1,009 research outputs found
Double Poisson brackets on free associative algebras
We discuss double Poisson structures in sense of M. Van den Bergh on free
associative algebras focusing on the case of quadratic Poisson brackets. We
establish their relations with an associative version of Young-Baxter
equations, we study a bi-hamiltonian property of the linear-quadratic pencil of
the double Poisson structure and propose a classification of the quadratic
double Poisson brackets in the case of the algebra with two free generators.
Many new examples of quadratic double Poisson brackets are proposed.Comment: 19 pages, late
Hilbert Schemes, Separated Variables, and D-Branes
We explain Sklyanin's separation of variables in geometrical terms and
construct it for Hitchin and Mukai integrable systems. We construct Hilbert
schemes of points on for \Sigma = {\IC}, {\IC}^{*} or elliptic
curve, and on and show that their complex deformations
are integrable systems of Calogero-Sutherland-Moser type. We present the
hyperk\"ahler quotient constructions for Hilbert schemes of points on cotangent
bundles to the higher genus curves, utilizing the results of Hurtubise,
Kronheimer and Nakajima. Finally we discuss the connections to physics of
-branes and string duality.Comment: harvmac, 27 pp. big mode; v2. typos and references correcte
Algebraic properties of Manin matrices 1
We study a class of matrices with noncommutative entries, which were first
considered by Yu. I. Manin in 1988 in relation with quantum group theory. They
are defined as "noncommutative endomorphisms" of a polynomial algebra. More
explicitly their defining conditions read: 1) elements in the same column
commute; 2) commutators of the cross terms are equal: (e.g. ). The basic claim
is that despite noncommutativity many theorems of linear algebra hold true for
Manin matrices in a form identical to that of the commutative case. Moreover in
some examples the converse is also true. The present paper gives a complete
list and detailed proofs of algebraic properties of Manin matrices known up to
the moment; many of them are new. In particular we present the formulation in
terms of matrix (Leningrad) notations; provide complete proofs that an inverse
to a M.m. is again a M.m. and for the Schur formula for the determinant of a
block matrix; we generalize the noncommutative Cauchy-Binet formulas discovered
recently [arXiv:0809.3516], which includes the classical Capelli and related
identities. We also discuss many other properties, such as the Cramer formula
for the inverse matrix, the Cayley-Hamilton theorem, Newton and
MacMahon-Wronski identities, Plucker relations, Sylvester's theorem, the
Lagrange-Desnanot-Lewis Caroll formula, the Weinstein-Aronszajn formula, some
multiplicativity properties for the determinant, relations with
quasideterminants, calculation of the determinant via Gauss decomposition,
conjugation to the second normal (Frobenius) form, and so on and so forth. We
refer to [arXiv:0711.2236] for some applications.Comment: 80 page
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