An Introduction to Nonassociative Algebras

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In order to extend this classification to semisimple Lie algebras, it is necessary to study the relationship between fine gradings by abelian groups on a direct sum of simple algebras and fine gradings on the simple components. In this talk, it will be shown that the classification of gradings by abelian groups on finite direct sums of simple finite-dimensional nonassociative algebras over an algebraically closed field can be reduced, by means of the use of loop algebras, to the corresponding problem for simple algebras. This requires a good definition of free products of group-gradings.

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  • An Introduction to Nonassociative Algebras?

Hom-Lie algebras were first introduced in the study of q-deformations of Lie algebras appearing in physics. Here the Jacobi identity is twisted by a morphism called structure map. Later other types of Hom-algebras such as Hom-associative algebras have been defined and studied. In this talk, I want to define a generalization of the Hochschild cohomology for associative algebras to the Hom-setting. This new cohomology is called alpha-type Hochschild cohomology. It can be used to study formal deformations in the sense of Gerstenhaber, where the multiplication and the structure map are deformed.

This is joint work with A. Shkarin [J. Algebra, ]. A-infinity algebras are a class of in general non-associative graded algebras, which are however associative up to a system of higher homotopies. A particular example is provided by the Ext-algebra of a module over an associative algebra. In this talk, I will explain how a version of Koszul duality for these A-infinty algebras provides the construction of a unique exact Borel subalgebra of each quasi-hereditary algebra.

Quasi-hereditary algebras are a class of finite dimensional associative algebras that frequently appear in representation theory, e. This is based on joint work with S.

Koenig, S. Ovsienko, and V. Makhlouf - abstract PDF. We will consider the known superconformal algebras, giving their constructions and the open conjectures. We will also discuss what is known about their representations and some work in progress.

1.2.2 Linear Lie Algebras, Part 2

Taking the exponential of a derivation is a classical tool for producing an automorphism of an algebra, under suitable conditions. Over fields of positive characteristic modified exponentials such as a truncated exponential have been used, notably in the fundamental but rather specialized technique technique of toral switching for modular Lie algebras.

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A crucial aspect of toral switching is that the application of such modified exponentials to a derivation produces a linear map which transforms a certain grading of the algebra into another one. I will analyze the mechanism which makes this work and describe how generalized exponentials based on certain Laguerre polynomials achieve a type of grading switching for nonassociative algebras which goes beyond the reach of toral switching. We introduce a new class of non-associative algebras coming from binary linear codes, called code algebras. This is inspired by the link between binary codes and vertex operator algebras VOAs , where the code governs the representation theory of the code VOA.

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  5. VOAs themselves are important objects: Borcherds used them to show monstrous moonshine - the unexpected link between the Monster simple sporadic group and modular forms - which won him a Fields Medal, but they are also mysterious and difficult to work with. Code algebras have several nice properties: except for one example, they are always simple algebras.

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    There are some obvious idempotents, but we also give a construction for finding other idempotents. This allows us to show when they are axial algebras. These are another type of non-associative algebra, generated by idempotents, with a Peirce-like decomposition of the algebra satisfying some fusion law. When this fusion law is graded it leads to automorphisms of the algebra.

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    I will describe a pair of adjoint functors between the category of Leibniz algebras and the category of differential graded Lie algebras; a non-linear version of these functors relates the category of racks to that of "cubical products". Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Ask Question. Asked 6 years, 5 months ago. Active 6 years, 5 months ago. Viewed times. Schafer starts with a chapter at that generality.

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    For example, Leibniz algebras, Evolution algebras, Bernstein algebras, antiflexible algebras. A generalisation called Hom-power associative algebras has been introduced by Yau. Cohn, Universal algebra, Harper and Row, Chapt. Boris Novikov Boris Novikov 2, 1 1 gold badge 10 10 silver badges 16 16 bronze badges.

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    An Introduction to Nonassociative Algebras

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