Spherical functions on a complex classical quantum group
Spherical functions on a family of quantum 3-spheres
Spin chain connected to the quantum superalgebra .
Squared Hopf algebras and reconstruction theorems
Given an abelian 𝑉-linear rigid monoidal category 𝑉, where 𝑉 is a perfect field, we define squared coalgebras as objects of cocompleted 𝑉 ⨂ 𝑉 (Deligne's tensor product of categories) equipped with the appropriate notion of comultiplication. Based on this, (squared) bialgebras and Hopf algebras are defined without use of braiding. If 𝑉 is the category of 𝑉-vector spaces, squared (co)algebras coincide with conventional ones. If 𝑉 is braided, a braided Hopf algebra can be obtained from a squared...
Statistics and quantum group symmetries
Using 'twisted' realizations of the symmetric groups, we show that Bose and Fermi statistics are compatible with transformations generated by compact quantum groups of Drinfel'd type.
Strange phenomena related to ordering problems in quantizations.
Symmetric functions for the generating matrix of the Yangian of .
Symmetric quantum Weyl algebras
We study the symmetric powers of four algebras: -oscillator algebra, -Weyl algebra, -Weyl algebra and . We provide explicit formulae as well as combinatorial interpretation for the normal coordinates of products of arbitrary elements in the above algebras.
Symplectic models of groups with noncommutative spaces
-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.