Representations of the Laurent series Rota-Baxter algebras and regular-singular decompositions

Abstract

There is a Rota-Baxter algebra structure on the field A=k((t)) with P being the projection map from A=k[[t]]⊕t−1k[t−1] onto k[[t]]. We study the representation theory and regular-singular decompositions of any finite dimensional A-vector space. The main result shows that the category of finite dimensional representations is semisimple and consists of exactly three isomorphism classes of irreducible representations which are all one-dimensional. As a consequence, the number of GLA(V)-orbits in the set of all regular-singular decompositions of an n-dimensional A-vector space V is (n+1)(n+2)/2. We also use the result to compute the generalized class number, i.e., the number of the GLn(A)-isomorphism classes of finitely generated k[[t]]-submodules of An.