A double algebra is a linear space V equipped with linear map V ⊗ V → V ⊗ V . Additional conditions on this map lead to the notions of Lie and associative double algebras. We prove that simple finite-dimensional Lie double algebras do not exist over an arbitrary field, and all simple finite-dimensional associative double algebras over an algebraically closed field are trivial. Over an arbitrary...

In this paper we compute the deformation theory of a special class of algebras, namely of Azumaya algebras on a manifold (C or complex analytic). Deformation theory of associative algebras was initiated by Gerstenhaber in [G]. A deformation of an associative algebra A over an Artinian ring a is an a-linear associative algebra structure on A⊗ a such that, for the maximal ideal m of a, A ⊗ m is a...

Over a field IF of any characteristic, for a commutative associative algebra A with an identity element and for the polynomial algebra IF [D] of a commutative derivation subalgebra D of A, the associative and the Lie algebras of Weyl type on the same vector space A[D] = A ⊗ IF [D] are defined. It is proved that A[D], as a Lie algebra (modular its center) or as an associative algebra, is simple ...

A commutative associative algebra A over C with a derivation is one of the simplest examples of a vertex algebra. However, the differences between the modules for A as a vertex algebra and the modules for A as an associative algebra are not well understood. In this paper, I give the classification of finite-dimensional indecomposable untwisted or twisted modules for the polynomial ring in one v...

In this brief note we would like to give the construction of a free commutative unital associative Nijenhuis algebra on a commutative unital associative algebra based on an augmented modified quasi-shuffle product. —————————————

We compactify the spaces K(m,n) introduced by Maxim Kontsevich. The initial idea was to construct an L∞ algebra governing the deformations of a (co)associative bialgebra. However, this compactification leads not to a resolution of the PROP of (co)associative bialgebras, but to a new algebraic structure we call here a CROC. It turns out that these constructions are related to the non-commutative...

Based on results about open string correlation functions, a nonassociative algebra was proposed in a recent paper for D-branes in a background with nonvanishing H. We show that our associative algebra obtained by quantizing the endpoints of an open string in an earlier work can also be used to reproduce the same correlation functions. The novelty of this algebra is that functions on the D-brane...

This paper is an exposition of the representation theory of vertex operator algebras in terms of associative algebras An(V ) and their bimodules. A new result on the rationality is given. That is, a simple vertex operator algebra V is rational if and only if its Zhu algebra A(V ) is a semisimple associative algebra. 2000MSC:17B69