Classification of two and three dimensional Lie superbialgebras

Abstract

Using adjoint representation of Lie superalgebras, we write the matrix form of super Jacobi and mixed super Jacobi identities of Lie super-bialgebras. Then through direct calculations of these identities and use of automorphism supergroups of two and three dimensional Lie superalgebras, we obtain and classify all two and three dimensional Lie

Lie superbialgebra structures on the Lie superalgebra C3+A and deformation of related integrable Hamiltonian systems

“…As mentioned in [14], because of tensorial form of super Jacobi and mixed super Jacobi identities (2.21) and (2.22), working with them is not so easy and we suggest writing these equations as matrix forms using the following adjoint representations for Lie superalgebras g andg…”

“…Inserting the transformation (3.2) into (3.3), we obtain the following matrix equation for the elements of automorphism supergroup [14]:…”

“…Until now there were distinguished and non-systematic ways for obtaining low-dimensional Lie superbialgebras (see, for instance, [12,13]). We have recently presented a systematic way for obtaining and classifying low-dimensional Lie superbialgebras by using the adjoint representation of Lie superalgebras [14] and then have applied this method to classify the Lie superbialgebras gl(1|1) [15]. In the present paper, we will try to perform the classification of the Lie superbialgebras (C 3 + A).…”

Classification of four-dimensional real Lie bialgebras of symplectic type and their Poisson–Lie groups

“…Recently, non-semisimple Lie algebras has shown to play important role in physical problems. Of course there are attempts for the classification of low dimensional non-semisimple Lie bialgebras [[6] - [11]] and Lie superbialgebras [12]. In this paper we will try to classify four dimensional real Lie bialgebras of symplectic type such that on the Lie algebras g and their dualsg we have symplectic structures.…”

“…In section two, we briefly review the definitions and notations. In section three, after giving the list of four dimensional real Lie algebra of symplectic type [13], [14] based in to [15] (classification of real four dimensional Lie algebras); we classify four dimensional real Lie bialgebras of symplectic type according to the method given in [12]. In section four, we determine the coboundary Lie bialgebras from the list obtained in section three.…”