The starting point is the nonsemisimple, inhomogeneous Lie algebra Un×I2n [denoted also as IU(n)], where I2n represents an Abelian subalgebra in semidirect product with the homogeneous part U(n). This is realized by explicitly giving the matrix elements of the generators on a modified Gelfand–Zetlin basis that allows representations of infinite dimensions. The enveloping algebra is q lifted by introducing q brackets in the matrix elements giving Uq(IU(n)). The deformation of the Abelian structure of I2n is studied for q≠1. Some implications are pointed out. The important invariants are constructed for arbitrary n. The results are compared to the corresponding ones for Jimbo’s construction of Uq (U(n+1)) on a Gelfeld–Zetlin basis. Finally, the related construction of Uq(U(n,1)) is presented and discussed. Here, Uq(SU(1,1)), the q‐analog of relativistic motion in a plane, is analyzed in the context of this formalism.

Topics
Lie algebras, Group theory
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© 1991 American Institute of Physics.
1991
American Institute of Physics
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