124 research outputs found
Algebraic properties of Manin matrices II: q-analogues and integrable systems
We study a natural q-analogue of a class of matrices with noncommutative
entries, which were first considered by Yu. I. Manin in 1988 in relation with
quantum group theory, (called Manin Matrices in [5]) . These matrices we shall
call q-Manin matrices(qMMs). They are defined, in the 2x2 case, by the
relations M_21 M_12 = q M_12 M_21; M_22 M_12 = q M_12 M_22; [M_11;M_22] = 1/q
M_21 M_12 - q M_12 M_21: They were already considered in the literature,
especially in connection with the q-Mac Mahon master theorem [16], and the
q-Sylvester identities [25]. The main aim of the present paper is to give a
full list and detailed proofs of algebraic properties of qMMs known up to the
moment and, in particular, to show that most of the basic theorems of linear
algebras (e.g., Jacobi ratio theorems, Schhur complement, the Cayley-Hamilton
theorem and so on and so forth) have a straightforward counterpart for q-Manin
matrices. We also show how this classs of matrices ?ts within the theory of
quasi-determninants of Gel'fand-Retakh and collaborators (see, e.g., [17]). In
the last sections of the paper, we frame our definitions within the tensorial
approach to non-commutative matrices of the Leningrad school, and we show how
the notion of q-Manin matrix is related to theory of Quantum Integrable
Systems.Comment: 62 pages, v.2 cosmetic changes, typos fixe
Groupes quantiques associes aux courbes rationnelles et elliptiques et leurs applications
The thesis was defended by the author in University of Angers (France). It
consists of four parts. The fist part (in French) is introductory and is
devoted to relation between quantum groups, integrable systems and statistical
models. In the second part (in English) the transition function of the periodic
Toda chain is interpreted in terms of the formalism of rational Lax operators.
In the third part (in French) one compares two elliptic quantum groups and one
conclude that they belong to two different bialgebra categories. The fourth
part (in English) contains a construction of the partition function of the SOS
model in terms of the projections of an elliptic quantum group.Comment: PhD thesis. Languages: French, Englis
Classical elliptic current algebras. I
In this paper we discuss classical elliptic current algebras and show that there are two different choices of commutative test function algebras related to a complex torus leading totwo different elliptic current algebras. Quantization of these classical current algebras gives rise to two classes of quantized dynamical quasi-Hopf current algebras studied by Enriquez, Felder and Rubtsov and by Arnaudon, Buffenoir, Ragoucy, Roche, Jimbo, Konno, Odake and Shiraishi
Algebraic properties of Manin matrices II: q-analogues and integrable systems
We study a natural q-analogue of a class of matrices with non-commutative entries, which were first considered by Yu.I. Manin in 1988 in relation with quantum group theory, (called Manin matrices in [5]). We call these q-analogues q-Manin matrices . These matrices are defined, in the 2×22×2 case by the following relations among their matrix entries:
M21M12=qM12M21, M22M12 = qM12M22
[M11,M22]=q-1M21M12-qM12M21
They were already considered in the literature, especially in connection with the q-MacMahon master theorem [10], and the q-Sylvester identities [22]. The main aim of the present paper is to give a full list and detailed proofs of the algebraic properties of q-Manin matrices known up to the moment and, in particular, to show that most of the basic theorems of linear algebras (e.g., Jacobi ratio theorems, Schur complement, the Cayley–Hamilton theorem and so on and so forth) have a straightforward counterpart for such a class of matrices. We also show how q-Manin matrices fit within the theory of quasideterminants of Gelfand–Retakh and collaborators (see, e.g., [11]). We frame our definitions within the tensorial approach to non-commutative matrices of the Leningrad school in the last sections. We finally discuss how the notion of q-Manin matrix is related to theory of Quantum Integrable Systems
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