By using the Ringel-Hall algebra approach, we investigate the structure of the Lie algebra L(Λ) generated by indecomposable constructible sets in the varieties of modules for any finite dimensional Calgebra Λ. We obtain a geometric realization of the universal enveloping algebra R(Λ) of L(Λ). This generalizes the main result of Riedtmann in [19]. We also obtain Green’s theorem in [6] in a geome...

In this paper the W -algebra W (2, 2) and its representation theory are studied. It is proved that a simple vertex operator algebra generated by two weight 2 vectors is either a vertex operator algebra associated to a highest irreducible W (2, 2)-module or a tensor product of two irreducible Virasoro vertex operator algebras. Furthermore, any rational, C2-cofinite and simple vertex operator alg...

Cohomology and deformation theories are developed for Poisson algebras starting with the more general concept of a Leibniz pair, namely of an associative algebra A together with a Lie algebra L mapped into the derivations of A. A bicomplex (with both Hochschild and Chevalley-Eilenberg cohomologies) is essential. The importance of Poisson algebras in classical mechanics makes it useful to have a...

A conformal vector with central charge c in a vertex operator algebra is an element of weight two whose component operators satisfy the Virasoro algebra relation with central charge c. Then the vertex operator subalgebra generated by the vector is isomorphic to a highest weight module for the Virasoro algebra with central charge c and highest weight 0 (cf. [M]). Let V2Al be the vertex operator ...

The property of degeneration of modular graded Lie algebras, first investigated by B. Weisfeiler, is analyzed. Transitive irreducible graded Lie algebras L = ∑ i∈Z Li, over algebraically closed fields of characteristic p > 2, with classical reductive component L0 are considered. We show that if a non-degenerate Lie algebra L contains a transitive degenerate subalgebra L′ such that dimL1 > 1, th...

and [[xy]z] + [[yz]x] + [[zx]y] = 0. Left multiplication in a Lie algebra is denoted by ad(x): ad(x)(y) = [x, y]. An associative algebra A becomes a Lie algebra A− under the product, [xy] = xy − yx. The first axiom implies that [xy] = −[yx] and the second (called the Jacobi identity) implies that x 7→ adx is a homomorphism of L into the Lie algebra (EndL)−, that is, ad [xy] = [adx, ad y]. Assum...

We give a simple proof of Bourgain's disc algebra version of Grothendieck's theorem, i.e. that every operator on the disc algebra with values in L 1 or L 2 is 2-absolutely summing and hence extends to an operator defined on the whole of C. This implies Bourgain's result that L 1 /H 1 is of cotype 2. We also prove more generally that L r /H r is of cotype 2 for 0 < r < 1.

We consider the problems of determining the feasibility of a linear congruence, producing a solution to a linear congruence, and finding a spanning set for the nullspace of an integer matrix, where each of these problems are considered modulo an arbitrary constant k > 2. These problems are known to be complete for the logspace modular counting classes coModkL in special case that k is prime [4]...

We introduce the notion of a Lie-like algebra⋄ (superalgebra⋄) for ⋄ ∈ { , , 3−rd }. By bundling a family of algebras, we introduce six new generalizations of Lie algebras which are called Lie-like algebras or Lie-like superalgebras in this paper. The examples of the new generalizations of Lie algebras come from an invariant algebra (or an invariant superalgebra), where the notion of an invaria...

We show that to cubic order double field theory is encoded in Yang-Mills theory. To this end we use algebraic structures from string as follows: The ${L}_{\ensuremath{\infty}}$-algebra of the tensor product $\mathcal{K}\ensuremath{\bigotimes}\mathfrak{g}$ Lie algebra $\mathfrak{g}$ gauge group and a ``kinematic algebra'' $\mathcal{K}$ ${C}_{\ensuremath{\infty}}$-algebra. This structure induces ...