Bethe subalgebras in affine Birman--Murakami--Wenzl algebras and flat connections for q-KZ equations

Commutative sets of Jucys-Murphyelements for affine braid groups of $A^{(1)},B^{(1)},C^{(1)},D^{(1)}$ types were defined. Construction of $R$-matrix representations of the affine braid group of type $C^{(1)}$ and its distinguish commutative subgroup generated by the $C^{(1)}$-type Jucys--Murphy elements are given. We describe a general method to produce flat connections for the two-boundary quantum Knizhnik-Zamolodchikov equations as necessary conditions for Sklyanin's type transfer matrix associated with the two-boundary multicomponent Zamolodchikov algebra to be invariant under the action of the $C^{(1)}$-type Jucys--Murphy elements. We specify our general construction to the case of the Birman--Murakami--Wenzl algebras. As an application we suggest a baxterization of the Dunkl--Cherednik elements $Y's$ in the double affine Hecke algebra of type $A$

On quantum matrix algebras satisfying the Cayley-Hamilton-Newton identities

The Cayley-Hamilton-Newton identities which generalize both the characteristic identity and the Newton relations have been recently obtained for the algebras of the RTT-type. We extend this result to a wider class of algebras M(R,F) defined by a pair of compatible solutions of the Yang-Baxter equation. This class includes the RTT-algebras as well as the Reflection equation algebras

On R-matrix representations of Birman-Murakami-Wenzl algebras

We show that to every local representation of the Birman-Murakami-Wenzl algebra defined by a skew-invertible R-matrix $R\in Aut(V\otimes V)$ one can associate pairings $V\otimes V\to C$ and $V^*\otimes V^*\to C$, where V is the representation space. Further, we investigate conditions under which the corresponding quantum group is of SO or Sp type.Comment: 9 page

Modified Affine Hecke Algebras and Drinfeldians of Type A

We introduce a modified affine Hecke algebra \h{H}^{+}_{q\eta}({l}) (\h{H}_{q\eta}({l})) which depends on two deformation parameters $q$ and $\eta$. When the parameter $\eta$ is equal to zero the algebra \h{H}_{q\eta=0}(l) coincides with the usual affine Hecke algebra \h{H}_{q}(l) of type $A_{l-1}$, if the parameter q goes to 1 the algebra \h{H}^{+}_{q=1\eta}(l) is isomorphic to the degenerate affine Hecke algebra \Lm_{\eta}(l) introduced by Drinfeld. We construct a functor from a category of representations of $H_{q\eta}^{+}(l)$ into a category of representations of Drinfeldian $D_{q\eta}(sl(n+1))$ which has been introduced by the first author.Comment: 11 pages, LATEX. Contribution to Proceedings "Quantum Theory and Symmetries" (Goslar, July 18-22, 1999) (World Scientific, 2000

Modified Braid Equations for SO_q (3) and noncommutative spaces

General solutions of the $\hat{R}TT$ equation with a maximal number of free parameters in the specrtal decomposition of vector $SO_q (3)$ $\hat{R}$ matrices are implemented to construct modified braid equations (MBE). These matrices conserve the given, standard, group relations of the nine elements of T, but are not constrained to satisfy the standard braid equation (BE). Apart from q and a normalisation factor our $\hat{R}$ contains two free parameters, instead of only one such parameter for deformed unitary algebras studied in a previous paper [1] where the nonzero right hand side of the $MBE$ had a linear term proprotional to $(\hat{R}_{(12)} - \hat{R}_{(23)})$. In the present case the r.h.s. is, in general, nonliear. Several particular solutions are given (Sec.2) and the general structure is analysed (App.A). Our formulation of the problem in terms of projectors yield also two new solutions of standard (nonmodified) braid equation (Sec.2) which are further discussed (App.B). The noncommutative 3-spaces obtained by implementing such generalized $\hat{R}$ matrices are studied (Sec.3). The role of coboundary $\hat{R}$ matrices (not satisfying the standard BE) is explored. The MBE and Baxterization are presented as complementary facets of the same basic construction, namely, the general solution of $\hat{R}TT$ equation (Sec.4). A new solution is presented in this context. As a simple but remarkable particular case a nontrivial solution of BE is obtained (App.B) for q=1. This solution has no free parameter and is not obtainable by twisting the identity matrix. In the concluding remarks (Sec.5), among other points, generalisation of our results to $SO_{q}(N)$ is discussed.Comment: 18 pages, no figure