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Leftsymmetric algebras, or preLie algebras in geometry and physics
 Cent. Eur. J. Math
"... Abstract. In this survey article we discuss the origin, theory and applications of leftsymmetric 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 preLie algebras is ..."
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Abstract. In this survey article we discuss the origin, theory and applications of leftsymmetric 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 preLie 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.
Leftinvariant affine structures on reductive Lie groups
 J. Algebra
, 1996
"... We describe leftinvariant affine structures (that is, leftinvariant flat torsionfree affine connections ∇) on reductive linear Lie groups G. They correspond bijectively to LSAstructures on the Lie algebra g of G. Here LSA stands for leftsymmetric algebra, see [BUR], [SE2]. If g has trivial or o ..."
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Cited by 10 (3 self)
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We describe leftinvariant affine structures (that is, leftinvariant flat torsionfree affine connections ∇) on reductive linear Lie groups G. They correspond bijectively to LSAstructures on the Lie algebra g of G. Here LSA stands for leftsymmetric algebra, see [BUR], [SE2]. If g has trivial or one dimensional center z then the affine representation α = λ ⊕ 1 of g, induced by any LSAstructure gλ on g is radiant, i.e., the radiance obstruction cα ∈ H1(g,gλ) vanishes. If dim z = 1 we prove that g = s ⊕ z, where s is split simple, admits LSAstructures if and only if s is of type A ` , that is g = gln. Here we have the associative LSAstructure given by ordinary matrix multiplication corresponding to the biinvariant affine structure on GL(n) , which was believed to be essentially the only possible LSAstructure on gln. We exhibit interesting LSAstructures different from the associative one. They arise as certain deformations of the matrix algebra. Then we classify all LSAstructures on gln using a result of [BAU]. For n = 2 we compute all structures explicitely over the complex numbers. 1
Faithful representation of minimal dimension of current Heisenberg Lie algebras
 Int. J. Math
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Affine cohomology classes for filiform Lie algebras
 Contemporary Mathematics
"... Abstract. We classify the cohomology spaces H2(g,K) for all filiform nilpotent Lie algebras of dimension n ≤ 11 over K and for certain classes of algebras of dimension n ≥ 12. The result is applied to the determination of affine cohomology classes [ω] ∈ H2(g,K). We prove the general result that the ..."
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Abstract. We classify the cohomology spaces H2(g,K) for all filiform nilpotent Lie algebras of dimension n ≤ 11 over K and for certain classes of algebras of dimension n ≥ 12. The result is applied to the determination of affine cohomology classes [ω] ∈ H2(g,K). We prove the general result that the existence of an affine cohomology class implies an affine structure of canonical type on g, hence a canonical leftinvariant affine structure on the corresponding nilpotent Lie group. For certain filiform algebras the absence of an affine cohomology class implies the nonexistence of any affine structure. Of particular interest are algebras g with minimal Betti numbers b1(g) = b2(g) = 2. 1.
Deschamps: Affine actions on nilpotent Lie groups
, 2009
"... Abstract. To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in tra ..."
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Abstract. To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence question of such a simply transitive affine action to a corresponding question on the Lie algebra level. As an example of the possible use of this translation, we then consider the case where dim(G) = dim(N)≤5. Finally, we specialize to the case of abelian simply transitive affine actions on a given connected and simply connected nilpotent Lie group. It turns out that such a simply transitive abelian affine action on N corresponds to a particular Lie compatible bilinear product on the Lie algebra n of N, which we call an LRstructure. 1. NILaffine actions In 1977 [12], J. Milnor asked whether or not any connected and simply connected solvable Lie group G of dimension n admits a representation ρ: G → Aff(R n) into the group of invertible affine mappings, letting G operate simply transitively on R n.
LEFTSYMMETRIC ALGEBRAS FOR gl(n)
, 1999
"... Abstract. We study the classification problem for leftsymmetric algebras with commutation Lie algebra gl(n) in characteristic 0. The problem is equivalent to the classification of étale affine representations of gl(n). Algebraic invariant theory is used to characterize those modules for the algebra ..."
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Abstract. We study the classification problem for leftsymmetric algebras with commutation Lie algebra gl(n) in characteristic 0. The problem is equivalent to the classification of étale affine representations of gl(n). Algebraic invariant theory is used to characterize those modules for the algebraic group SL(n) which belong to affine étale representations of gl(n). From the classification of these modules we obtain the solution of the classification problem for gl(n). As another application of our approach, we exhibit leftsymmetric algebra structures on certain reductive Lie algebras with a onedimensional center and a nonsimple semisimple ideal. 1.
COMPLETE LRSTRUCTURES ON SOLVABLE LIE ALGEBRAS
, 2009
"... An LRstructure on a Lie algebra g is a bilinear product on g, satisfying certain commutativity relations, and which is compatible with the Lie product. LRstructures arise in the study of simply transitive affine actions on Lie groups. In particular one is interested in the question which Lie alge ..."
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Cited by 4 (3 self)
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An LRstructure on a Lie algebra g is a bilinear product on g, satisfying certain commutativity relations, and which is compatible with the Lie product. LRstructures arise in the study of simply transitive affine actions on Lie groups. In particular one is interested in the question which Lie algebras admit a complete LRstructure. In this paper we show that a Lie algebra admits a complete LRstructure if and only if it admits any LRstructure.
Polynomial structures for nilpotent groups
 Transactions
, 1996
"... Abstract. If a polycyclicbyfinite rankK group Γ admits a faithful affine representation making it acting on RK properly discontinuously and with compact quotient, we say that Γ admits an affine structure. In 1977, John Milnor questioned the existence of affine structures for such groups Γ. Very r ..."
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Abstract. If a polycyclicbyfinite rankK group Γ admits a faithful affine representation making it acting on RK properly discontinuously and with compact quotient, we say that Γ admits an affine structure. In 1977, John Milnor questioned the existence of affine structures for such groups Γ. Very recently examples have been obtained showing that, even for torsionfree, finitely generated nilpotent groups N, affine structures do not always exist. It looks natural to consider affine structures as examples of polynomial structures of degree one. We introduce the concept of a canonical type polynomial structure for polycyclicbyfinite groups. Using the algebraic framework of the Seifert Fiber Space construction and a nice cohomology vanishing theorem, we prove the existence and uniqueness (up to conjugation) of canonical type polynomial structures for virtually finitely generated nilpotent groups. Applying this uniqueness to a result going back to Mal ′ cev, it follows that, for torsionfree, finitely generated nilpotent groups, each canonical polynomial structure is expressed in polynomials of limited degree. The minimal degree needed for obtaining a polynomial structure will determine the “affine defect number”. We prove that the known counterexamples to Milnor’s question have the smallest possible affine defect, i.e. affine defect number equal to one. 1. Introduction and
THE AUSLANDER CONJECTURE FOR NILAFFINE CRYSTALLOGRAPHIC GROUPS
, 2004
"... Abstract. We study subgroups Γ in Aff(N) = N ⋊ Aut(N) acting properly discontinuously and cocompactly on N. Here N is a simply connected, connected real nilpotent Lie group of finite dimension n. This situation is a natural generalization of the socalled affine crystallographic groups. We prove th ..."
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Abstract. We study subgroups Γ in Aff(N) = N ⋊ Aut(N) acting properly discontinuously and cocompactly on N. Here N is a simply connected, connected real nilpotent Lie group of finite dimension n. This situation is a natural generalization of the socalled affine crystallographic groups. We prove that for all dimensions 1 ≤ n ≤ 5 the generalized Auslander conjecture holds, i.e., that such subgroups are virtually polycyclic. 1.