Zero Modes’ Fusion Ring and Braid Group Representations for the Extended Chiral su(2) WZNW Model

Abstract

A zero modes' Fock space F q is constructed for the extended chiral su(2) WZNW model. It gives room to a realization of the fusion ring of representations of the restricted quantum universal enveloping algebra U q = U q sl(2) at an even root of unity, q h = −1 , and of its infinite dimensional extensionŨ q by the Lusztig operators E (h) , F (h) . We provide a streamlined derivation of the characteristic equation for the Casimir invariant from the defining relations of U q . A central result is the characterization of the Grothendieck ring of both U q and U q in Theorem 3.1. The properties of theŨ q fusion ring in F q are related to the braiding properties of correlation functions of primary fields of the conformal su(2) h−2 current algebra model.

A differential U-module algebra for U = Ū q sℓ(2) at an even root of unity

“…5 But this particular version of the quantum s (2) actually made its first appearance much earlier; a regrettable omission in (the arXiv version of) [7] was [21], where the regular representation of U was elegantly described in terms of the even subalgebra of a matrix algebra times a Grassmann algebra on two generators for each block (also see [22]- [24] for a very closely related quantum group at p = 3). This quantum group was also the subject of attention in [25], [26].…”

“Spread” Restricted Young Diagrams from a 2D WZNW Dynamical Quantum Group

“…This has been done in a completely satisfactory way for n = 2 (in [24,21]; see also [18]) and the emerging picture is easy to describe. It turns out that in this case the diagonal elements of the matrix Q = (Q i j ) commute with the off-diagonal ones and both generate two copies of the (finite dimensional) restricted quantum group U q (s (2)) of [13,14,20]. The corresponding Fock space representations are however quite different: while the one generated by the off-diagonal elements of Q is one dimensional, the di-…”

“…In the most interesting applications the deformation parameter q is a root of unity (in our case we take q = e −i π h , h = k + n). It has been shown, in particular, in [20] that the Fock representation of the chiral zero modes' algebra for n = 2 carries a representation of the restricted (finite dimensional) quantum group U q (s (2)) of [13,14] containing, as submodules or quotient modules, all irreducible representations of the latter.…”

On the 2D Zero Modes’ Algebra of the SU(n) WZNW Model

“…3 Q-algebra -the n = 2 case A great simplification in the n = 2 case comes from the fact that the exchange relations combine with the determinant condition (5), which in this case is also bilinear in the zero modes, to form powerful operator identities. For n = 2 and q = e ±i π h the chiral Fock space F q carries a representation of the 2h 3 -dimensional restricted quantum group U q = U q (sℓ(2)) generated by E, F, K such that E h = 0 = F h , K 2h = 1 [14]. The restricted Fock space F…”