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Pseudo-Euclidean Novikov algebras of arbitrary signature
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A Lorentzian flat Lie group is a Lie group G with a flat left invariant
metric with signature . The Lie algebra
of G endowed with is called flat Lorentzian Lie
algebra. It is known that the metric of a flat Lorentzian Lie group is
geodesically complete if and only if its Lie algebra is unimodular. In this
paper, we determine all nonunimodular flat Lorentzian Lie algebras leading to
the determination of all incomplete Lorentzian flat Lie groups.
A bilinear form f on a nonassociative algebra A is said to be invariant i f(ab,c) = f(a,bc) for all a,b,c 2 A. Finite-dimensional complex semisimple Lie algebras (with their Killing form) and certain associative algebras (with a trace) carry such a structure. We discuss the ideal structure of A if f is nondegenerate and introduce the notion of T -extension of an arbitrary algebra B (i.e. by its dual space B ) where the natural pairing gives rise to a nondegenerate invariant symmetric bilinear form on A := B B. The T -extension involves the third scalar cohomology H 3 (B, K ) if B is Lie and the second cyclic cohomology HC 2 (B) if B is associative in a natural way. Moreover, we show that every nilpotent finite- dimensional algebra A over an algebraically closed field carrying a nondegenerate invariant symmetric bilinear form is a suitable T -extension. As a Corollary, we prove that every complex Lie algebra carrying a nondegenerate invariant symmetric bilinear form is always a special type of Manin pair in the sense of Drinfel'd but not always isomorphic to a Manin triple. Examples involving the Heisenb erg and filiform Lie algebras (whose third scalar cohomology is computed) are discussed.
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.
A flat pseudo-Euclidean Lie algebra is a left-symmetric algebra endowed with a non-degenerate symmetric bilinear form such that left multiplications are skew-symmetric. We observe that, in many classes of flat pseudo-Euclidean Lie algebras, the left-symmetric product is actually of Novikov. This leads us to study pseudo-Euclidean Novikov algebras (A,〈,〉), that is, Novikov algebra A endowed with a non-degenerate symmetric bilinear form such that left multiplications are skew-symmetric. We show that a Lorentzian Novikov algebra (A,〈,〉) must be transitive. This implies that the underlying Lie algebra AL must be unimodular. In geometrical terms, the left-invariant metric on a corresponding Lie group is geodesically complete. In this case, we show that the restriction of the product to [AL,AL]⊥ is trivial. Using the double extension process, we prove that (A,〈,〉) is a pseudo-Euclidean Novikov algebra such that [AL,AL] is Lorentzian if and only if AL splits as AL=[AL,AL]⊥⊕[AL,AL] where [AL,AL] and [AL,AL]⊥ are abelian, [AL,AL] is Lorentzian and adx is skew-symmetric for any x∈[AL,AL]⊥. This also implies, in this case, that A is transitive, AL is unimodular and two-solvable. This result solves the problem for Lorentzian Novikov algebras such that [AL,AL] is non-degenerate. If [AL,AL] is degenerate, we show that such Lorentzian Novikov algebra is obtained by the double extension process from a flat Euclidean algebra, and we give some applications in low dimensions.
A flat pseudo-Euclidean Lie algebra is a real Lie algebra with a non degenerate symmetric bilinear form and a left symmetric product whose the commutator is the Lie bracket and such that the left multiplications are skew-symmetric. We show that the center of a flat pseudo-Euclidean nilpotent Lie algebra of signature must be degenerate and all flat pseudo-Euclidean nilpotent Lie algebras of signature can be obtained by using the double extension process from flat Lorentzian nilpotent Lie algebras. We show also that the center of a flat pseudo-Euclidean 2-step nilpotent Lie algebra is degenerate and all these Lie algebras are obtained by using a sequence of double extension from an abelian Lie algebra. In particular, we determine all flat pseudo-Euclidean 2-step nilpotent Lie algebras of signature . The paper contains also some examples in low dimension.
In this note, we shall classify Novikov algebras that admit an invariant Lorentzian symmetric bilinear form.
We give a complete classification of finite-dimensional simple Novikov algebras and their irreducible modules over an algebraically closed field with prime characteristic. Moreover, we introduce “Novikov–Poisson algebras” and their tensor theory. Our tensor theory enables us to understand better certain finite-dimensional simple Novikov algebras and their irreducible modules.
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. On the other hand, there can be geometry and Lagrangian mechanics on homogenous spaces related to Novikov algebras. The nondegenerate symmetric bilinear forms on Novikov algebras can be regarded as the pseudometrics, and some additional identities for these forms correspond to some “conserved quantities.” In particular, there is an important kind of “conserved” nondegenerate symmetric bilinear forms that correspond to the pseudo-Riemannian connections such that parallel translation preserves the bilinear form on the tangent spaces. Moreover, the fact that the left multiplication operators form a Lie algebra for a Novikov algebra is compatible with such a form. However, we show in this note that there are no such forms on most Novikov algebras in low dimensions.
This article outlines what is known to the author about the Riemannian geometry of a Lie group which has been provided with a Riemannian metric invariant under left translation.
We call the Lie algebra of a Lie group with a left invariant
pseudo-Riemannian flat metric pseudo-Riemannian flat Lie algebra. We give a new
proof of a classical result of Milnor on Riemannian flat Lie algebras. We
reduce the study of Lorentzian flat Lie algebras to those with trivial center
or those with degenerate center. We show that the double extension process can
be used to construct all Lorentzian flat Lie algebras with degenerate center
generalizing a result of Aubert-Medina on Lorentzian flat nilpotent Lie
algebras. Finally, we give the list of Lorentzian flat Lie algebras with
degenerate center up to dimension 6.
The determination of affine Lie groups (i.e., which carry a left-invariant affine structure) is an open problem. In this work we begin the study of Lie groups with a left-invariant, flat pseudo-Riemannian metric (flat pseudo-Riemannian groups). We show that in such groups the left-invariant affine structure defined by the Levi-Civita connection is geodesically complete if and only if the group is unimodular. We also show that the cotangent manifold of an affine Lie group is endowed with an affine Lie group structure and a left-invariant, flat hyperbolic metric. We describe a double extension process which allows us to construct all nilpotent, flat Lorentzian groups. We give examples and prove that the only Heisenberg group which carries a left invariant, flat pseudo-Riemannian metric is the three dimensional one.
In this paper, we first present a classification theorem of infinite-dimensional simple Novikov algebras over an algebraically closed field with characteristic 0. Then we classify all the irreducible modules of a certain infinite-dimensional simple Novikov algebras with an idempotent element whose left action is locally finite.
On a class of local translation invariant Lie algebras
- Zelmanov