57 research outputs found
Extending invariant complex structures
We study the problem of extending a complex structure to a given Lie algebra
g, which is firstly defined on an ideal h of g. We consider the next
situations: h is either complex or it is totally real. The next question is to
equip g with an additional structure, such as a (non)-definite metric or a
symplectic structure and to ask either h is non-degenerate, isotropic, etc.
with respect to this structure, by imposing a compatibility assumption. We show
that this implies certain constraints on the algebraic structure of g.
Constructive examples illustrating this situation are shown, in particular
computations in dimension six are given.Comment: 22 pages, plus an Addendu
Non-solvable contractions of semisimple Lie algebras in low dimension
The problem of non-solvable contractions of Lie algebras is analyzed. By
means of a stability theorem, the problem is shown to be deeply related to the
embeddings among semisimple Lie algebras and the resulting branching rules for
representations. With this procedure, we determine all deformations of
indecomposable Lie algebras having a nontrivial Levi decomposition onto
semisimple Lie algebras of dimension , and obtain the non-solvable
contractions of the latter class of algebras.Comment: 21 pages. 2 Tables, 2 figure
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
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
The Null Decomposition of Conformal Algebras
We analyze the decomposition of the enveloping algebra of the conformal
algebra in arbitrary dimension with respect to the mass-squared operator. It
emerges that the subalgebra that commutes with the mass-squared is generated by
its Poincare subalgebra together with a vector operator. The special cases of
the conformal algebras of two and three dimensions are described in detail,
including the construction of their Casimir operators.Comment: 31 page
Invariants of Triangular Lie Algebras
Triangular Lie algebras are the Lie algebras which can be faithfully
represented by triangular matrices of any finite size over the real/complex
number field. In the paper invariants ('generalized Casimir operators') are
found for three classes of Lie algebras, namely those which are either strictly
or non-strictly triangular, and for so-called special upper triangular Lie
algebras. Algebraic algorithm of [J. Phys. A: Math. Gen., 2006, V.39, 5749;
math-ph/0602046], developed further in [J. Phys. A: Math. Theor., 2007, V.40,
113; math-ph/0606045], is used to determine the invariants. A conjecture of [J.
Phys. A: Math. Gen., 2001, V.34, 9085], concerning the number of independent
invariants and their form, is corroborated.Comment: LaTeX2e, 16 pages; misprints are corrected, some proofs are extende
Computation of Invariants of Lie Algebras by Means of Moving Frames
A new purely algebraic algorithm is presented for computation of invariants
(generalized Casimir operators) of Lie algebras. It uses the Cartan's method of
moving frames and the knowledge of the group of inner automorphisms of each Lie
algebra. The algorithm is applied, in particular, to computation of invariants
of real low-dimensional Lie algebras. A number of examples are calculated to
illustrate its effectiveness and to make a comparison with the same cases in
the literature. Bases of invariants of the real solvable Lie algebras up to
dimension five, the real six-dimensional nilpotent Lie algebras and the real
six-dimensional solvable Lie algebras with four-dimensional nilradicals are
newly calculated and listed in tables.Comment: 17 pages, extended versio
All solvable extensions of a class of nilpotent Lie algebras of dimension n and degree of nilpotency n-1
We construct all solvable Lie algebras with a specific n-dimensional
nilradical n_(n,2) (of degree of nilpotency (n-1) and with an (n-2)-dimensional
maximal Abelian ideal). We find that for given n such a solvable algebra is
unique up to isomorphisms. Using the method of moving frames we construct a
basis for the Casimir invariants of the nilradical n_(n,2). We also construct a
basis for the generalized Casimir invariants of its solvable extension s_(n+1)
consisting entirely of rational functions of the chosen invariants of the
nilradical.Comment: 19 pages; added references, changes mainly in introduction and
conclusions, typos corrected; submitted to J. Phys. A, version to be
publishe
Invariants of Lie Algebras with Fixed Structure of Nilradicals
An algebraic algorithm is developed for computation of invariants
('generalized Casimir operators') of general Lie algebras over the real or
complex number field. Its main tools are the Cartan's method of moving frames
and the knowledge of the group of inner automorphisms of each Lie algebra.
Unlike the first application of the algorithm in [J. Phys. A: Math. Gen., 2006,
V.39, 5749; math-ph/0602046], which deals with low-dimensional Lie algebras,
here the effectiveness of the algorithm is demonstrated by its application to
computation of invariants of solvable Lie algebras of general dimension
restricted only by a required structure of the nilradical.
Specifically, invariants are calculated here for families of real/complex
solvable Lie algebras. These families contain, with only a few exceptions, all
the solvable Lie algebras of specific dimensions, for whom the invariants are
found in the literature.Comment: LaTeX2e, 19 page
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