A brief review of abelian categorifications.
A characterization of coboundary Poisson Lie groups and Hopf algebras
We show that a Poisson Lie group (G,π) is coboundary if and only if the natural action of G×G on M=G is a Poisson action for an appropriate Poisson structure on M (the structure turns out to be the well known ). We analyze the same condition in the context of Hopf algebras. A quantum analogue of the structure on SU(N) is described in terms of generators and relations as an example.
A classification of Poisson homogeneous spaces of complex reductive Poisson-Lie groups
Let G be a complex reductive connected algebraic group equipped with the Sklyanin bracket. A classification of Poisson homogeneous G-spaces with connected isotropy subgroups is given. This result is based on Drinfeld's correspondence between Poisson homogeneous G-spaces and Lagrangian subalgebras in the double D𝖌 (here 𝖌 = Lie G). A geometric interpretation of some Poisson homogeneous G-spaces is also proposed.
A cluster algebra approach to -characters of Kirillov–Reshetikhin modules
We describe a cluster algebra algorithm for calculating -characters of Kirillov–Reshetikhin modules for any untwisted quantum affine algebra . This yields a geometric -character formula for tensor products of Kirillov–Reshetikhin modules. When is of type , this formula extends Nakajima’s formula for -characters of standard modules in terms of homology of graded quiver varieties.
A family of tridiagonal pairs related to the quantum affine algebra .
A Frobenius formula for the characters of the Hecke algebras.
A Littlewood-Richardson rule for evaluation representations of .
A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras.
A note on coalgebra gauge theory
A generalisation of quantum principal bundles in which a quantum structure group is replaced by a coalgebra is proposed.
A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems
We obtain a presentation by generators and relations of any Nichols algebra of diagonal type with finite root system. We prove that the defining ideal is finitely generated. The proof is based on Kharchenko’s theory of PBW bases of Lyndon words. We prove that the lexicographic order on Lyndon words is convex for PBW generators and so the PBW basis is orthogonal with respect to the canonical non-degenerate form associated to the Nichols algebra.
À propos du calcul différentiel sur les groupes quantiques
A -analogue of the centralizer construction and skew representations of the quantum affine algebra.
A -representation with no quantum symmetric algebra
We show by explicit calculations in the particular case of the 4-dimensional irreducible representation of that it is not always possible to generalize to the quantum case the notion of symmetric algebra of a Lie algebra representation.
Abelian extensions of Lie bialgebras. (Extensions abéliennes des bigèbres de Lie.)
Addendum to "Generalized radical rings, unknotted biquandles, and quantum groups" (Colloq. Math. 109 (2007), 85-100)
Affine quivers and canonical bases
An Analogue of the Classical Yang-Baxter Equation for General Algebraic Structures.
An homotopy formula for the Hochschild cohomology
An idempotent for a Jordanian quantum complex sphere
A new Jordanian quantum complex 4-sphere together with an instanton-type idempotent is obtained as a suspension of the Jordanian quantum group .
Analogues de la forme de Killing et du théorème d'Harish-Chandra pour les groupes quantiques