On Bernstein polynomials for compact Lie groups
type: Item
instance of: scholarly article
title: On Bernstein polynomials for compact Lie groups
zbMATH Open document ID: 0920.22007
author: Carina Boyallian
published in: Journal of Algebra
publication date: 1999-09-09T00:00:00Z
review text: For a compact connected Lie group \(G\), let \(A(G)\) be the ring of functions on \(G\) generated by all matrix coefficients of all irreducible finite dimensional representations of \(G\). For \(f\in A(G)\), let \(A_f\) be the ring of rational functions whose denominators are nonnegative powers of \(f\). For an independent variable \(s\), we consider the ring of polynomials \(A_f[s]\). Let \(A_f[s]\cdot f^s\) be the \(A_f[s]\) free module generated by \(f^s\). We extend formally the action of the Lie algebra of \(G\) on \(f^s\) as \(X(f^s)={s\over f}X(f)\cdot f^s\). Then we extend this action to \(A_f[s]\cdot f^s\). Let \({\mathcal D}(G)\) be the ring of differential operators generated by \(A(G)\) and the Lie algebra of \(G\). The action of the Lie algebra of \(G\) can be extended to \({\mathcal D}(G)\). Allowing \(s\) to act multiplicatively, we have an action of \({\mathcal D}(G)[s]\). By the theory of \({\mathcal D}\)-modules, we have the functional equation \(D(g,s)(f(g))^{s+1}=b(s)(f(g))^s\), where \(D(g,s)\) belongs to the ring of differential operators \({\mathcal D}(G)[s]\). The polynomial \(b(s)\) is not unique. There exists a monic polynomial of minimal degree that satisfies the above equation. This polynomial is called the Bernstein polynomial attached to \(f\). The explicit calculation of Bernstein polynomials has been the goal of many papers. The author gives the Bernstein polynomials for basic matrix entries of irreducible unitary representations of the compact Lie group \(G=SU(2)\). Their application to the analytic continuation of certain distributions on \(SU(2)\) is given. The Bernstein polynomials for \(B_-\times B_-\)-semi-invariant functions on a semisimple complex Lie group are briefly described.
reviewed by: A. U. Klimyk
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 1247577
zbMATH Keywords: Bernstein polynomials
zbMATH Keywords: matrix entries of irreducible representations
zbMATH Keywords: Lie group \(SU(2)\)
zbMATH Keywords: irreducible unitary representations
zbMATH Keywords: compact Lie group
zbMATH Keywords: semisimple complex Lie group
MaRDI profile type: Publication
full work available at URL: jabr.1998.7576
OpenAlex ID: W2002166890
Wikidata QID: Q115395846
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DOI: 10.1006/JABR.1998.7576
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description: scientific article; zbMATH DE number 1247577