9,162 research outputs found
Dirac operators and the Very Strange Formula for Lie superalgebras
Using a super-affine version of Kostant's cubic Dirac operator, we prove a
very strange formula for quadratic finite-dimensional Lie superalgebras with a
reductive even subalgebra.Comment: Latex file, 25 pages. A few misprints corrected. To appear in the
forthcoming volume "Advances in Lie Superalgebras", Springer INdAM Serie
Cluster algebras of type
In this paper we study cluster algebras \myAA of type . We solve
the recurrence relations among the cluster variables (which form a T--system of
type ). We solve the recurrence relations among the coefficients of
\myAA (which form a Y--system of type ). In \myAA there is a
natural notion of positivity. We find linear bases \BB of \myAA such that
positive linear combinations of elements of \BB coincide with the cone of
positive elements. We call these bases \emph{atomic bases} of \myAA. These
are the analogue of the "canonical bases" found by Sherman and Zelevinsky in
type . Every atomic basis consists of cluster monomials together
with extra elements. We provide explicit expressions for the elements of such
bases in every cluster. We prove that the elements of \BB are parameterized
by \ZZ^3 via their --vectors in every cluster. We prove that the
denominator vector map in every acyclic seed of \myAA restricts to a
bijection between \BB and \ZZ^3. In particular this gives an explicit
algorithm to determine the "virtual" canonical decomposition of every element
of the root lattice of type . We find explicit recurrence relations
to express every element of \myAA as linear combinations of elements of
\BB.Comment: Latex, 40 pages; Published online in Algebras and Representation
Theory, springer, 201
On classical finite and affine W-algebras
This paper is meant to be a short review and summary of recent results on the
structure of finite and affine classical W-algebras, and the application of the
latter to the theory of generalized Drinfeld-Sokolov hierarchies.Comment: 12 page
Equivariant map superalgebras
Suppose a group acts on a scheme and a Lie superalgebra
. The corresponding equivariant map superalgebra is the Lie
superalgebra of equivariant regular maps from to . We
classify the irreducible finite dimensional modules for these superalgebras
under the assumptions that the coordinate ring of is finitely generated,
is finite abelian and acts freely on the rational points of , and
is a basic classical Lie superalgebra (or ,
, if is trivial). We show that they are all (tensor products
of) generalized evaluation modules and are parameterized by a certain set of
equivariant finitely supported maps defined on . Furthermore, in the case
that the even part of is semisimple, we show that all such
modules are in fact (tensor products of) evaluation modules. On the other hand,
if the even part of is not semisimple (more generally, if
is of type I), we introduce a natural generalization of Kac
modules and show that all irreducible finite dimensional modules are quotients
of these. As a special case, our results give the first classification of the
irreducible finite dimensional modules for twisted loop superalgebras.Comment: 27 pages. v2: Section numbering changed to match published version.
Other minor corrections. v3: Minor corrections (see change log at end of
introduction
Dynamics of the chiral phase transition from AdS/CFT duality
We use Lorentzian signature AdS/CFT duality to study a first order phase
transition in strongly coupled gauge theories which is akin to the chiral phase
transition in QCD. We discuss the relation between the latent heat and the
energy (suitably defined) of the component of a D-brane which lies behind the
horizon at the critical temperature. A numerical simulation of a dynamical
phase transition in an expanding, cooling Quark-Gluon plasma produced in a
relativistic collision is carried out.Comment: 30 pages, 5 figure
Unifying N=5 and N=6
We write the Lagrangian of the general N=5 three-dimensional superconformal
Chern-Simons theory, based on a basic Lie superalgebra, in terms of our
recently introduced N=5 three-algebras. These include N=6 and N=8
three-algebras as special cases. When we impose an antisymmetry condition on
the triple product, the supersymmetry automatically enhances, and the N=5
Lagrangian reduces to that of the well known N=6 theory, including the ABJM and
ABJ models.Comment: 19 pages. v2: Published version. Minor typos corrected, references
adde
Flavor-symmetry Breaking with Charged Probes
We discuss the recombination of brane/anti-brane pairs carrying brane
charge in . These configurations are dual to co-dimension one
defects in the super-Yang-Mills description. Due to their
charge, these defects are actually domain walls in the dual gauge theory,
interpolating between vacua of different gauge symmetry. A pair of unjoined
defects each carry localized dimensional fermions and possess a global
flavor symmetry while the recombined brane/anti-brane pairs
exhibit only a diagonal U(N). We study the thermodynamics of this
flavor-symmetry breaking under the influence of external magnetic field.Comment: 21 pages, 10 figure
Investigation of the complex dynamics and regime control in Pierce diode with the delay feedback
In this paper the dynamics of Pierce diode with overcritical current under
the influence of delay feedback is investigated. The system without feedback
demonstrates complex behaviour including chaotic regimes. The possibility of
oscillation regime control depending on the delay feedback parameter values is
shown. Also the paper describes construction of a finite-dimensional model of
electron beam behaviour, which is based on the Galerkin approximation by linear
modes expansion. The dynamics of the model is close to the one given by the
distributed model.Comment: 18 pages, 6 figures, published in Int. J. Electronics. 91, 1 (2004)
1-1
Second-order -regularity in nonlinear elliptic problems
A second-order regularity theory is developed for solutions to a class of
quasilinear elliptic equations in divergence form, including the -Laplace
equation, with merely square-integrable right-hand side. Our results amount to
the existence and square integrability of the weak derivatives of the nonlinear
expression of the gradient under the divergence operator. This provides a
nonlinear counterpart of the classical -coercivity theory for linear
problems, which is missing in the existing literature. Both local and global
estimates are established. The latter apply to solutions to either Dirichlet or
Neumann boundary value problems. Minimal regularity on the boundary of the
domain is required. If the domain is convex, no regularity of its boundary is
needed at all
- …