Let g be a Lie algebra over a field F of characteristic zero, let C be a certain tensor category of representations of g, and C a certain category of duals. In [M 4] we associated to C and C by a Tannaka reconstruction a monoid M with a coordinate ring F [M ] of matrix coefficients, as well as a Lie algebra Lie(M). We interpreted the Tannaka monoid M algebraic geometrically as a weak algebraic ...

One of the algebraic structures that has emerged recently in the study of the operator product expansions of chiral fields in conformal field theory is that of a Lie conformal algebra [K]. A Lie pseudoalgebra is a generalization of the notion of a Lie conformal algebra for which C[∂] is replaced by the universal enveloping algebra H of a finite-dimensional Lie algebra [BDK]. The finite (i.e., f...

The multiplication in a quasiassociative algebra R satisfies the property a ∗ (b ∗ c)− (a ∗ b) ∗ c = b ∗ (a ∗ c)− (b ∗ a) ∗ c, a, b, c ∈ R. (∗∗) This property is necessary and sufficient for the Lie algebra Lie(R) to have a phase space. The above formulae are put into a cohomological framework, with the relevant complex being different from the Hochschild one even when the relevant quasiassocia...

The discrete symmetries P, C and T are discussed in terms of Lie algebra extensions of the Poincare Lie algebra. This formulation leads to certain problems of Lie algebra theory which will be presented in this and succeeding papers. (To be submitted to Communications in Mathematical Physics) * Work supported by the U. S. Atomic Energy Commission.

We consider the universal central extension of the Lie algebra Vect(S 1)⋉C ∞ (S 1). The coadjoint representation of this Lie algebra has a natural geometric interpretation by matrix analogues of the Sturm-Liouville operators. This approach leads to new Lie superalgebras generalizing the well-known Neveu-Schwartz algebra.

We consider the lower central filtration of the free associative algebra An with n generators as a Lie algebra. We consider the associated graded Lie algebra. It is shown that this Lie algebra has a huge center which belongs to the cyclic words, and on the quotient Lie algebra by the center there acts the Lie algebraWn of polynomial vector fields on C. We compute the space [An, An]/[An, [An, An...

A Riemann-Lie algebra is a Lie algebra G such that its dual G∗ carries a Riemannian metric compatible (in the sense introduced by the author in C. R. Acad. Sci. Paris, t. 333, Série I, (2001) 763–768) with the canonical linear Poisson structure of G∗ . The notion of Riemann-Lie algebra has its origins in the study, by the author, of Riemann-Poisson manifolds (see Differential Geometry and its A...

Let H be the quaternion algebra. Let g be a complex Lie algebra and let U(g) be the enveloping algebra of g. The quaternification g = (H ⊗ U(g), [ , ]gH ) of g is defined by the bracket [ z ⊗ X , w ⊗ Y ] gH = (z · w) ⊗ (XY ) − (w · z) ⊗ (Y X) , for z, w ∈ H and the basis vectors X and Y of U(g). Let SH be the ( non-commutative) algebra of H-valued smooth mappings over S and let Sg = SH ⊗ U(g). ...

In this paper, we classify the indecomposable non-nilpotent solvable Lie algebras with $N(R_n,m,r)$ nilradical,by using the derivation algebra and the automorphism group of $N(R_n,m,r)$.We also prove that these solvable Lie algebras are complete and unique, up to isomorphism.