Generalized Cayley-Hamilton-Newton identities

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

Q-multilinear Algebra

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.Comment: 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, German

Two-component abelian sandpile models

In one-component abelian sandpile models, the toppling probabilities are independent quantities. This is not the case in multi-component models. The condition of associativity of the underlying abelian algebras impose nonlinear relations among the toppling probabilities. These relations are derived for the case of two-component quadratic abelian algebras. We show that abelian sandpile models with two conservation laws have only trivial avalanches.Comment: Final version. To appear in Phys.Rev.

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

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

Cayley-Hamilton Theorem for Symplectic Quantum Matrix Algebras

We establish the analogue of the Cayley--Hamilton theorem for the quantum matrix algebras of the symplectic type.Comment: arXiv admin note: text overlap with arXiv:math/051161