Abstract. 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.
Ž .Let L L ރ be the variety of complex n-dimensional Lie algebras. The group n Ž . Ž . GL ރ acts on it via change of basis. An orbit O under this action consists of n all structures isomorphic to . The aim of this paper is to give a complete classification of orbit closures of 4-dimensional Lie algebras, i.e., determining all Ž . g O where g L L ރ . Starting with a classification of complex Lie algebras Ž . 4 of dimension n F 4, we study the behavior of several Lie algebra invariants under degeneration, i.e., under transition to the orbit closure. As a corollary, we will show Ž . that all degenerations in L L ރ can be realized via a one-parameter subgroup, but 3 Ž . this is not the case in L L ރ . ᮊ 1999 Academic Press 4 729
We investigate the existence of affine structures on nilmanifolds Γ\G in the case where the Lie algebra g of the Lie group G is filiform nilpotent of dimension less or equal to 11. Here we obtain examples of nilmanifolds without any affine structure in dimensions 10, 11. These are new counterexamples to the Milnor conjecture. So far examples in dimension 11 were known where the proof is complicated, see [5] and [4]. Using certain 2-cocycles we realize the filiform Lie algebras as deformation algebras from a standard graded filiform algebra. Thus we study the affine algebraic variety of complex filiform nilpotent Lie algebra structures of a given dimension ≤11. This approach simplifies the calculations, and the counterexamples in dimension 10 are less complicated than the known ones. We also obtain results for the minimal dimension µ(g) of a faithful g-module for these filiform Lie algebras g.
This survey on crystallographic groups, geometric structures on Lie groups and associated algebraic structures is based on a lecture given in the Ostrava research seminar in 2017.
Abstract. We study the varieties of Lie algebra laws and their subvarieties of nilpotent Lie algebra laws. We classify all degenerations of (almost all) five-step and six-step nilpotent seven-dimensional complex Lie algebras. One of the main tools is the use of trivial and adjoint cohomology of these algebras. In addition, we give some new results on the varieties of complex Lie algebra laws in low dimension. IntroductionLet g be an n-dimensional vector space over a field k and consider the set L n (k) of all possible Lie brackets µ on g. This is an algebraic subset of the variety Λ 2 g * ⊗ g of all alternating bilinear maps from g × g to g. Indeed, for a fixed basis (x 1 , . . . , x n ) of g the Lie bracket µ is determined by the point (c ijr ) ∈ k n 3 of structure constants withc ijr x r satisfying the polynomial conditionsThe variety L n (k) is often called the variety of Lie algebra laws. The general linear group GL n (k) acts on L n (k) by base change:One denotes by O(µ) the orbit of µ under the action of GL n (k), and by O(µ) the closure of the orbit with respect to the Zariski topology. The orbits in L n (k) correspond to isomorphism classes of n-dimensional Lie algebras. However, the orbit space is no longer an algebraic set. It makes sense to take out the zero point and to viewas the moduli space. There are many questions on the structure of the varieties L n (k). In particular one is interested in the irreducible components of L n (k) and in the open orbits.In that case the corresponding Lie algebra g is algebraic and does not admit any non-trivial deformation (1), . . . , r(7)) = (1, 1, 2, 4, 7, 17, 49) and (s(1), . . . , s(7)) = (1, 1, 1, 2, 3, 6, 14).These numbers grow very fast in n. One has the following estimates for n big enough [10]:In studying the orbit closures the concept of Lie algebra degenerations is of great interest.In that case we also say that λ degenerates to µ, which is denoted by λ → deg µ.Since C is closed relative to the Zariski topology, the orbit closure O(µ) is contained in C. Hence any irreducible component containing µ also contains all degenerations of µ. Remark 1.4. The concept of degenerations was first introduced by theoretical physicists in the special case of contractions [9]. Often the limit procedures considered in physics can be described by Lie algebra contractions. As an example, classical mechanics is a limit of quantum mechanics given by the contraction h → deg t 2n+1 , where h is the Weyl-Heisenberg algebra and t 2n+1 is the abelian Lie algebra of the same dimension.
Left-symmetric algebras (LSAs) are Lie admissible algebras arising from geometry. The leftinvariant affine structures on a Lie group G correspond bijectively to LSA-structures on its Lie algebra. Moreover if a Lie group acts simply transitively as affine transformations on a vector space, then its Lie algebra admits a complete LSA-structure. In this paper we study simple LSAs having only trivial two-sided ideals. Some natural examples and deformations are presented. We classify simple LSAs in low dimensions and prove results about the Lie algebra of simple LSAs using a canonical root space decomposition. A special class of complete LSAs is studied.
Abstract. We describe a method for classifying the Novikov algebras with a given associated Lie algebra. Subsequently we give the classification of the Novikov algebras of dimension 3 over R and C, as well as the classification of the 4-dimensional Novikov algebras over C whose associated Lie algebra is nilpotent. In particular this includes a list of all 4-dimensional commutative associative algebras over C.
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