47 research outputs found
Projective Representations of the Inhomogeneous Hamilton Group: Noninertial Symmetry in Quantum Mechanics
Symmetries in quantum mechanics are realized by the projective
representations of the Lie group as physical states are defined only up to a
phase. A cornerstone theorem shows that these representations are equivalent to
the unitary representations of the central extension of the group. The
formulation of the inertial states of special relativistic quantum mechanics as
the projective representations of the inhomogeneous Lorentz group, and its
nonrelativistic limit in terms of the Galilei group, are fundamental examples.
Interestingly, neither of these symmetries includes the Weyl-Heisenberg group;
the hermitian representations of its algebra are the Heisenberg commutation
relations that are a foundation of quantum mechanics. The Weyl-Heisenberg group
is a one dimensional central extension of the abelian group and its unitary
representations are therefore a particular projective representation of the
abelian group of translations on phase space. A theorem involving the
automorphism group shows that the maximal symmetry that leaves invariant the
Heisenberg commutation relations are essentially projective representations of
the inhomogeneous symplectic group. In the nonrelativistic domain, we must also
have invariance of Newtonian time. This reduces the symmetry group to the
inhomogeneous Hamilton group that is a local noninertial symmetry of Hamilton's
equations. The projective representations of these groups are calculated using
the Mackey theorems for the general case of a nonabelian normal subgroup
Obtainment of internal labelling operators as broken Casimir operators by means of contractions related to reduction chains in semisimple Lie algebras
We show that the In\"on\"u-Wigner contraction naturally associated to a
reduction chain of semisimple Lie algebras
induces a decomposition of the Casimir operators into homogeneous polynomials,
the terms of which can be used to obtain additional mutually commuting missing
label operators for this reduction. The adjunction of these scalars that are no
more invariants of the contraction allow to solve the missing label problem for
those reductions where the contraction provides an insufficient number of
labelling operators
Contractions and deformations of quasi-classical Lie algebras preserving a non-degenerate quadratic Casimir operator
By means of contractions of Lie algebras, we obtain new classes of
indecomposable quasi-classical Lie algebras that satisfy the Yang-Baxter
equations in its reformulation in terms of triple products. These algebras are
shown to arise naturally from non-compact real simple algebras with non-simple
complexification, where we impose that a non-degenerate quadratic Casimir
operator is preserved by the limiting process. We further consider the converse
problem, and obtain sufficient conditions on integrable cocycles of
quasi-classical Lie algebras in order to preserve non-degenerate quadratic
Casimir operators by the associated linear deformations.Comment: 12 pages. LATEX with revtex4; Proceedings of the XII International
Conference on Symmetry Methods in Physics, (Yerevan, 2006) eds. G.S. Pogosyan
et al
Unexpected Features of Supersymmetry with Central Charges
It is shown that N=2 supersymmetric theories with central charges present
some hidden quartic symmetry. This enables us to construct representations of
the quartic structure induced by superalgebra representations.Comment: 14 pages, more details have been given, to appear in J. Phys.
On the structure of maximal solvable extensions and of Levi extensions of nilpotent algebras
We establish an improved upper estimate on dimension of any solvable algebra
s with its nilradical isomorphic to a given nilpotent Lie algebra n. Next we
consider Levi decomposable algebras with a given nilradical n and investigate
restrictions on possible Levi factors originating from the structure of
characteristic ideals of n. We present a new perspective on Turkowski's
classification of Levi decomposable algebras up to dimension 9.Comment: 21 pages; major revision - one section added, another erased;
author's version of the published pape
Composition algebras and the two faces of
We consider composition and division algebras over the real numbers: We note
two r\^oles for the group : as automorphism group of the octonions and
as the isotropy group of a generic 3-form in 7 dimensions. We show why they are
equivalent, by means of a regular metric. We express in some diagrams the
relation between some pertinent groups, most of them related to the octonions.
Some applications to physics are also discussed.Comment: 11 pages, 3 figure
Dynamical systems embedded into Lie algebras
Analytical and geometrical information on certain dynamical systems X is obtained under the assumption that X is embedded into a certain real Lie algebra
Contractions of Low-Dimensional Lie Algebras
Theoretical background of continuous contractions of finite-dimensional Lie
algebras is rigorously formulated and developed. In particular, known necessary
criteria of contractions are collected and new criteria are proposed. A number
of requisite invariant and semi-invariant quantities are calculated for wide
classes of Lie algebras including all low-dimensional Lie algebras.
An algorithm that allows one to handle one-parametric contractions is
presented and applied to low-dimensional Lie algebras. As a result, all
one-parametric continuous contractions for the both complex and real Lie
algebras of dimensions not greater than four are constructed with intensive
usage of necessary criteria of contractions and with studying correspondence
between real and complex cases.
Levels and co-levels of low-dimensional Lie algebras are discussed in detail.
Properties of multi-parametric and repeated contractions are also investigated.Comment: 47 pages, 4 figures, revised versio
Non-geometric flux vacua, S-duality and algebraic geometry
The four dimensional gauged supergravities descending from non-geometric
string compactifications involve a wide class of flux objects which are needed
to make the theory invariant under duality transformations at the effective
level. Additionally, complex algebraic conditions involving these fluxes arise
from Bianchi identities and tadpole cancellations in the effective theory. In
this work we study a simple T and S-duality invariant gauged supergravity, that
of a type IIB string compactified on a orientifold with
O3/O7-planes. We build upon the results of recent works and develop a
systematic method for solving all the flux constraints based on the algebra
structure underlying the fluxes. Starting with the T-duality invariant
supergravity, we find that the fluxes needed to restore S-duality can be simply
implemented as linear deformations of the gauge subalgebra by an element of its
second cohomology class. Algebraic geometry techniques are extensively used to
solve these constraints and supersymmetric vacua, centering our attention on
Minkowski solutions, become systematically computable and are also provided to
clarify the methods.Comment: 47 pages, 10 tables, typos corrected, Accepted for Publication in
Journal of High Energy Physic