On the commutative quotient of Fomin-Kirillov algebras

The Fomin-Kirillov algebra $\mathcal E_n$ is a noncommutative algebra with a generator for each edge in the complete graph on $n$ vertices. For any graph $G$ on $n$ vertices, let $\mathcal E_G$ be the subalgebra of $\mathcal E_n$ generated by the edges in $G$. We show that the commutative quotient of $\mathcal E_G$ is isomorphic to the Orlik-Terao algebra of $G$. As a consequence, the Hilbert series of this quotient is given by $(-t)^n \chi_G(-t^{-1})$, where $\chi_G$ is the chromatic polynomial of $G$. We also give a reduction algorithm for the graded components of $\mathcal E_G$ that do not vanish in the commutative quotient and show that their structure is described by the combinatorics of noncrossing forests.Comment: 11 pages, 3 figure

PBW deformations of a Fomin-Kirillov algebra and other examples

We begin the study of PBW deformations of graded algebras relevant to the theory of Hopf algebras. One of our examples is the Fomin-Kirillov algebra FK3. Another one appeared in a paper of Garc\'ia Iglesias and Vay. As a consequence of our methods, we determine when the deformations are semisimple and we are able to produce PBW bases and polynomial identities for these deformations.Comment: 22 pages. Accepted for publication in Algebr. Represent. Theor

PBW deformations of a Fomin–Kirillov Algebra and other examples

We begin the study of PBW deformations of graded algebras relevant to the theory of Hopf algebras. One of our examples is the Fomin–Kirillov algebra E3. Another one appeared in a paper of García Iglesias and Vay. As a consequence of our methods, we determine when the deformations are semisimple and we are able to produce PBW bases and polynomial identities for these deformations.Fil: Heckenberger, I.. Philipps-Universität Marburg; AlemaniaFil: Vendramin, Claudio Leandro. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

On Some Quadratic Algebras I 12\frac{1}{2}: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials

We study some combinatorial and algebraic properties of certain quadratic algebras related with dynamical classical and classical Yang-Baxter equations. One can find more details about the content of present paper in Extended Abstract.Comment: Dedicated to the memory of Alain Lascoux (1944-2013). Preprint RIMS-1817, 172 page