Representations of Rota-Baxter algebras and regular singular decompositions

Abstract

There is a Rota-Baxter algebra structure on the field $A=\mathbf{k}((t))$ with $P$ being the projection map $A=\mathbf{k}[[t]]\oplus t^{-1}\mathbf{k}[t^{-1}]$ 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 $G{L}_A(V)$-orbits in the set of all regular-singular decompositions of an $n$-dimensional $A$-vector space $V$ is $(n+2)(n+1)/2$. We also use the result to compute the generalized class number, i.e., the number of the $G{L}_n(A)$-isomorphism classes of finitely generated $\mathbf{k}[[t]]$-submodules of $A^n$.

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Cited by in corpus (6)

• Representations and cohomologies of relative Rota-Baxter Lie algebras and applications
• Representations and Modules of Rota-Baxter Algebras
• Modules of polynomial Rota-Baxter Algebras and matrix equations
• Rota-Baxter paired modules and their constructions from Hopf algebras
• Representations of Polynomial Rota-Baxter Algebras
• Rota-Baxter $C^{\ast }$-algebras