S. G. Mihov

Spectral Decomposition and Baxterization of Exotic Bialgebras and Associated Noncommutative Geometries

We study the geometric aspects of two exotic bialgebras S03 and S14 introduced in math.QA/0206053. These bialgebras are obtained by the Faddeev–Reshetikhin–Takhtajan RTT prescription with non-triangular R-matrices which are denoted R03 and R14 in the classification of Hietarinta, and they are not deformations of either GL(2) or GL(1|1). We give the spectral decomposition which involves two, respectively, three, projectors. These projectors are then used to provide the baxterization procedure with one, respectively, two, parameters. Further, the projectors are used to construct the noncommutative planes together with the corresponding differentials following the Wess–Zumino prescription. In all these constructions there appear nonstandard features which are noted. Such features show the importance of systematic study of all bialgebras of four generators.

Higher dimensional unitary braid matrices: Construction, associated structures, and entanglements

We construct (2n)2×(2n)2 unitary braid matrices R for n⩾2 generalizing the class known for n=1. A set of (2n)×(2n) matrices (I,J,K,L) is defined. R is expressed in terms of their tensor products (such as K⊗J), leading to a canonical formulation for all n. Complex projectors P± provide a basis for our real, unitary R. Baxterization is obtained. Diagonalizations and block diagonalizations are presented. The loss of braid property when R (n>1) is block diagonalized in terms of R (n=1) is pointed out and explained. For odd dimension (2n+1)2×(2n+1)2, a previously constructed braid matrix is complexified to obtain unitarity. RLL and RTT algebras, chain Hamiltonians, potentials for factorizable S matrices, and complex noncommutative spaces are all studied briefly in the context of our unitary braid matrices. Turaev construction of link invariants is formulated for our case. We conclude with comments concerning entanglements.

Duality and representations for new exotic bialgebras

We find the exotic matrix bialgebras which correspond to the two nontriangular nonsingular 4×4 R-matrices in the classification of Hietarinta, namely, RS0,3 and RS1,4. We find two new exotic bialgebras S03 and S14 which are not deformations of the classical algebras of functions on GL(2) or GL(1|1). With this we finalize the classification of the matrix bialgebras which unital associative algebras generated by four elements. We also find the corresponding dual bialgebras of these new exotic bialgebras and study their representation theory in detail. We also discuss in detail a special case of RS1,4 in which the corresponding algebra turns out to be a special case of the two-parameter quantum group deformation GLp,q(2).

On combined standard–nonstandard or hybrid (q,h)-deformations

Combined (q,h)-deformations proposed by Kupershmidt and Ballesteros–Herranz–Parashar are studied. In each case a transformation is shown to lead to an equivalent, standard q-deformation. We briefly indicate that appropriate singular limits of the same type of transformations can however lead from standard biparametric (p,q)-deformations to nonhybrid but biparametric nonstandard (g,h) ones. Finally a case of hybrid (q,h)-deformation is recalled, related to the superalgebra GL(1|1).

Higher dimensional multiparameter unitary and nonunitary braid matrices: Even dimensions

A class of (2n)2×(2n)2 multiparameter braid matrices are presented for all n(n⩾1). Apart from the spectral parameter θ, they depend on 2n2 free parameters mij(±), i,j=1,…,n. For real parameters, the matrices R(θ) are nonunitary. For purely imaginary parameters, they became unitary. Thus, a unification is achieved with odd dimensional multiparameter solutions presented before.

Exotic Bialgebra S03: Representations, Baxterisation and Applications

Abstract.The exotic bialgebra S03, defined by a solution of the Yang–Baxter equation, which is not a deformation of the trivial, is considered. Its FRT dual algebra s03F is studied. The Baxterisation of the dual algebra is given in two different parametrizations. The finite-dimensional representations of s03F are considered. Diagonalizations of the braid matrices are used to yield remarkable insights concerning representations of the L-algebra and to formulate the fusion of finite-dimensional representations. Possible applications are considered, in particular, an exotic eight-vertex model and an integrable spin-chain model.Communicated by Petr Kulish

Duality for Exotic Bialgebras

In the classification of Hietarinta, three triangular 4 × 4 R-matrices lead, via the FRT formalism, to matrix bialgebras which are not deformations of the trivial one. In this paper, we find the bialgebras which are in duality with these three exotic matrix bialgebras. We note that the L − T duality of FRT is not sufficient for the construction of the bialgebras in duality. We find also the quantum planes corresponding to these bialgebras both by the Wess-Zumino R-matrix method and by Manin’s method.

Duality for the Jordanian matrix quantum group

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

Exotic bialgebras from 9 × 9 unitary braid matrices

The exotic bialgebras that arise from a 9 × 9 unitary braid matrix are constructed. The dual bialgebra of one of these exotic bialgebras is presented.

Exotic bialgebras from 9×9 braid matrices

We construct exotic bialgebras that arise from multiparameter 9 × 9 R-matrices, some of which are new. We also construct the dual bialgebras of two of these exotic bialgebras.