2,413 research outputs found
Multigraded Hilbert Series of noncommutative modules
In this paper, we propose methods for computing the Hilbert series of
multigraded right modules over the free associative algebra. In particular, we
compute such series for noncommutative multigraded algebras. Using results from
the theory of regular languages, we provide conditions when the methods are
effective and hence the sum of the Hilbert series is a rational function.
Moreover, a characterization of finite-dimensional algebras is obtained in
terms of the nilpotency of a key matrix involved in the computations. Using
this result, efficient variants of the methods are also developed for the
computation of Hilbert series of truncated infinite-dimensional algebras whose
(non-truncated) Hilbert series may not be rational functions. We consider some
applications of the computation of multigraded Hilbert series to algebras that
are invariant under the action of the general linear group. In fact, in this
case such series are symmetric functions which can be decomposed in terms of
Schur functions. Finally, we present an efficient and complete implementation
of (standard) graded and multigraded Hilbert series that has been developed in
the kernel of the computer algebra system Singular. A large set of tests
provides a comprehensive experimentation for the proposed algorithms and their
implementations.Comment: 28 pages, to appear in Journal of Algebr
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
Morita Duality and Noncommutative Wilson Loops in Two Dimensions
We describe a combinatorial approach to the analysis of the shape and
orientation dependence of Wilson loop observables on two-dimensional
noncommutative tori. Morita equivalence is used to map the computation of loop
correlators onto the combinatorics of non-planar graphs. Several
nonperturbative examples of symmetry breaking under area-preserving
diffeomorphisms are thereby presented. Analytic expressions for correlators of
Wilson loops with infinite winding number are also derived and shown to agree
with results from ordinary Yang-Mills theory.Comment: 32 pages, 9 figures; v2: clarifying comments added; Final version to
be published in JHE
Morita Duality and Large-N Limits
We study some dynamical aspects of gauge theories on noncommutative tori. We
show that Morita duality, combined with the hypothesis of analyticity as a
function of the noncommutativity parameter Theta, gives information about
singular large-N limits of ordinary U(N) gauge theories, where the large-rank
limit is correlated with the shrinking of a two-torus to zero size. We study
some non-perturbative tests of the smoothness hypothesis with respect to Theta
in theories with and without supersymmetry. In the supersymmetric case this is
done by adapting Witten's index to the present situation, and in the
nonsupersymmetric case by studying the dependence of energy levels on the
instanton angle. We find that regularizations which restore supersymmetry at
high energies seem to preserve Theta-smoothness whereas nonsupersymmetric
asymptotically free theories seem to violate it. As a final application we use
Morita duality to study a recent proposal of Susskind to use a noncommutative
Chern-Simons gauge theory as an effective description of the Fractional Hall
Effect. In particular we obtain an elegant derivation of Wen's topological
order.Comment: 41 pages, Harvmac. Some corrections to section 6.3. Comments added on
Hall Effec
Computing with rational symmetric functions and applications to invariant theory and PI-algebras
Let the formal power series f in d variables with coefficients in an
arbitrary field be a symmetric function decomposed as a series of Schur
functions, and let f be a rational function whose denominator is a product of
binomials of the form (1 - monomial). We use a classical combinatorial method
of Elliott of 1903 further developed in the Partition Analysis of MacMahon in
1916 to compute the generating function of the multiplicities (i.e., the
coefficients) of the Schur functions in the expression of f. It is a rational
function with denominator of a similar form as f. We apply the method to
several problems on symmetric algebras, as well as problems in classical
invariant theory, algebras with polynomial identities, and noncommutative
invariant theory.Comment: 37 page
Geometry of free loci and factorization of noncommutative polynomials
The free singularity locus of a noncommutative polynomial f is defined to be
the sequence of hypersurfaces. The main
theorem of this article shows that f is irreducible if and only if is
eventually irreducible. A key step in the proof is an irreducibility result for
linear pencils. Apart from its consequences to factorization in a free algebra,
the paper also discusses its applications to invariant subspaces in
perturbation theory and linear matrix inequalities in real algebraic geometry.Comment: v2: 32 pages, includes a table of content
Instantons, Fluxons and Open Gauge String Theory
We use the exact instanton expansion to illustrate various string
characteristics of noncommutative gauge theory in two dimensions. We analyse
the spectrum of the model and present some evidence in favour of Hagedorn and
fractal behaviours. The decompactification limit of noncommutative torus
instantons is shown to map in a very precise way, at both the classical and
quantum level, onto fluxon solutions on the noncommutative plane. The
weak-coupling singularities of the usual Gross-Taylor string partition function
for QCD on the torus are studied in the instanton representation and its double
scaling limit, appropriate for the mapping onto noncommutative gauge theory, is
shown to be a generating function for the volumes of the principal moduli
spaces of holomorphic differentials. The noncommutative deformation of this
moduli space geometry is described and appropriate open string interpretations
are proposed in terms of the fluxon expansion.Comment: 70 pages, 6 figure
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