65 research outputs found
Generators of simple modular Lie superalgebras
Let be one of the finite-dimensional simple graded Lie superalgebras of
Cartan type or over an algebraically closed
field of characteristic . In this paper we prove that can be generated
by one element except the ones of type , or in certain
exceptional cases, in which can be generated by two elements. As a
subsidiary result, we also prove that certain classical Lie superalgebras or
their relatives can be generated by one or two elements
Generators of simple Lie algebras in arbitrary characteristics
In this paper we study the minimal number of generators for simple Lie
algebras in characteristic 0 or p > 3. We show that any such algebra can be
generated by 2 elements. We also examine the 'one and a half generation'
property, i.e. when every non-zero element can be completed to a generating
pair. We show that classical simple algebras have this property, and that the
only simple Cartan type algebras of type W which have this property are the
Zassenhaus algebras.Comment: 26 pages, final version, to appear in Math. Z. Main improvements and
corrections in Section 4.
Differential systems associated with tableaux over Lie algebras
We give an account of the construction of exterior differential systems based
on the notion of tableaux over Lie algebras as developed in [Comm. Anal. Geom
14 (2006), 475-496; math.DG/0412169]. The definition of a tableau over a Lie
algebra is revisited and extended in the light of the formalism of the Spencer
cohomology; the question of involutiveness for the associated systems and their
prolongations is addressed; examples are discussed.Comment: 16 pages; to appear in: "Symmetries and Overdetermined Systems of
Partial Differential Equations" (M. Eastwood and W. Miller, Jr., eds.), IMA
Volumes in Mathematics and Its Applications, Springer-Verlag, New Yor
PU(2) monopoles and links of top-level Seiberg-Witten moduli spaces
This is the first of two articles in which we give a proof - for a broad
class of four-manifolds - of Witten's conjecture that the Donaldson and
Seiberg-Witten series coincide, at least through terms of degree less than or
equal to c-2, where c is a linear combination of the Euler characteristic and
signature of the four-manifold. This article is a revision of sections 1-3 of
an earlier version of the article dg-ga/9712005, now split into two parts,
while a revision of sections 4-7 of that earlier version appears in a recently
updated dg-ga/9712005. In the present article, we construct virtual normal
bundles for the Seiberg-Witten strata of the moduli space of PU(2) monopoles
and compute their Chern classes.Comment: Journal fur die Reine und Angewandte Mathematik, to appear; 64 page
Involution and Constrained Dynamics I: The Dirac Approach
We study the theory of systems with constraints from the point of view of the
formal theory of partial differential equations. For finite-dimensional systems
we show that the Dirac algorithm completes the equations of motion to an
involutive system. We discuss the implications of this identification for field
theories and argue that the involution analysis is more general and flexible
than the Dirac approach. We also derive intrinsic expressions for the number of
degrees of freedom.Comment: 28 pages, latex, no figure
Multimode solutions of first-order elliptic quasilinear systems obtained from Riemann invariants
Two new approaches to solving first-order quasilinear elliptic systems of
PDEs in many dimensions are proposed. The first method is based on an analysis
of multimode solutions expressible in terms of Riemann invariants, based on
links between two techniques, that of the symmetry reduction method and of the
generalized method of characteristics. A variant of the conditional symmetry
method for constructing this type of solution is proposed. A specific feature
of that approach is an algebraic-geometric point of view, which allows the
introduction of specific first-order side conditions consistent with the
original system of PDEs, leading to a generalization of the Riemann invariant
method for solving elliptic homogeneous systems of PDEs. A further
generalization of the Riemann invariants method to the case of inhomogeneous
systems, based on the introduction of specific rotation matrices, enables us to
weaken the integrability condition. It allows us to establish a connection
between the structure of the set of integral elements and the possibility of
constructing specific classes of simple mode solutions. These theoretical
considerations are illustrated by the examples of an ideal plastic flow in its
elliptic region and a system describing a nonlinear interaction of waves and
particles. Several new classes of solutions are obtained in explicit form,
including the general integral for the latter system of equations
Jacobson generators, Fock representations and statistics of sl(n+1)
The properties of A-statistics, related to the class of simple Lie algebras
sl(n+1) (Palev, T.D.: Preprint JINR E17-10550 (1977); hep-th/9705032), are
further investigated. The description of each sl(n+1) is carried out via
generators and their relations, first introduced by Jacobson. The related Fock
spaces W_p (p=1,2,...) are finite-dimensional irreducible sl(n+1)-modules. The
Pauli principle of the underlying statistics is formulated. In addition the
paper contains the following new results: (a) The A-statistics are interpreted
as exclusion statistics; (b) Within each W_p operators B(p)_1^\pm, ...,
B(p)_n^\pm, proportional to the Jacobson generators, are introduced. It is
proved that in an appropriate topology the limit of B(p)_i^\pm for p going to
infinity is equal to B_i^\pm, where B_i^\pm are Bose creation and annihilation
operators; (c) It is shown that the local statistics of the degenerated
hard-core Bose models and of the related Heisenberg spin models is p=1
A-statistics.Comment: LaTeX-file, 33 page
Holomorphic potentials for graded D-branes
We discuss gauge-fixing, propagators and effective potentials for topological
A-brane composites in Calabi-Yau compactifications. This allows for the
construction of a holomorphic potential describing the low-energy dynamics of
such systems, which generalizes the superpotentials known from the ungraded
case. Upon using results of homotopy algebra, we show that the string field and
low energy descriptions of the moduli space agree, and that the deformations of
such backgrounds are described by a certain extended version of `off-shell
Massey products' associated with flat graded superbundles. As examples, we
consider a class of graded D-brane pairs of unit relative grade. Upon computing
the holomorphic potential, we study their moduli space of composites. In
particular, we give a general proof that such pairs can form acyclic
condensates, and, for a particular case, show that another branch of their
moduli space describes condensation of a two-form.Comment: 47 pages, 7 figure
Compactifications and algebraic completions of Limit groups
In this paper we consider the existence of dense embeddings of Limit groups
in locally compact groups generalizing earlier work of Breuillard, Gelander,
Souto and Storm [GBSS] where surface groups were considered. Our main results
are proved in the context of compact groups and algebraic groups over local
fields. In addition we prove a generalization of the classical Baumslag lemma
which is a useful tool for generating eventually faithful sequences of
homomorphisms. The last section is dedicated to correct a mistake from [BGSS]
and to get rid of the even genus assumption.Comment: v2: Substantial changes to sections 7 and 8.2. Typos corrected.
References added. v3: Acknowledgement correcte
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