Mickelsson algebras and Zhelobenko operators

We construct a family of automorphisms of Mickelsson algebra, satisfying braid group relations. The construction uses 'Zhelobenko cocycle' and includes the dynamical Weyl group action as a particular case

Central Extension of the Yangian Double

Central extension \DYg of the Double of the Yangian is defined for a simple Lie algebra ${\bf g}$ with complete proof for ${\bf g} =sl_2$. Basic representations and intertwining operators are constructed for \DY2.Comment: 12 pages, latex, no figure

Weight function for the quantum affine algebra Uq(A2(2))U_q(A_2^{(2)})

In this article, we give an explicit formula for the universal weight function of the quantum twisted affine algebra $U_q(A_2^{(2)})$. The calculations use the technique of projecting products of Drinfeld currents onto the intersection of Borel subalgebras of different types.Comment: 25 page

Quillen homology for operads via Gr\"obner bases

The main goal of this paper is to present a way to compute Quillen homology of operads. The key idea is to use the notion of a shuffle operad we introduced earlier; this allows to compute, for a symmetric operad, the homology classes and the shape of the differential in its minimal model, although does not give an insight on the symmetric groups action on the homology. Our approach goes in several steps. First, we regard our symmetric operad as a shuffle operad, which allows to compute its Gr\"obner basis. Next, we define a combinatorial resolution for the "monomial replacement" of each shuffle operad (provided by the Gr\"obner bases theory). Finally, we explain how to "deform" the differential to handle every operad with a Gr\"obner basis, and find explicit representatives of Quillen homology classes for a large class of operads. We also present various applications, including a new proof of Hoffbeck's PBW criterion, a proof of Koszulness for a class of operads coming from commutative algebras, and a homology computation for the operads of Batalin-Vilkovisky algebras and of Rota-Baxter algebras.Comment: 41 pages, this paper supersedes our previous preprint arXiv:0912.4895. Final version, to appear in Documenta Mat

Diagonal reduction algebras of \gl type

Several general properties, concerning reduction algebras - rings of definition and algorithmic efficiency of the set of ordering relations - are discussed. For the reduction algebras, related to the diagonal embedding of the Lie algebra $gl_n$ into $gl_n \oplus gl_n$, we establish a stabilization phenomenon and list the complete sets of defining relations.Comment: 24 pages, no figure