Symmetric functions of two noncommuting variables

2013/07/05 by Agler, Jim, Young, N. J. · 1 citation
#30G30 #32A30 #47N99 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1307.1588

Abstract

We prove a noncommutative analogue of the fact that every symmetric analytic function of (z,w) in the bidisc \D² can be expressed as an analytic function of the variables z+w and zw. We construct an analytic nc-map S from the biball to an infinite-dimensional nc-domain Ω with the property that, for every bounded symmetric function \ph of two noncommuting variables that is analytic on the biball, there exists a bounded analytic nc-function Φ on Ω such that \ph=Φ∘ S. We also establish a realization formula for Φ, and hence for \ph, in terms of operators on Hilbert space.

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• Invariants of finite groups acting on (free) skew fields

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