45 research outputs found
Noncommutative marked surfaces
The aim of the paper is to attach a noncommutative cluster-like structure to
each marked surface . This is a noncommutative algebra generated by "noncommutative geodesics" between marked points
subject to certain triangle relations and noncommutative analogues of
Ptolemy-Pl\"ucker relations. It turns out that the algebra exhibits a noncommutative Laurent Phenomenon with respect to any
triangulation of , which confirms its "cluster nature". As a surprising
byproduct, we obtain a new topological invariant of , which is a free
or a 1-relator group easily computable in terms of any triangulation of
. Another application is the proof of Laurentness and positivity of
certain discrete noncommutative integrable systems.Comment: 49 pages, AmsLaTex, some typos are corrected and pictures updated, to
appear in Advances in Mathematic
Quasideterminants
The determinant is a main organizing tool in commutative linear algebra. In
this review we present a theory of the quasideterminants defined for matrices
over a division algebra. We believe that the notion of quasideterminants should
be one of main organizing tools in noncommutative algebra giving them the same
role determinants play in commutative algebra.Comment: amstex; final version; to appear in Advances in Mat
Cohomology of Lie superalgebras and of their generalizations
The cohomology groups of Lie superalgebras and, more generally, of color Lie
algebras, are introduced and investigated. The main emphasis is on the case
where the module of coefficients is non-trivial. Two general propositions are
proved, which help to calculate the cohomology groups. Several examples are
included to show the peculiarities of the super case. For L = sl(1|2), the
cohomology groups H^1(L,V) and H^2(L,V), with V a finite-dimensional simple
graded L-module, are determined, and the result is used to show that
H^2(L,U(L)) (with U(L) the enveloping algebra of L) is trivial. This implies
that the superalgebra U(L) does not admit of any non-trivial formal
deformations (in the sense of Gerstenhaber). Garland's theory of universal
central extensions of Lie algebras is generalized to the case of color Lie
algebras.Comment: 50 pages, Latex, no figures. In the revised version the proof of
Lemma 5.1 is greatly simplified, some references are added, and a pertinent
result on sl(m|1) is announced. To appear in the Journal of Mathematical
Physic
Manin matrices and Talalaev's formula
We study special class of matrices with noncommutative entries and
demonstrate their various applications in integrable systems theory. They
appeared in Yu. Manin's works in 87-92 as linear homomorphisms between
polynomial rings; more explicitly they read: 1) elements in the same column
commute; 2) commutators of the cross terms are equal: (e.g. ). We claim
that such matrices 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) holds true for them.
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
hep-th/0404153 remarkable formulas: ,
det(1-e^{-\p}T_{Yangian}(z)) for the "quantum spectral curve", etc. 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 (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.Comment: 40 pages, V2: exposition reorganized, some proofs added, misprints
e.g. in Newton id-s fixed, normal ordering convention turned to standard one,
refs. adde
Fredholm determinants and pole-free solutions to the noncommutative Painleve' II equation
We extend the formalism of integrable operators a' la
Its-Izergin-Korepin-Slavnov to matrix-valued convolution operators on a
semi-infinite interval and to matrix integral operators with a kernel of the
form E_1^T(x) E_2(y)/(x+y) thus proving that their resolvent operators can be
expressed in terms of solutions of some specific Riemann-Hilbert problems. We
also describe some applications, mainly to a noncommutative version of
Painleve' II (recently introduced by Retakh and Rubtsov), a related
noncommutative equation of Painleve' type. We construct a particular family of
solutions of the noncommutative Painleve' II that are pole-free (for real
values of the variables) and hence analogous to the Hastings-McLeod solution of
(commutative) Painleve' II. Such a solution plays the same role as its
commutative counterpart relative to the Tracy-Widom theorem, but for the
computation of the Fredholm determinant of a matrix version of the Airy kernel.Comment: 46 pages, no figures (oddly
Factorizations of Elements in Noncommutative Rings: A Survey
We survey results on factorizations of non zero-divisors into atoms
(irreducible elements) in noncommutative rings. The point of view in this
survey is motivated by the commutative theory of non-unique factorizations.
Topics covered include unique factorization up to order and similarity, 2-firs,
and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and
Jordan and generalizations thereof. We recall arithmetical invariants for the
study of non-unique factorizations, and give transfer results for arithmetical
invariants in matrix rings, rings of triangular matrices, and classical maximal
orders as well as classical hereditary orders in central simple algebras over
global fields.Comment: 50 pages, comments welcom
Local BRST cohomology in (non-)Lagrangian field theory
Some general theorems are established on the local BRST cohomology for not
necessarily Lagrangian gauge theories. Particular attention is given to the
BRST groups with direct physical interpretation. Among other things, the groups
of rigid symmetries and conservation laws are shown to be still connected,
though less tightly than in the Lagrangian theory. The connection is provided
by the elements of another local BRST cohomology group whose elements are
identified with Lagrange structures. This extends the cohomological formulation
of the Noether theorem beyond the scope of Lagrangian dynamics. We show that
each integrable Lagrange structure gives rise to a Lie bracket in the space of
conservation laws, which generalizes the Dickey bracket of conserved currents
known in Lagrangian field theory. We study the issues of existence and
uniqueness of the local BRST complex associated with a given set of field
equations endowed with a compatible Lagrange structure. Contrary to the usual
BV formalism, such a complex does not always exist for non-Lagrangian dynamics,
and when exists it is by no means unique. The ambiguity and obstructions are
controlled by certain cohomology classes, which are all explicitly identified.Comment: 37 pages, 1 figure, minor corrections, references adde
Lie algebras and Lie groups over noncommutative rings
The aim of this paper is to introduce and study Lie algebras and Lie groups
over noncommutative rings. For any Lie algebra sitting inside an
associative algebra and any associative algebra \FF we introduce and
study the algebra (\gg,A)(\FF), which is the Lie subalgebra of \FF \otimes
A generated by \FF \otimes \gg. In many examples is the universal
enveloping algebra of . Our description of the algebra (\gg,A)(\FF) has
a striking resemblance to the commutator expansions of \FF used by M.
Kapranov in his approach to noncommutative geometry. To each algebra (\gg,
A)(\FF) we associate a ``noncommutative algebraic'' group which naturally acts
on (\gg,A)(\FF) by conjugations and conclude the paper with some examples of
such groups.Comment: Introduction is improved and some typos corrected. To appear in
"Advances