A Lie-admissible algebra gives a Lie algebra by anticommutativity. In this work we describe remarkable types of Lie-admissible algebras such as Vinberg algebras, pre-Lie algebras or Lie algebras. We compute the corresponding binary quadratic operads and study their duality. Considering Lie algebras as Lie-admissible algebras, we can define for each Lie algebra a cohomology with values in an Lie...

To any manifold equipped with a higher degree closed form, one can associate an L∞-algebra of local observables that generalizes the Poisson algebra of a symplectic manifold. Here, by means of an explicit homotopy equivalence, we interpret this L∞-algebra in terms of infinitesimal autoequivalences of higher prequantum bundles. By truncating the connection data on the prequantum bundle, we produ...

We realize the current algebra of a Kac-Moody algebra as a quotient of a semi-direct product of the Kac-Moody Lie algebra and the free Lie algebra of the Kac-Moody algebra. We use this realization to study the representations of the current algebra. In particular we see that every ad-invariant ideal in the symmetric algebra of the Kac-Moody algebra gives rise in a canonical way to a representat...

It is well known that the standard bracketings of Lyndon words in an alphabet A form a basis for the free Lie algebra Lie(A) generated by A . Suppose that g 2 Lie(A)/J is a Lie algebra given by a generating set A and a Lie ideal J of relations. Using a Grobner basis type approach we define a set of "standard" Lyndon words, a subset of the set Lyndon words, such that the standard bracketings of ...

We define a remarkable Lie algebra of infinite dimension, and conjecture that it may be related to the Fischer-Griess Monster group. The idea was mooted in [C-N] that there might be an infinite-dimensional Lie algebra (or superalgebra) L that in some sense “explains” the Fischer-Griess ‘Monster” group M . In this chapter we produce some candidates for L based on properties of the Leech lattice ...

A (2k+1)-dimensional Lie algebra is called contact if it admits a one-form φ such that φ∧(dφ)k≠0. Here, we extend recent work to describe combinatorial procedure for generating contact, type-A poset algebras whose associated posets have chains of arbitrary cardinality, and conjecture our construction leads complete characterization.