697 research outputs found
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
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
On the spatial structure of the Perseids meteor stream
The analysis of radar observations of the Perseid meteor stream conducted in an ionospherical laboratory in the period from 1964 to 1981 is presented. The Perseids meteor rates were determined by the fluctuation method. Analysis of their hourly distributions showed that the stream maximum position is different for different years, i.e., the stream nodal position is constantly changing. The results of the analysis are presented and discussed
SOS model partition function and the elliptic weight functions
We generalize a recent observation [arXiv:math/0610433] that the partition
function of the 6-vertex model with domain-wall boundary conditions can be
obtained by computing the projections of the product of the total currents in
the quantum affine algebra in its current
realization. A generalization is proved for the the elliptic current algebra
[arXiv:q-alg/9703018,arXiv:q-alg/9601022]. The projections of the product of
total currents are calculated explicitly and are represented as integral
transforms of the product of the total currents. We prove that the kernel of
this transform is proportional to the partition function of the SOS model with
domain-wall boundary conditions.Comment: 21 pages, 5 figures, requires iopart packag
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