Color Lie bialgebras: big bracket, cohomology and deformations (Q1625124)

From the introduction: In recent years, color Lie algebras have become an interesting subject of mathematics and physics. The cohomology groups of color Lie algebras were introduced and investigated by \textit{M. Scheunert} [Zbl 0407.17001; Zbl 0928.17023] and the representations were explicitly described by \textit{J. Feldvoss} [Zbl 0998.17032]. Some other properties have been studied. The original deformation theory was developed by \textit{M. Gerstenhaber} for rings and algebras using formal power series in [Zbl 0123.03101]. It is closely related to Hochschild cohomology. Then, it was extended to Lie algebras, using Chevalley-Eilenberg cohomology, by \textit{A. Nijenhuis} and \textit{R. W. Richardson jun.} [Zbl 0153.04402]. Since then, this approach was used for different algebraic structures and in different contexts. We aim in this paper to study color Lie bialgebras which are a natural generalization of Lie bialgebras and Lie superbialgebras. In the first section, we summarize the basics and review preliminaries about color algebraic structures over an abelian group and a fixed commutation factor. In Sect. 3, we provide a functor associating to any color (Lie) algebra a (Lie) superalgebra. It turns out that in this procedure one looses the finer structure related to the grading. In Sect. 4, we study representations and consider semidirect product constructions. Manin triples and r-matrices which are strongly related to Lie bialgebras are considered for color case in Sect. 5. In Sect. 6, we extend to the color case the construction and properties of big bracket, from which we derive in Sect. 7 a cohomology complex. Section 8 is dedicated to establish one-parameter deformation theory, introduced by Gerstenhaber for associative algebras, in the case of color Lie bialgebras. Moreover, we describe quantum universal enveloping algebra associated to color Lie algebra and its color Lie bialgebra structure. In the last section, we provide some explicit examples by giving a classification of 3-dimensional color Lie bialgebras and computing some cohomology groups. For the entire collection see [Zbl 1392.42001].