Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 26 (2016), No. 1, 219--225Copyright 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 [ Fulltext-pdf  (263  KB)] for subscribers only.