Contractions of Lie Algebras and Algebraic Groups

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. 1. Contractions, degenerations and deformations of Lie algebras 1.1. Basic definitions and properties. The notion of Lie algebra and Lie group contractions was first introduced by I. E. Segal [14] and E. Inönü, E. P. Wigner [11]. The usual definition of a continuous contraction of a Lie algebra is as follows. Definition 1.1. Let V be a vector space over R or C and g : (0, 1] → GL(V ) be a continuous function. Let [ , ] be a Lie bracket on V . A parametrized family of Lie brackets on V is defined by [x, y]ε = gε ( [g ε (x), g −1 ε (y)] ) . If the limit Jx, yK = lim ε→0 [x, y]ε exists, then J, K is a Lie bracket on V and (V, J, K) is called a contraction of (V, [, ]). For 0 < ε ≤ 1 the Lie algebras (V, [, ]ε) are all isomorphic to (V, [, ]). Hence to obtain a new Lie algebra via contraction one needs det(gε) = 0 for ε = 0. This is a necessary condition, but not a sufficient one. A contraction can be viewed as a special case of a so called degeneration. We need some notations to explain this. Let V be an n-dimensional vector space over a field k. Denote by k[t] the polynomial ring in one variable, by k(t) its field of fractions, and by k[[t]] the ring of formal power series with coefficients in k. 2000 Mathematics Subject Classification. 14Lxx, 17Bxx, 81R05.