Quantum Algebra And Topology

We define an action of the symmetric group on the set of domino tableaux, and prove that the number of domino tableaux of a given weight does not depend on the permutation of components of the last. A bijective proof of the well-known result due to J. Stembridge that the number of self-evacuating tableaux of a given shape is equal to that of domino tableaux of the same shape is given.

The construction of Lie bialgebra from double Lie algebra is presented. It is used to relate some types of cobracket on inhomogenous so(p,q) algebras with double Lie algebra structures on so(p+1,q) or so(p,q+1). Also it is shown that the cobracket corresponding to kappa-deformation gives rise to complete Poisson-Lie Euclidean groups and non-complete Poincare groups.

For $\g=sl(n)$ we construct a two parametric $U_h(\g)$-invariant family of algebras, $(S\g)_{t,h}$, which defines a quantization of the function algebra $S\g$ on the coadjoint representation and in the parameter t gives a quantization of the Lie bracket. The family induces a two parametric deformation of the function algebra of any maximal orbit which is a quantization of the Kirillov-Kostant-Souriau bracket in the parameter t . In addition we construct a quantum de Rham complex on $\g^*$.

Drinfeld showed that any finite dimensional Hopf algebra \G extends to a quasitriangular Hopf algebra \D(\G), the quantum double of \G. Based on the construction of a so--called diagonal crossed product developed by the authors, we generalize this result to the case of quasi--Hopf algebras \G. As for ordinary Hopf algebras, as a vector space the ``quasi--quantum double'' \D(\G) is isomorphic to the tensor product of \G and its dual \dG. We give explicit formulas for the product, the coproduct, the R--matrix and the antipode on \D(\G) and prove that they fulfill Drinfeld's axioms of a quasitriangular quasi--Hopf algebra. In particular \D(\G) becomes an associative algebra containing \G as a quasi--Hopf subalgebra. On the other hand, \dG \otimes 1 is not a subalgebra of \D(\G) unless the coproduct on \G is strictly coassociative. It is shown that the category of finite dimensional representations of \D(\G) coincides with what has been called the double category of \G--modules by S. Majid [M2]. Thus our construction gives a concrete realization of Majid's abstract definition of quasi--quantum doubles in terms of a Tannaka--Krein--like reconstruction procedure. The whole construction is shown to generalize to weak quasi--Hopf algebras with \D(\G) now being linearly isomorphic to a subspace of \dG \otimes \G.

We obtain Drinfel'd's realization of quantum affine superalgebra U q (gl(1|1)) ^ based on the super version of RS construction method and Gauss decomposition.

Any deformation of a Weyl or Clifford algebra can be realized through a change of generators in the undeformed algebra. q-Deformations of Weyl or Clifford algebrae that were covariant under the action of a simple Lie algebra g are characterized by their being covariant under the action of the quantum group U q g . We present a systematic procedure for determining all possible corresponding changes of generators, together with the corresponding realizations of the U q g -action. The intriguing relation between g-invariants and U q g -invariants suggests that these changes of generators might be employed to simplify the dynamics of some g-covariant quantum physical systems.

For the current realization of the quantum affine algebras, Drinfeld gave a simple comultiplication of the quantum current operators. With this comultiplication, we study the related vertex operators for the case of $U_q(\hgtsl_n)$ and give an explicit bosonization of these new vertex operators. We use these vertex operators to construct the quantum current operators of $U_q(\hgtsl_n)$ and discuss its connection with quantum boson-fermion correspondence.

We give a detailed description of the adjoint representation of Drinfeld's twist element, as well as of its coproduct, for s u q (2) . We also discuss, as applications, the computation of the universal R-matrix in this representation and the problem of symmetrization of identical-particle states with quantum su(2) symmetry.

We propose a q-difference version of the Drinfeld-Sokolov reduction scheme, which gives us q-deformations of the classical W-algebras by reduction from Poisson-Lie loop groups. We consider in detail the case of SL(2). The nontrivial consistency conditions fix the choice of the classical r-matrix defining the Poisson-Lie structure on the loop group LSL(2), and this leads to a new elliptic classical r-matrix. The reduced Poisson algebra coincides with the deformation of the classical Virasoro algebra previously defined in q-alg/9505025. We also consider a discrete analogue of this Poisson algebra. In the second part (q-alg/9702016) the construction is generalized to the case of an arbitrary semisimple Lie algebra.

The paper is the sequel to q-alg/9704011. We extend the Drinfeld-Sokolov reduction procedure to q-difference operators associated with arbitrary semisimple Lie algebras. This leads to a new elliptic deformation of the Lie bialgebra structure on the associated loop algebra. The related classical r-matrix is explicitly described in terms of the Coxeter transformation. We also present a cross-section theorem for q-gauge transformations which generalizes a theorem due to R.Steinberg.