Generalized adjoint actions

The aim of this paper is to generalize the classical formula $e^xye^{-x}=\sum\limits_{k\ge 0} \frac{1}{k!} (ad~x)^k(y)$ by replacing $e^x$ with any formal power series $\displaystyle {f(x)=1+\sum_{k\ge 1} a_kx^k}$. We also obtain combinatorial applications to $q$-exponentials, $q$-binomials, and Hall-Littlewood polynomials.Comment: 5 pages, LaTeX, typos corrected, to appear in Journal of Lie Theor

Algebras of Quasi-Pl\"ucker Coordinates are Koszul

Motivated by the theory of quasi-determinants, we study non-commutative algebras of quasi-Pl\"ucker coordinates. We prove that these algebras provide new examples of non-homogeneous quadratic Koszul algebras by showing that their quadratic duals have quadratic Gr\"obner bases.Comment: 15 pages, AMS tex, tikz figures. V2: small corrections based on referee report. V3: Further minor changes. To appear in J. Pure Appl. Algebr

Noncommutative Catalan numbers

The goal of this paper is to introduce and study noncommutative Catalan numbers $C_n$ which belong to the free Laurent polynomial algebra in $n$ generators. Our noncommutative numbers admit interesting (commutative and noncommutative) specializations, one of them related to Garsia-Haiman $(q,t)$-versions, another -- to solving noncommutative quadratic equations. We also establish total positivity of the corresponding (noncommutative) Hankel matrices $H_m$ and introduce accompanying noncommutative binomial coefficients.Comment: 12 pages AM LaTex, a picture and proof of Lemma 3.6 are added, misprints correcte

Algebras associated to acyclic directed graphs

We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices. Each finite directed acyclic graph admits countably many structures of a generalized layered graph. We construct linear bases in such algebras and compute their Hilbert series. Our interest to generalized layered graphs and algebras associated to those graphs is motivated by their relations to factorizations of polynomials over noncommutative rings.Comment: 20 pages, Latex; an expanded and corrected version; to appear in "Advances of Applied Mathematics