Representations of the Laurent series Rota-Baxter algebras and regular-singular decompositions. (English) Zbl 1537.16022
Summary: There is a Rota-Baxter algebra structure on the field \(A = \mathbf{k}((t))\) with \(P\) being the projection map from \(A = \mathbf{k} [[t]] \oplus t^{- 1} \mathbf{k} [ t^{- 1}]\) onto \(\mathbf{k} [[t]]\). 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 \(\mathrm{G L}_A(V)\)-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 \(\mathrm{G L}_n(A)\)-isomorphism classes of finitely generated \(\mathbf{k} [[t]]\)-submodules of \(A^n\).
MSC:
| 16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
| 17B38 | Yang-Baxter equations and Rota-Baxter operators |
| 13F25 | Formal power series rings |
| 14H60 | Vector bundles on curves and their moduli |
| 16S32 | Rings of differential operators (associative algebraic aspects) |
| 16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |
| 11R29 | Class numbers, class groups, discriminants |
Keywords:
Rota-Baxter algebrasReferences:
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