97 research outputs found
Generalized adjoint actions
The aim of this paper is to generalize the classical formula
by replacing
with any formal power series . We
also obtain combinatorial applications to -exponentials, -binomials, and
Hall-Littlewood polynomials.Comment: 5 pages, LaTeX, typos corrected, to appear in Journal of Lie Theor
Noncommutative Catalan numbers
The goal of this paper is to introduce and study noncommutative Catalan
numbers which belong to the free Laurent polynomial algebra in
generators. Our noncommutative numbers admit interesting (commutative and
noncommutative) specializations, one of them related to Garsia-Haiman
-versions, another -- to solving noncommutative quadratic equations. We
also establish total positivity of the corresponding (noncommutative) Hankel
matrices and introduce accompanying noncommutative binomial coefficients.Comment: 12 pages AM LaTex, a picture and proof of Lemma 3.6 are added,
misprints correcte
Mystic Reflection Groups
This paper aims to systematically study mystic reflection groups that emerged
independently in the paper [Selecta Math. (N.S.) 14 (2009), 325-372,
arXiv:0806.0867] by the authors and in the paper [Algebr. Represent. Theory 13
(2010), 127-158, arXiv:0806.3210] by Kirkman, Kuzmanovich and Zhang. A detailed
analysis of this class of groups reveals that they are in a nontrivial
correspondence with the complex reflection groups . We also prove
that the group algebras of corresponding groups are isomorphic and classify all
such groups up to isomorphism
Cocycle Twists and Extensions of Braided Doubles
It is well known that central extensions of a group G correspond to
2-cocycles on G. Cocycles can be used to construct extensions of G-graded
algebras via a version of the Drinfeld twist introduced by Majid. We show how
2-cocycles can be defined for an abstract monoidal category C, following
Panaite, Staic and Van Oystaeyen. A braiding on C leads to analogues of Nichols
algebras in C, and we explain how the recent work on twists of Nichols algebras
by Andruskiewitsch, Fantino, Garcia and Vendramin fits in this context.
Furthermore, we propose an approach to twisting the multiplication in braided
doubles, which are a class of algebras with triangular decomposition over G.
Braided doubles are not G-graded, but may be embedded in a double of a Nichols
algebra, where a twist may be carried out if careful choices are made. This is
a source of new algebras with triangular decomposition. As an example, we show
how to twist the rational Cherednik algebra of the symmetric group by the
cocycle arising from the Schur covering group, obtaining the spin Cherednik
algebra introduced by Wang.Comment: 60 pages, LaTeX; v2: references added, misprints correcte
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