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Journal of Lie Theory 26 (2016), No. 1, 219--225
Copyright Heldermann Verlag 2016

Generalized Adjoint Actions

Arkady Berenstein
Dept. of Mathematics, University of Oregon, Eugene, OR 97403, U.S.A.

Vladimir Retakh
Dept. of Mathematics, Rutgers University, Piscataway, NJ 08854, U.S.A.


The aim of this paper is to generalize the classical formula $$ e^xye^{-x} = \sum_{k\ge 0}{1\over k!}\,({\rm ad}~x)^k (y) $$ by replacing $e^x$ with any formal power series $$ f(x)=1+\sum_{k\ge 1} a_k t^k. $$ We also obtain combinatorial applications to $q$-exponentials, $q$-binomials, and Hall-Littlewood polynomials.

Keywords: Adjoint action, commutator, q-exponential, Hall-Littlewood polynomial.

MSC: 20F40, 05E05

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