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Finite and infinite dimensional Lie group structures on Riordan groups
Cheon, Gi-Sang,Luzó,n, Ana,Moró,n, Manuel A.,Prieto-Martinez, L. Felipe,Song, Minho Academic Press 2017 Advances in Mathematics Vol. No.
<P><B>Abstract</B></P> <P>We introduce a Frechet Lie group structure on the Riordan group. We give a description of the corresponding Lie algebra as a vector space of infinite lower triangular matrices. We describe a natural linear action induced on the Frechet space <SUP> K N </SUP> by any element in the Lie algebra. We relate this to a certain family of bivariate linear partial differential equations. We obtain the solutions of such equations using one-parameter groups in the Riordan group. We show how a particular semidirect product decomposition in the Riordan group is reflected in the Lie algebra. We study the stabilizer of a formal power series under the action induced by Riordan matrices. We get stabilizers in the fraction field K ( ( x ) ) using bi-infinite representations. We provide some examples. The main tool to get our results is the paper where the Riordan group was described using inverse sequences of groups of finite matrices.</P>