On the solutions of Knizhnik-Zamolodchikov system

Sakhnovich, Lev

doi:10.2478/s11533-008-0059-z

Cited by 3 publications

(2 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular it is important to establish that its solutions are rational functions (see [MTV05]). The proposition above relates this question to the rationality of KZ-solutions which is a more natural question (see [Sa06]).…”

Section: Application To the Knizhnik-zamolodchikov Equationmentioning

confidence: 99%

Manin matrices and Talalaev's formula

Chervov¹,

Falqui²

2008

J. Phys. A: Math. Theor.

105

View full text Add to dashboard Cite

In this paper we study properties of Lax and Transfer matrices associated with quantum integrable systems. Our point of view stems from the fact that their elements satisfy special commutation properties, considered by Yu. I. Manin some twenty years ago at the beginning of Quantum Group Theory. They are the commutation properties of matrix elements of linear homomorphisms between polynomial rings; more explicitly they read: 1) elements in the same column commute; 2) commutators of the cross terms are equal:The main aim of this paper is twofold: on the one hand we observe and prove that such matrices (which we call Manin matrices for short) behave almost as well as matrices with commutative elements. Namely theorems of linear algebra (e.g., a natural definition of the determinant, the Cayley-Hamilton theorem, the Newton identities and so on and so forth) have a straightforward counterpart in the case of Manin matrices.On the other hand, we remark that such matrices are somewhat ubiquitous in the theory of quantum integrability. For instance, Manin matrices (and their q-analogs) include matrices satisfying the Yang-Baxter relation "RTT=TTR" and the so-called Cartier-Foata matrices. Also, they enter Talalaev's remarkable formulas: det(∂ z − L Gaudin (z)), det(1 − e −∂z T Y angian (z)) for the "quantum spectral curve", and appear in separation of variables problem and Capelli identities.We show that theorems of linear algebra, after being established for such matrices, have various applications to quantum integrable systems and Lie algebras, e.g in the construction of new generators in Z(U crit ( gl n )) (and, in general, in the construction of quantum conservation laws), in the Knizhnik-Zamolodchikov equation, and in the problem of Wick ordering.We also discuss applications to the separation of variables problem, new Capelli identities and the Langlands correspondence.

show abstract