Quantum Algebra And Topology

Let g ^ be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let h ^ be the dual Lie bialgebra. By dualizing the quantum double construction - via formal Hopf algebras - we construct a new quantum group U q ( h ^ ) , dual of U q ( g ^ ) . Studying its restricted and unrestricted integer forms and their specializations at roots of 1 (in particular, their classical limits), we prove that U q ( h ^ ) yields quantizations of h ^ and G ^ ∞ (the formal group attached to g ^ ), and we construct new quantum Frobenius morphisms. The whole picture extends to the untwisted affine case the results known for quantum groups of finite type.

We introduce a tensor category O_+ (resp. O_{-}) of certain modules of gl_{\infty} with non-negative (resp. non-positive) integral central charges with the usual tensor product. We also introduce a tensor category O_f consisting of certain modules over GL(N) for all N. We show that the tensor categories O_+, O_{-} and O_f are semisimple abelian and all equivalent to each other. We give a formula to decompose a tensor product of two modules in each of these categories. We also introduce a tensor category O^w of certain modules over W_{1 +\infty} with non-negative integral central charges. We show that O^w is semisimple abelian and give an explicit formula to decompose a tensor product of two modules in O^w.

We consider nonquasiclassical solutions to the quantum Yang-Baxter equation and the corresponding quantum cogroups $\Fun(SL(S))$ constructed earlier by one of the authors . We give a criterion of the existence of a dual quasitriangular structure in the algebra $\Fun(SL(S))$ and describe a large class of such objects related to the Temperley-Lieb algebra satisfying this criterion. We show also that this dual quasitriangular structure is in some sense nondegenerate.

We find the Hopf algebra U g,h dual to the Jordanian matrix quantum group G L g,h (2) . As an algebra it depends only on the sum of the two parameters and is split in two subalgebras: U ′ g,h (with three generators) and U(Z) (with one generator). The subalgebra U(Z) is a central Hopf subalgebra of U g,h . The subalgebra U ′ g,h is not a Hopf subalgebra and its coalgebra structure depends on both parameters. We discuss also two one-parameter special cases: g=h and g=−h . The subalgebra U ′ h,h is a Hopf algebra and coincides with the algebra introduced by Ohn as the dual of S L h (2) . The subalgebra U ′ −h,h is isomorphic to U(sl(2)) as an algebra but has a nontrivial coalgebra structure and again is not a Hopf subalgebra of U −h,h .

We construct and study in detail various dual pairs acting on some Fock representations between a finite dimensional Lie group and a completed infinite rank affine algebra associated to an infinite affine Cartan matrix. We give explicit decompositions of a Fock representation into a direct sum of irreducible isotypic subspaces with respect to the action of a dual pair, present explicit formulas for the common highest weight vectors and calculate the corresponding highest weights. We further outline applications of these dual pairs to the study of tensor products of modules of such an infinite dimensional Lie algebra.

We extend the notion of dually conjugate Hopf (super)algebras to the coloured Hopf (super)algebras H c that we recently introduced. We show that if the standard Hopf (super)algebras H q that are the building blocks of H c have Hopf duals H ∗ q , then the latter may be used to construct coloured Hopf duals H c∗ , endowed with coloured algebra and antipode maps, but with a standard coalgebraic structure. Next, we review the case where the H q 's are quantum universal enveloping algebras of Lie (super)algebras U q (g) , so that the corresponding H ∗ q 's are quantum (super)groups G q . We extend the Fronsdal and Galindo universal T -matrix formalism to the coloured pairs ( U c (g), G c ) by defining coloured universal T -matrices. We then show that together with the coloured universal R -matrices previously introduced, the latter provide an algebraic formulation of the coloured RTT-relations, proposed by Basu-Mallick. This establishes a link between the coloured extensions of Drinfeld-Jimbo and Faddeev-Reshetikhin-Takhtajan pictures of quantum groups and quantum algebras. Finally, we illustrate the construction of coloured pairs by giving some explicit results for the two-parameter deformations of (U(gl(2)),Gl(2)) , and (U(gl(1/1)),Gl(1/1)) .

We express Hitchin's systems on curves in Schottky parametrization, and construct dynamical r -matrices attached to them.

The generalized Cremmer-Gervais R-matrix being a twist of the standard R-matrix of S L q (3) , depends on two extra parameters. Properties of this R-matrix are discussed and two dynamical systems, the quantum group covariant q -oscillator and an integrable spin chain with a non-hermitian Hamiltonian, are constructed.

In this paper, we propose an elliptic algebra A q,p; π ^ ( g l 2 ^ ) which is based on the relations RLL=LL R ∗ , where R and R ∗ are the dynamical R-maxtrices of A (1) 1 type face model with the elliptic moduli shifted by the center of the algebra. From the Ding-Frenkel correspondence, we find that its corresponding (Drinfeld) current algebra at level one is the algebra of screening currents for q-deformed Virasoro algebra.We realize the elliptic algebra at level one by Miki's construction from the bosonization for the type I and type II vertex operators.We also show that the algebra A q,p; π ^ ( g l 2 ^ ) is related with the algebra A q,p ( g l 2 ^ ) by a dynamically twisting.

Suppose X is a 1-connected simplicial set with finitely many nondegenerate simplices. We give an effective algorithm to calculate a simplicial set with the n -type of the loop space ΩX . Iterating gives an algorithm to calculate the π i (X) , different from the algorithms already known due to E. Brown and Kan-Curtis. The method is an effective version of the generalized Seifert-Van Kampen theorem of alg-geom/9704006. This can be viewed as a Van Kampen statement concerning the loop space ΩX with its delooping structure. We use Segal's delooping machinery but at the end we speculate on extensions to other delooping machinery.