15,024 research outputs found
On the Construction of Simply Connected Solvable Lie Groups
Let be a Lie algebra valued differential -form on a
manifold satisfying the structure equations where
is solvable. We show that the problem of finding a smooth map
, where is an -dimensional solvable Lie group with Lie
algebra and left invariant Maurer-Cartan form , such that
can be solved by quadratures and the matrix
exponential. In the process we give a closed form formula for the vector fields
in Lie's third theorem for solvable Lie algebras. A further application
produces the multiplication map for a simply connected -dimensional solvable
Lie group using only the matrix exponential and quadratures. Applications
to finding first integrals for completely integrable Pfaffian systems with
solvable symmetry algebras are also given.Comment: 22 pages. Fixed typos from version 1, and added more details in the
example
On the classification of 2-solvable Frobenius Lie algebras
We discuss the classification of 2-solvable Frobenius Lie algebras. We prove
that every 2-solvable Frobenius Lie algebra splits as a semidirect sum of an
n-dimensional vector space V and an n-dimensional maximal Abelian subalgebra
(MASA) of the full space of endomorphisms of V. We supply a complete
classification of 2-solvable Frobenius Lie algebras corresponding to
nonderogatory endomorphisms, as well as those given by maximal Abelian
nilpotent subalgebras (MANS) of class 2, hence of Kravchuk signature (n-1,0,1).
In low dimensions, we classify all 2-solvable Frobenius Lie algebras in general
up to dimension 8. We correct and complete the classification list of MASAs of
sl(4, R) by Winternitz and Zassenhaus. As a biproduct, we give a simple proof
that every nonderogatory endormorphism of a real vector space admits a Jordan
form and also provide a new characterization of Cartan subalgebras of sl(n, R).Comment: V2: 26 pages, Latex. Title slightly changed. Typos corrected. More
details on the proof that every nonderogatory real endomorphism/matrix admits
a Jordan form. To appear at Journal of Lie Theor
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
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