We introduce a class of noncommutatative algebras called representation complete intersections (RCI). A graded associative algebra A is said to be RCI provided there exist arbitrarily large positive integers n such that the scheme Rep n A, of n-dimensional representations of A, is a complete intersection. We discuss examples of RCI algebras, including those arising from quivers. There is anothe...

The deformation equation and its integrability condition (Bianchi identity) of a non-(co)associative deformation in operad algebra are found. Based on physical analogies, cogravity equations are proposed.

1 For multiple Fourier integrals in the place of Fourier series this formula has been previously estabLished and made use of in the paper: S. Bochner, "Ein Konvergenzsatz fur mehrvariablige Fouriersche Integrale," Math. Zeit., 34, 440-447 (1931); see also A. C. Berry, "The Fourier Transform Theorem," Jour. Math. Phys. Mass. Inst. Technology, 8, 106-118 (1929). 2 See Bochner, S., "Properties of ...

We present new, e cient algorithms for some fundamental computations with nite dimensional (but not necessarily commutative) associative algebras over nite elds. For a semisimple algebra A we show how to compute a complete Wedderburn decomposition of A as a direct sum of simple algebras, an isomorphism between each simple component and a full matrix algebra, and a basis for the centre of A. If ...

We continue the study of pairs of associative algebra and a linear functional on it. New results together with corrected proofs for previously published material are presented.

The notion of pre-Lie algebra can be seen as a weakened form of associative algebra, still giving a Lie algebra by anti-symmetrization. It has been introduced by Gerstenhaber in his work on deformations of algebras [Ger64]. More recently, it has been studied from the point of view of operad theory and seen to be related to rooted trees [CL01]. The motivating problem for this paper is the classi...

The deformation equation and its integrability condition (Bianchi identity) of a non-associative deformation in operad algebra are found. Their relation to the theory of gravity is discussed.

We compute the stable homology of necklace Lie algebras associated with quivers and give a construction of stable homology classes from certain A∞-categories. Our construction is a generalization of the construction of homology classes of moduli spaces of curves due to M. Kontsevich. In the second part of the paper we produce a Moyal-type quantization of the symmetric algebra of a necklace Lie ...

A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite loop spaces. The purpose of this article is to prove that the associative algebra structure on the normalized cochain complex of a simplici...