QUANTUM MINORS ASSOCIATED TO YANG-BAXTER OPERATORS

Abstract

The graded algebra of invariant elements under the representations induced by the Yang-Baxter operators constructed in our previous work is used here to generalize the notion of minors associated to a matrix with coefficients in any -commutative algebra . We call them quantum minors to be reminded that they are obtained in a non commutative context. Quantum minors are elements of A appearing as coefficients of the algebra endomorphism induced on the graded algebra of invariant elements by a given A-module endomorphism. It turns out that quantum minors are multiplicative and satisfy a Laplace-like expansion. It is shown that the usual determinant, the various -graded determinants, and the quantum and multiparametric quantum determinants are special cases of quantum minors, depending on the choice of the commutation factor .

Acknowledgments

OASV has been partially supported by CONACyT grants 28491-E, and E130.1880, and MBRLM grant 1411-99. CVM has been partially supported by DGICyT grant PB94-1196 and Acció Integrada 1998-99 Comissionat per a Universitats i Recerca, Generalitat de Catalunya. Both authors would like to thank the referee for his/her enormously valuable comments, as they led us to a clearer and more simplified exposition.