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The ice Ansatz on matrix solutions of the Yang-Baxter equation is weakened to a condition which we call rime. Generic rime solutions of the Yang-Baxter equation are described. We prove that the rime non-unitary (respectively, unitary) R-matrix is equivalent to the Cremmer-Gervais (respectively, boundary Cremmer-Gervais) solution. Generic rime classical r-matices satisfy the (non-)homogeneous associative classical Yang-Baxter equation., 4 pages, talk given at the VII International Workshop "Supersymmetries and Quantum Symmetries", Dubna 2007

We define quantum matrix groups GL(3) by their coaction on appropriate quantum planes and the requirement that the Poincare series coincides with the classical one. It is shown that this implies the existence of a Yang-Baxter operator. Exploiting stronger equations arising at degree four of the algebra, we classify all quantum matrix groups GL(3). We find 26 classes of solutions, two of which do not admit a normal ordering. The corresponding R-matrices are given., Comment: 28 pages, Latex

The ring Diff_{h}(n) of h-deformed differential operators appears in the theory of reduction algebras. In this thesis, we construct the rings of generalized differential operators on the h-deformed vector spaces of gl-type. In contrast to the q-deformed vector spaces for which the ring of differential operators is unique up to an isomorphism, the general ring of h-deformed differential operators Diff_{h,σ}(n) is labeled by a rational function σ in n variables, satisfying an over-determined system of finite-difference equations. We obtain the general solution of the system. We show that the center of Diff_{h,σ}(n) is a ring of polynomials in n variables. We construct an isomorphism between certain localizations of Diff_{h,σ}(n) and the Weyl algebra Wn extended by n indeterminates. We present some conditions for the irreducibility of the finite dimensional Diff_{h,σ}(n)-modules. Finally, we discuss difficulties for finding analogous constructions for the ring Diff_{h}(n,N) formed by several copies of Diff_{h}(n).; L'anneau Diff_{h}(n) des opérateurs différentiels h-déformés apparaît dans la théorie des algèbres de réduction.Dans cette thèse, nous construisons les anneaux des opérateurs différentiels généralisés sur les espaces vectoriels h-déformés de type gl. Contrairement aux espaces vectoriels q-déformés pour lequel l'anneau des opérateurs différentiels est unique à isomorphisme près, l'anneau généralisé des opérateurs différentiels h-déformés Diff_{h,σ}(n) est indexée par une fonction rationnelle σ en n variables, solution d'un système dégénéré d'équations aux différences finies. Nous obtenons la solution générale de ce système. Nous montrons que le centre de Diff_{h,σ}(n) est un anneau des polynômes en n variables. Nous construisons un isomorphisme entre des localisations de l'anneau Diff_{h,σ}(n) et de l’algèbre de Weyl Wn étendue par n indéterminés. Nous présentons des conditions irréductibilité des modules de dimension fini de Diff_{h,σ}(n). Finalement, nous discutons des difficultés a trouver les constructions analogues pour l'anneau Diff_{h}(n,N) correspondant à N copies de Diff_{h}(n).

International audience

We suggest a formula for quantum universal $R$-matrices corresponding to quasitriangular classical $r$-matrices classified by Belavin and Drinfeld for all simple Lie algebras. The $R$-matrices are obtained by twisting the standard universal $R$-matrix., Comment: 12 pages, LaTeX

The quantum dynamical Yang-Baxter (or Gervais-Neveu-Felder) equation defines an R-matrix R(p), where $p$ stands for a set of mutually commuting variables. A family of SL(n)-type solutions of this equation provides a new realization of the Hecke algebra. We define quantum antisymmetrizers, introduce the notion of quantum determinant and compute the inverse quantum matrix for matrix algebras of the type R(p) a_1 a_2 = a_1 a_2 R. It is pointed out that such a quantum matrix algebra arises in the operator realization of the chiral zero modes of the WZNW model., 28 pages, LaTeX

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.

We construct for all $N$ a solution of the Frenkel--Moore $N$--simplex equation which generalizes the $R$--matrix for the Jordanian quantum group., Comment: 6 pages

We review some facts about the representation theory of the Hecke algebra. We adapt for the Hecke algebra case the approach of Okounkov and Vershik which was developed for the representation theory of symmetric groups. We justify an explicit construction of the idempotents in the Hecke algebra in terms of Jucys-Murphy elements. Ocneanu's traces for these idempotents (which can be interpreted as q-dimensions of corresponding irreducible representations of quantum linear groups) are presented., Comment: 11 pages, Proceedings of International Workshop "Supersymmetries and Quantum Symmetries", Dubna, 2006

The Birman-Wenzl-Murakami algebra, considered as the quotient of the braid group algebra, possesses the commutative set of Jucys--Murphy elements. We show that the set of Jucys--Murphy elements is maximal commutative for the generic Birman-Wenzl-Murakami algebra and reconstruct the representation theory of the tower of Birman-Wenzl-Murakami algebras., Comment: Proceedings of International Workshop "Supersymmetries and Quantum Symmetries", Dubna, 2009

We consider the integrable open chain models formulated in terms of generators of the Hecke algebra. The spectrum of the Hamiltonians for the open Hecke chains of finite size with free boundary conditions is deduced for special (corner type) irreducible representations of the Hecke algebra., Comment: 7 pages, To appear in Rep. Math. Phys., in the proceedings of XVIth International Colloquium "Integrable Systems and Quantum Symmetries", Prague, Czech Republic, 14 - 16 June 2007

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., 9 pages

For a Hopf algebra A, we define the structures of differential complexes on two dual exterior Hopf algebras: 1) an exterior extension of A and 2) an exterior extension of the dual algebra A^*. The Heisenberg double of these two exterior Hopf algebras defines the differential algebra for the Cartan differential calculus on A. The first differential complex is an analog of the de Rham complex. In the situation when A^* is a universal enveloping of a Lie (super)algebra the second complex coincides with the standard complex. The differential is realized as an (anti)commutator with a BRST- operator Q. A recurrent relation which defines uniquely the operator Q is given. The BRST and anti-BRST operators are constructed explicitly and the Hodge decomposition theorem is formulated for the case of the quantum Lie algebra U_q(gl(N))., Comment: 20 pages, LaTeX, Lecture given at the Workshop on "Classical and Quantum Integrable Systems", 8 - 11 January, Protvino, Russia; corrected some typos

For a quantum Lie algebra $\Gamma$, let $\Gamma^\wedge$ be its exterior extension (the algebra $\Gamma^\wedge$ is canonically defined). We introduce a differential on the exterior extension algebra $\Gamma^\wedge$ which provides the structure of a complex on $\Gamma^{\wedge}$. In the situation when $\Gamma$ is a usual Lie algebra this complex coincides with the "standard complex". The differential is realized as a commutator with a (BRST) operator $Q$ in a larger algebra $\Gamma^\wedge[\Omega]$, with extra generators canonically conjugated to the exterior generators of $\Gamma^{\wedge}$. A recurrent relation which defines uniquely the operator $Q$ is given., Comment: 10 pages, LaTeX. Report given at XXIII Int. Colloquium on Group Theoretical Methods in Physics, July 31 - August 05, 2000, Dubna (Russia)

The zero modes of the chiral SU(n) WZNW model give rise to an intertwining quantum matrix algebra A generated by an n x n matrix a=(a^i_\alpha) (with noncommuting entries) and by rational functions of n commuting elements q^{p_i}. We study a generalization of the Fock space (F) representation of A for generic q (q not a root of unity) and demonstrate that it gives rise to a model of the quantum universal enveloping algebra U_q(sl_n), each irreducible representation entering F with multiplicity 1. For an integer level k the complex parameter q is an even root of unity, q^h=-1 (h=k+n) and the algebra A has an ideal I_h such that the factor algebra A_h = A/I_h is finite dimensional., Comment: 48 pages, LaTeX, uses amsfonts; final version to appear in J. Phys. A

The Cayley-Hamilton-Newton theorem - which underlies the Newton identities and the Cayley-Hamilton identity - is reviewed, first, for the classical matrices with commuting entries, second, for two q-matrix algebras, the RTT-algebra and the RLRL-algebra. The Cayley-Hamilton-Newton identities for these q-algebras are related by the factorization map. A class of algebras M(R,F) is presented. The algebras M(R,F) include the RTT-algebra and the RLRL-algebra as particular cases. The algebra M(R,F) is defined by a pair of compatible matrices R and F. The Cayley-Hamilton-Newton theorem for the algebras M(R,F) is stated. A nontrivial example of a compatible pair is given., LaTeX, 12 pages. Lecture given at the 3rd International Workshop on "Lie Theory and Its Applications in Physics - Lie III", 11 - 14 July 1999, Clausthal, Germany

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)

In the framework of the Drinfeld theory of twists in Hopf algebras we construct quantum matrix algebras which generalize the Reflection Equation and the RTT algebras. Finite-dimensional representations of these algebras related to the theory of nonultralocal spin chains are presented. The Cayley-Hamilton-Newton identities are demonstrated. These identities allow to define the quantum spectrum for the quantum matrices. We mention possible applications of the new quantum matrix algebras to constructions of noncommutative analogs of Minkowski space and quantum Poincar\'e algebras., Comment: 11 pages, LaTeX. Submitted to the Proceedings of the Intern. Seminar "Supersymmetries and Quantum Symmetries" (27-31 July, 1999, Dubna, Russia)

The q-generalizations of the two fundamental statements of matrix algebra -- the Cayley-Hamilton theorem and the Newton relations -- to the cases of quantum matrix algebras of an "RTT-" and of a "Reflection equation" types have been obtained in [2]-[6]. We construct a family of matrix identities which we call Cayley-Hamilton-Newton identities and which underlie the characteristic identity as well as the Newton relations for the RTT- and Reflection equation algebras, in the sence that both the characteristic identity and the Newton relations are direct consequences of the Cayley-Hamilton-Newton identities., Comment: 6 pages, submitted to the Proceedings of 7-th International Colloquium "Quantum Groups and Integrable Systems" (Prague, 18-20 June 1998)