We study the notion of Γ-graded commutative algebra for an arbitrary abelian group Γ. The main examples are the Clifford algebras already treated in [2]. We prove that the Clifford algebras are the only simple finitedimensional associative graded commutative algebras over R or C. Our approach also leads to non-associative graded commutative algebras extending the Clifford algebras.

5 Abstract Algebra 1 5.1 Binary Operations on Sets . . . . . . . . . . . . . . . . . . . . 1 5.2 Commutative Binary Operations . . . . . . . . . . . . . . . . 2 5.3 Associative Binary Operations . . . . . . . . . . . . . . . . . . 2 5.4 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5.5 The General Associative Law . . . . . . . . . . . . . . . . . . 4 5.6 Identity elements...

algebra. First, we recall some definitions in the theory of associative algebras. LetA 6= 0 be an algebra over the fieldK. If the equations ax = b, ya = b, ∀a, b ∈ A, a 6= 0, have unique solutions, then the algebra A is called a division algebra. If A is a finite-dimensional algebra, then A is a division algebra if and only if A is without zero divisors (x 6= 0, y 6= 0 ⇒ xy 6= 0).(see [9]) Let ...

Let V =V1 ⊗ V2 be a tensor product of VOAs. Using Zhu theory we discuss the theory of representations of V (associative algebra, modules and ’fusion rules’). We prove that this theory is more or less the same as representation theory of tensor product of the associative algebras.

Algebraic scheme for constructing deformations of structure constants for associative algebras generated by a deformation driving algebras (DDAs) is discussed. An ideal of left divisors of zero plays a central role in this construction. Deformations of associative three-dimensional algebras with the DDA being a three-dimensional Lie algebra and their connection with integrable systems are studied.

By considering representation theory for non-associative algebras we construct the fundamental and adjoint representations of the octonion algebra. We then show how these representations by associative matrices allow a consistent octonionic gauge theory to be realized. We find that non-associativity implies the existence of new terms in the transformation laws of fields and the kinetic term of ...

7 Abstract Algebra 1 7.1 Binary Operations on Sets . . . . . . . . . . . . . . . . . . . . 1 7.2 Commutative Binary Operations . . . . . . . . . . . . . . . . 2 7.3 Associative Binary Operations . . . . . . . . . . . . . . . . . . 2 7.4 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7.5 The General Associative Law . . . . . . . . . . . . . . . . . . 4 7.6 Identity elements...

2 Abstract Algebra 24 2.1 Binary Operations on Sets . . . . . . . . . . . . . . . . . . . . 24 2.2 Commutative Binary Operations . . . . . . . . . . . . . . . . 24 2.3 Associative Binary Operations . . . . . . . . . . . . . . . . . . 24 2.4 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 The General Associative Law . . . . . . . . . . . . . . . . . . 26 2.6 Identity el...

Algebraic scheme for constructing deformations of structure constants for associative algebras generated by a deformation driving algebras (DDAs) is proposed. An ideal of left divisors of zero plays a central role in this construction. Deformations of associative commutative three-dimensional algebras with the DDA being a three-dimensional Lie algebra and their connection with integrable system...

Dendriform algebras are certain splitting of associative and arise naturally from Rota-Baxter operators, shuffle planar binary trees. In this paper, we first consider involutive dendriform algebras, their cohomology homotopy analogs. The an algebra splits the Hochschild algebra. next, introduce a more general notion oriented algebras. We develop theory for that closely related to extensions gov...