-systems and -systems for quantum affinizations of quantum Kac-Moody algebras.
Tetrahedron equation and the algebraic geometry
The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl (2, C).
The Azéma martingales as components of quantum independent increment processes
The bar automorphism in quantum groups and geometry of quiver representations
Two geometric interpretations of the bar automorphism in the positive part of a quantized enveloping algebra are given. The first is in terms of numbers of rational points over finite fields of quiver analogues of orbital varieties; the second is in terms of a duality of constructible functions provided by preprojective varieties of quivers.
The Brauer category and invariant theory
A category of Brauer diagrams, analogous to Turaev’s tangle category, is introduced, a presentation of the category is given, and full tensor functors are constructed from this category to the category of tensor representations of the orthogonal group O or the symplectic group Sp over any field of characteristic zero. The first and second fundamental theorems of invariant theory for these classical groups are generalised to the category theoretic setting. The major outcome is that we obtain presentations...
The catenarity of the positive part of the quantized enveloping algebra of a simple Lie algebra of type . (La caténarité de la partie positive de l'algèbre enveloppante quantifiée de l'algèbre de Lie simple de type .)
The center of the Dipper Donkin quantized matrix algebra.
The cyclic homology of
The dynamical quantum group.
The elliptic dynamical quantum group as an -Hopf algebroid.
The garden of quantum spheres
A list of known quantum spheres of dimension one, two and three is presented.
The geometric reductivity of the quantum group
We introduce the concept of geometrically reductive quantum group which is a generalization of the Mumford definition of geometrically reductive algebraic group. We prove that if G is a geometrically reductive quantum group and acts rationally on a commutative and finitely generated algebra A, then the algebra of invariants is finitely generated. We also prove that in characteristic 0 a quantum group G is geometrically reductive if and only if every rational G-module is semisimple, and that in...
The Harish-Chandra homomorphism for a quantized classical hermitian symmetric pair
Let a noncompact symmetric space with Iwasawa decomposition . The Harish-Chandra homomorphism is an explicit homomorphism between the algebra of invariant differential operators on and the algebra of polynomials on that are invariant under the Weyl group action of the pair . The main result of this paper is a generalization to the quantum setting of the Harish-Chandra homomorphism in the case of being an hermitian (classical) symmetric space
The Khovanov-Lauda 2-category and categorifications of a level two quantum representation.
The Kontsevich integral and quantized Lie superalgebras.
The multisegment duality and the preprojective algebras of type .
The quantum double of a (locally) compact group.
The quantum duality principle
The “quantum duality principle” states that the quantization of a Lie bialgebra – via a quantum universal enveloping algebra (in short, QUEA) – also provides a quantization of the dual Lie bialgebra (through its associated formal Poisson group) – via a quantum formal series Hopf algebra (QFSHA) — and, conversely, a QFSHA associated to a Lie bialgebra (via its associated formal Poisson group) yields a QUEA for the dual Lie bialgebra as well; more in detail, there exist functors and , inverse to...
The relationship between Zhedanov's algebra and the double affine Hecke algebra in the rank one case.