QUANTUM
   INTEGRABLE
   SYSTEMS and
   GEOMETRY
   2012









 3-7 SEPTEMBER 2012
 OLHÃO, PORTUGAL

SL(n,R) - Decontraction formula and Unitary Irreducible Representations

Đorđe Šijački

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Lie algebra, topology and unitary representations issues of the covering groups of the SL(n,R) and Diff(n,R) groups w.r.t. their maximal compact SO(n) subgroups are considered. Topological properties determining spinorial representations of these groups are reviewed. Special attention is paid to the fact that, contrary to other Classical Lie algebras, the SL(n,R), n>2 covering groups are necessarily defined in infinite-dimensional spaces, thus yielding the same feature to all their spinorial representations.

A notion of Lie algebra decontraction, also known as the Gell-Mann formula, that plays a role of an inverse to the Inonu-Wigner contraction is recalled. Contrary to the case of orthogonal type of algebras, the decotraction formula for sl(n,R) algebras is of limited value. The validity domain of this formula for sl(n,R) algebras contracted w.r.t. their so(n) subalgebras is outlined. A recent generalization of the decontraction formula, that applies to all SL(n,R) covering group representations, as well as an explicit closed expression of all non-compact sl(n,R) operators matrix elements for all representations is presented.

A construction of the unitary sl(n,R) representations is discussed within a framework that combines the Harish-Chandra results and a method of fulfilling the unitarity requirements in Hilbert spaces with non-trivial scalar product kernels. Relevance of these representations to the Diff(n,R) covering groups representations construction is considered.