Root vectors in quantum groups.
Semilinear relations and *-representations of deformations of so(3)
We study a family of commuting selfadjoint operators , which satisfy, together with the operators of the family , semilinear relations , (, , are fixed Borel functions). The developed technique is used to investigate representations of deformations of the universal enveloping algebra U(so(3)), in particular, of some real forms of the Fairlie algebra .
Separation of variables in the quantum integrable models related to the Yangian
link homology.
Smallness problem for quantum affine algebras and quiver varieties
The geometric small property (Borho-MacPherson [2]) of projective morphisms implies a description of their singularities in terms of intersection homology. In this paper we solve the smallness problem raised by Nakajima [37, 35] for certain resolutions of quiver varieties [37] (analogs of the Springer resolution): for Kirillov-Reshetikhin modules of simply-laced quantum affine algebras, we characterize explicitly the Drinfeld polynomials corresponding to the small resolutions. We use an elimination...
Snyder space-time: K-loop and Lie triple system.
Solvable lattice model and representation theory of quantum affine algebras.
Some remarks on quantum and braided group gauge theory
We clarify some aspects of quantum group gauge theory and its recent generalisations (by T. Brzeziński and the author) to braided group gauge theory and coalgebra gauge theory. We outline the diagrammatic version of the braided case. The bosonisation of any braided group provides us a trivial principal bundle in three ways.
Spin chain connected to the quantum superalgebra .
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.
nonstandard bases: case of mutually unbiased bases.
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.
Symmetries of an extended Hubbard Model
The Hamiltonian for an extended Hubbard model with phonons as introduced by A. Montorsi and M. Rasetti is considered on a D-dimensional lattice. The symmetries of the model are studied in various cases. It is shown that for a certain choice of the parameters a superconducting holds as a true quantum symmetry, but only for D=1.
Symmetries of spin Calogero models.
Tensor product construction of 2-freeness
From a sequence of m-fold tensor product constructions that give a hierarchy of freeness indexed by natural numbers m we examine in detail the first non-trivial case corresponding to m=2 which we call 2-freeness. We show that in this case the constructed tensor product of states agrees with the conditionally free product for correlations of order ≤ 4. We also show how to associate with 2-freeness a cocommutative *-bialgebra.
The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl (2, C).
The Azéma martingales as components of quantum independent increment processes
The closed Friedman world model with the initial and final singularities as a non-commutative space
The most elegant definition of singularities in general relativity as b-boundary points, when applied to the closed Friedman world model, leads to the disastrous situation: both the initial and final singularities form the single point of the b-boundary which is not Hausdorff separated from the rest of space-time. We apply Alain Connes' method of non-commutative geometry, defined in terms of a C*-algebra, to this case. It turns out that both the initial and final singularities can be analysed as...
The coribbon structures of some finite dimensional braided Hopf algebras generated by 2×2-matrix coalgebras
We determine the coribbon structures of some finite dimensional braided Hopf algebras generated by 2×2-matrix coalgebras constructed by S. Suzuki. As a consequence, we see that such a Hopf algebra has a coribbon structure if and only if it is of Kac-Paljutkin type.