In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs ...in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.
We study sympathetic (i.e., perfect and complete) Lie algebras. Among other topics they arise in the study of adjoint Lie algebra cohomology. Here a motivation comes from a conjecture of Pirashvili, ...which says that a finite-dimensional complex perfect Lie algebra is semisimple if and only if its adjoint cohomology vanishes. We prove several general results for sympathetic Lie algebras and for the adjoint Lie algebra cohomology of arbitrary finite-dimensional Lie algebras in characteristic zero using a result of Hochschild and Serre. Moreover, for certain semidirect products we obtain explicit results for the adjoint cohomology.
We study post-Lie algebra structures on pairs of Lie algebras (g,n), which describe simply transitive nil-affine actions of Lie groups. We prove existence results for such structures depending on the ...interplay of the algebraic structures of g and n. We consider the classes of simple, semisimple, reductive, perfect, solvable, nilpotent, abelian and unimodular Lie algebras. Furthermore we consider commutative post-Lie algebra structures on perfect Lie algebras. Using Lie algebra cohomology we can classify such structures in several cases. We also study commutative structures on low-dimensional Lie algebras and on nilpotent Lie algebras.
Rota-Baxter operators R of weight 1 on
are in bijective correspondence to post-Lie algebra structures on pairs
, where
is complete. We use such Rota-Baxter operators to study the existence and ...classification of post-Lie algebra structures on pairs of Lie algebras
, where
is semisimple. We show that for semisimple
and
, with
or
simple, the existence of a post-Lie algebra structure on such a pair
implies that
and
are isomorphic, and hence both simple. If
is semisimple, but
is not, it becomes much harder to classify post-Lie algebra structures on
, or even to determine the Lie algebras
which can arise. Here only the case
was studied. In this paper, we determine all Lie algebras
such that there exists a post-Lie algebra structure on
with
.
We give an explicit description of commutative post-Lie algebra structures on some classes of nilpotent Lie algebras. For non-metabelian filiform nilpotent Lie algebras and Lie algebras of strictly ...upper-triangular matrices we show that all CPA-structures are associative and induce an associated Poisson-admissible algebra.
Jacobi–Jordan algebras Burde, Dietrich; Fialowski, Alice
Linear algebra and its applications,
10/2014, Letnik:
459
Journal Article
Recenzirano
Odprti dostop
We study finite-dimensional commutative algebras, which satisfy the Jacobi identity. Such algebras are Jordan algebras. We describe some of their properties and give a classification in dimensions ...n<7 over algebraically closed fields of characteristic not 2 or 3.
We show that any CPA-structure (commutative post-Lie algebra structure) on a perfect Lie algebra is trivial. Furthermore we give a general decomposition of inner CPA-structures, and classify all ...CPA-structures on complete Lie algebras. As a special case we obtain the CPA-structures of parabolic subalgebras of semisimple Lie algebras.
We study disemisimple Lie algebras, i.e., Lie algebras which can be written as a vector space sum of two semisimple subalgebras. We show that a Lie algebra g is disemisimple if and only if its ...solvable radical coincides with its nilradical and is a prehomogeneous s-module for a Levi subalgebra s of g. We use the classification of prehomogeneous s-modules for simple Lie algebras s given by Vinberg to show that the solvable radical of a disemisimple Lie algebra with simple Levi subalgebra is abelian. We extend this result to disemisimple Lie algebras having no simple quotients of type A.
We prove nilpotency results for Lie algebras over an arbitrary field admitting a derivation, which satisfies a given polynomial identity r(t) = 0. In the special case of the polynomial
we obtain a ...uniform bound on the nilpotency class of Lie algebras admitting a periodic derivation of order n. We even find an optimal bound on the nilpotency class in characteristic p if p does not divide a certain invariant ρ
n
.
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