Degenerations, contractions and deformations of various algebraic structures play an important role in mathematics and physics. There are many different definitions and special cases of these notions. We try to give a general definition which unifies these notions and shows the connections among them. Here we focus on contractions of Lie algebras and algebraic groups.
Contractions of Poisson-Lie groups are introduced by using Lie bialgebra contractions. As an application, contractions of SL(2,R) Poisson-Lie groups leading to (1+1) Poincaré and Heisenberg structures are analysed. It is shown how the method here introduced allows a systematic construction of the Poisson structures associated to non-coboundary Lie bialgebras. Finally, it is sketched how contractions are also implemented after quantization by using the Lie bialgebra approach.
We seek to classify the full-rank left-invariant control affine systems evolving on solvable three-dimensional Lie groups. In this paper we consider only the cases corresponding to the solvable Lie algebras of types II, IV, and V in the Bianchi-Behr classification.
Given the notion of -structures without torsion on a real dimensional Lie algebra we study the problem of their classification when is a reductive algebra.
In this paper, we develop the crystal basis theory for the quantum queer superalgebra . We define the notion of crystal bases and prove the tensor product rule for -modules in the category . Our main theorem shows that every -module in the category has a unique crystal basis.
This survey on crystallographic groups, geometric structures on Lie groups and associated algebraic structures is based on a lecture given in the Ostrava research seminar in .
We define the -restriction and -induction functors on the category of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.
In this paper the concepts of mixed cartesian square and quasi-cocartesian square, already known in the category of groups, are adapted to the category of Lie algebras. These concepts can be used in the study of the obstructions of Lie algebra extensions in the same way that Wu has studied the obstructions of group extensions.