130 research outputs found
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 one can
associate pairings and , 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
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