Deformations of inhomogeneous classical Lie algebras to the algebras of the linear groups

Abstract

We study a new class of deformations of algebra representations, namely, i2so(n)⇒sl(n,R), i2u(n)⇒sl(n,C)⊕u(1) and i2sp(n)⊕sp(1)⇒sl(n,Q)⊕sp(1). The new generators are built as commutators between the Casimir invariant of the maximal compact subalgebra and a second-rank mixed tensor. These algebra deformations are related to multiplier representations and manifold mappings of the corresponding Lie groups. Behavior of the representations under Inönü-Wigner contractions is exhibited. Through the use of these methods we can construct a principal degenerate series of representations of the linear groups and their algebras.