9,273 research outputs found
Gradings of non-graded Hamiltonian Lie algebras
A thin Lie algebra is a Lie algebra graded over the positive integers
satisfying a certain narrowness condition. We describe several cyclic grading
of the modular Hamiltonian Lie algebras H(2\colon\n;\omega_2) (of dimension
one less than a power of ) from which we construct infinite-dimensional thin
Lie algebras. In the process we provide an explicit identification of
H(2\colon\n;\omega_2) with a Block algebra. We also compute its second
cohomology group and its derivation algebra (in arbitrary prime
characteristic).Comment: 36 pages, to be published in J. Austral. Math. Soc. Ser.
Nottingham Lie algebras with diamonds of finite and infinite type
We consider a class of infinite-dimensional, modular, graded Lie algebras,
which includes the graded Lie algebra associated to the Nottingham group with
respect to its lower central series. We identify two subclasses of Nottingham
Lie algebras as loop algebras of finite-dimensional simple Lie algebras of
Hamiltonian Cartan type. A property of Laguerre polynomials of derivations,
which is related to toral switching, plays a crucial role in our constructions.Comment: 17 pages; minor changes from the previous versio
Lie 2-algebra models
In this paper, we begin the study of zero-dimensional field theories with
fields taking values in a semistrict Lie 2-algebra. These theories contain the
IKKT matrix model and various M-brane related models as special cases. They
feature solutions that can be interpreted as quantized 2-plectic manifolds. In
particular, we find solutions corresponding to quantizations of R^3, S^3 and a
five-dimensional Hpp-wave. Moreover, by expanding a certain class of Lie
2-algebra models around the solution corresponding to quantized R^3, we obtain
higher BF-theory on this quantized space.Comment: 47 pages, presentation improved, version published in JHE
Lie n-algebras of BPS charges
We uncover higher algebraic structures on Noether currents and BPS charges.
It is known that equivalence classes of conserved currents form a Lie algebra.
We show that at least for target space symmetries of higher parameterized
WZW-type sigma-models this naturally lifts to a Lie (p+1)-algebra structure on
the Noether currents themselves. Applied to the Green-Schwarz-type action
functionals for super p-brane sigma-models this yields super Lie (p+1)-algebra
refinements of the traditional BPS brane charge extensions of supersymmetry
algebras. We discuss this in the generality of higher differential geometry,
where it applies also to branes with (higher) gauge fields on their
worldvolume. Applied to the M5-brane sigma-model we recover and properly
globalize the M-theory super Lie algebra extension of 11-dimensional
superisometries by 2-brane and 5-brane charges. Passing beyond the
infinitesimal Lie theory we find cohomological corrections to these charges in
higher analogy to the familiar corrections for D-brane charges as they are
lifted from ordinary cohomology to twisted K-theory. This supports the proposal
that M-brane charges live in a twisted cohomology theory.Comment: 19 pages, v2: references added, details of the main computation
spelled ou
Contraction of broken symmetries via Kac-Moody formalism
I investigate contractions via Kac-Moody formalism. In particular, I show how
the symmetry algebra of the standard 2-D Kepler system, which was identified by
Daboul and Slodowy as an infinite-dimensional Kac-Moody loop algebra, and was
denoted by , gets reduced by the symmetry breaking term,
defined by the Hamiltonian For this I
define two symmetry loop algebras , by
choosing the `basic generators' differently. These
can be mapped isomorphically onto subalgebras of , of
codimension 2 or 3, revealing the reduction of symmetry. Both factor algebras
, relative to the corresponding
energy-dependent ideals , are isomorphic to
and for , respectively, just as for the
pure Kepler case. However, they yield two different non-standard contractions
as , namely to the Heisenberg-Weyl algebra or to an abelian Lie algebra, instead of the Euclidean algebra
for the pure Kepler case. The above example suggests a
general procedure for defining generalized contractions, and also illustrates
the {\em `deformation contraction hysteresis'}, where contraction which involve
two contraction parameters can yield different contracted algebras, if the
limits are carried out in different order.Comment: 21 pages, 1 figur
Integrable Systems and Factorization Problems
The present lectures were prepared for the Faro International Summer School
on Factorization and Integrable Systems in September 2000. They were intended
for participants with the background in Analysis and Operator Theory but
without special knowledge of Geometry and Lie Groups. In order to make the main
ideas reasonably clear, I tried to use only matrix algebras such as
and its natural subalgebras; Lie groups used are either GL(n)
and its subgroups, or loop groups consisting of matrix-valued functions on the
circle (possibly admitting an extension to parts of the Riemann sphere). I hope
this makes the environment sufficiently easy to live in for an analyst. The
main goal is to explain how the factorization problems (typically, the matrix
Riemann problem) generate the entire small world of Integrable Systems along
with the geometry of the phase space, Hamiltonian structure, Lax
representations, integrals of motion and explicit solutions. The key tool will
be the \emph{% classical r-matrix} (an object whose other guise is the
well-known Hilbert transform). I do not give technical details, unless they may
be exposed in a few lines; on the other hand, all motivations are given in full
scale whenever possible.Comment: LaTeX 2.09, 69 pages. Introductory lectures on Integrable systems,
Classical r-matrices and Factorization problem
Real Formulations of Complex Gravity and a Complex Formulation of Real Gravity
Two gauge and diffeomorphism invariant theories on the Yang-Mills phase space
are studied. They are based on the Lie-algebras and
-- the loop-algebra of . Although the theories are
manifestly real, they can both be reformulated to show that they describe
complex gravity and an infinite number of copies of complex gravity,
respectively. The connection to real gravity is given. For these theories, the
reality conditions in the conventional Ashtekar formulation are represented by
normal constraint-like terms.Comment: 23 pages, CGPG-94/4-
- …