◽  
pp. 535-550 ◽  
Author(s):  
H. Kötz
Keyword(s):  
Differential Equations ◽  
Symmetry Group ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Classification Problem ◽  
Similarity Solutions ◽  
Point Symmetry ◽  
Point Symmetry Group ◽  
Relativistic Case ◽  
Lie Point Symmetry

"Optimal systems" of similarity solutions of a given system of nonlinear partial (integro-)differential equations which admits a finite-dimensional Lie point symmetry group Gare an effective systematic means to classify these group-invariant solutions since every other such solution can be derived from the members of the optimal systems. The classification problem for the similarity solutions leads to that of "constructing" optimal subalgebraic systems for the Lie algebra Gof the known symmetry group G. The methods for determining optimal systems of s-dimensional Lie subalgebras up to the dimension r of Gvary in case of 3 ≤ s ≤ r, depending on the solvability of G. If the r-dimensional Lie algebra Gof the infinitesimal symmetries is nonsolvable, in addition to the optimal subsystems of solvable subalgebras of Gone has to determine the optimal subsystems of semisimple subalgebras of Gin order to construct the full optimal systems of s-dimensional subalgebras of Gwith 3 ≤ s ≤ r. The techniques presented for this classification process are applied to the nonsolvable Lie algebra Gof the eight-dimensional Lie point symmetry group Gadmitted by the three-dimensional Vlasov-Maxwell equations for a multi-species plasma in the non-relativistic case.

Sign in / Sign up

Export Citation Format

Share Document