13 research outputs found
Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities. I. Generalizations of the Capelli and Turnbull identities
We prove, by simple manipulation of commutators, two noncommutative
generalizations of the Cauchy-Binet formula for the determinant of a product.
As special cases we obtain elementary proofs of the Capelli identity from
classical invariant theory and of Turnbull's Capelli-type identities for
symmetric and antisymmetric matrices.Comment: LaTeX2e, 43 pages. Version 2 corrects an error in the statements of
Propositions 1.4 and 1.5 (see new Remarks in Section 4) and includes a Note
Added at the end of Section 1 comparing our work with that of Chervov et al
(arXiv:0901.0235
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
Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities II. Grassmann and quantum oscillator algebra representation
We prove that, for , , and matrices with entries in a
non-commutative ring such that ,
satisfying suitable commutation relations (in particular, is a Manin
matrix), the following identity holds: . Furthermore,
if also is a Manin matrix, . Notations: , , are respectively the bra and the ket of the ground state,
and the creation and annihilation operators of a quantum
harmonic oscillator, while and are Grassmann
variables in a Berezin integral. These results should be seen as a
generalization of the classical Cauchy-Binet formula, in which and are
null matrices, and of the non-commutative generalization, the Capelli identity,
in which and are identity matrices and
.Comment: 40 page
Quantum algebra of multiparameter Manin matrices
Multiparametric quantum semigroups are
generalization of the one-parameter general linear semigroups
, where and are
parameters satisfying certain conditions. In this paper, we study the algebra
of multiparametric Manin matrices using the R-matrix method. The systematic
approach enables us to obtain several classical identities such as Muir
identities, Newton's identities, Capelli-type identities, Cauchy-Binet's
identity both for determinant and permanent as well as a rigorous proof of the
MacMahon master equation for the quantum algebra of multiparametric Manin
matrices. Some of the generalized identities are also generalized to
multiparameter -Yangians.Comment: 31 pages; final versio
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
Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians
The classic Cayley identity states that
where is an matrix of indeterminates and is the corresponding matrix of partial derivatives. In this paper we present straightforward algebraic/combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (= exterior algebra) and Grassmann–Berezin integration. Among the new identities proven here are a pair of “diagonal-parametrized” Cayley identities, a pair of “Laplacian-parametrized” Cayley identities, and the “product-parametrized” and “border-parametrized” rectangular Cayley identities
Representing some non-representable matroids
We extend the notion of representation of a matroid to algebraic structures
that we call skew partial fields. Our definition of such representations
extends Tutte's definition, using chain groups. We show how such
representations behave under duality and minors, we extend Tutte's
representability criterion to this new class, and we study the generator
matrices of the chain groups. An example shows that the class of matroids
representable over a skew partial field properly contains the class of matroids
representable over a skew field.
Next, we show that every multilinear representation of a matroid can be seen
as a representation over a skew partial field.
Finally we study a class of matroids called quaternionic unimodular. We prove
a generalization of the Matrix Tree theorem for this class.Comment: 29 pages, 2 figure