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.
arkadiy@math.uoregon.edu
Vladimir Retakh
Dept. of Mathematics, Rutgers University, Piscataway, NJ 08854, U.S.A.
vretakh@math.rutgers.edu
[Abstract-pdf]
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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