Generalized reductive algebras and a quantum example. (Q2496562)

The universal enveloping algebra of a semisimple Lie algebra is FCR. Recall that an algebra \(R\) is FCR (``finite-dimensional representations are completely reducible'') if it has enough finite-dimensional representations, in the sense that every nonzero element of \(R\) survives in some finite-dimensional image, and every finite-dimensional \(R\)-module is semisimple. Classical representation theory of Lie groups frequently requires the examination of a slightly larger class of Lie algebras -- the reductive ones. Intuitively, a reductive Lie algebra is ``semisimple up to a central subalgebra'': finite-dimensional modules on which the central subalgebra acts like a character will be completely reducible. The authors try to formulate the appropriate definition of ``reductive over a central subalgebra'' and generalize complete reducibility for finite-dimensional modules to encompass the representations of reductive Lie algebras. They begin by discussing notions related to separability of ring extensions, but with conditions restricted to finite-dimensional modules. Given \(R| S\) an extension of algebras, the extension \(R| S\) is said to be ``finitarily semisimple'' provided that every short exact sequence of finite-dimensional left \(R\)-modules which splits as \(S\)-modules must also split as \(R\)-modules; the extension \(R| S\) is said to be ``finitarily Wedderburn'' when every finite-dimensional left \(R\)-module which is semisimple as an \(S\)-module is also semisimple as an \(R\)-module. The authors prove that, if \(R| S\) is a central extension, both definitions are equivalent. Another notion is needed to formulate the desired definition: an algebra \(A\) is ``residually finite-dimensional'' if the zero ideal is the intersection of the ideals with finite codimension. The authors give the following definition of reductive extension of algebras by joining the former concepts: Given \(R| S\) a central extension of algebras, it is a ``reductive extension'' provided it is finitarily semisimple and \(R\) is residually finite-dimensional. The bulk of the paper studies an example related to the theory of quantum symmetric pairs developed by Letzter which is presented as a nontrivial illustration of these ideas.