Modular An(V) theory Ren, Li
Journal of algebra,
09/2017, Volume:
485
Journal Article
Peer reviewed
Open access
A series of associative algebras An(V) for a vertex operator algebra V over an arbitrary algebraically closed field and nonnegative integers n are constructed such that there is a one to one ...correspondence between irreducible An(V)-modules which are not An−1(V) modules and irreducible V-modules. Moreover, it is proved that V is rational if and only if An(V) are semisimple for all n. In particular, the homogeneous subspaces of any irreducible V-module are finite dimensional for rational vertex operator algebra V.
Cyclic homology for Hom-associative algebras Hassanzadeh, Mohammad; Shapiro, Ilya; Sütlü, Serkan
Journal of geometry and physics,
December 2015, 2015-12-00, Volume:
98
Journal Article
Peer reviewed
Open access
In the present paper we investigate the noncommutative geometry of a class of algebras, called the Hom-associative algebras, whose associativity is twisted by a homomorphism. We define the ...Hochschild, cyclic, and periodic cyclic homology and cohomology for this class of algebras generalizing these theories from the associative to the Hom-associative setting.
In this article, we give an extension of the Fundamental Theorem of finite dimensional algebras to the case of ℤ
2
-graded algebras. Essentially, the results are the same as in the classical case, ...except that the notion of a ℤ
2
-graded division algebra needs to be modified. We classify all finite dimensional ℤ
2
-graded division algebras over ℂ and ℝ.
By the Three Graces we refer, following J.-L. Loday, to the algebraic operads Ass, Com, and Lie, each generated by a single binary operation; algebras over these operads are respectively associative, ...commutative associative, and Lie. We classify all distributive laws (in the categorical sense of Beck) between these three operads. Some of our results depend on the computer algebra system Maple, especially its packages LinearAlgebra and Groebner.
Some general criteria to produce explicit free algebras inside the division ring of fractions of skew polynomial rings are presented. These criteria are applied to some special cases of division ...rings with natural involutions, yielding, for instance, free subalgebras generated by symmetric elements both in the division ring of fractions of the group algebra of a torsion free nilpotent group and in the division ring of fractions of the first Weyl algebra.
In this article we explore an approach to the problem of Albert described by U. Umirbaev. We characterize the space of invariant bilinear forms of the polarization algebra of a polynomial ...endomorphism F:Kn→Kn, where K is a field with characteristic different from two. We obtain some conjectures expressed in the language of polynomial endomorphisms, which are equivalent to the existence of invariant bilinear forms in a finite-dimensional commutative algebra. We give a characterization of the space of invariant bilinear forms in terms of differential forms in the ring Kx1,…,xn. We also introduce a new kind of algebra, we call them totally symmetric algebras, and we establish the relationship between these algebras and the existence of invariant bilinear forms in any commutative algebra.
Higman's PORC theory implies that the number Nd,r(q) of isomorphism types of nilpotent associative algebras of dimension d, rank r and class 2 over a finite field with q elements, considered as a ...function in q, can be described by a polynomial on residue classes in q. We describe an algorithm that, given a rank r, determines such polynomials for Nd,r(q) for all dimensions d. Using this, we determine Nd,r(q) for r∈{1,…,5} and arbitrary d.
Using the knowledge about the finite-dimensional irreducible modules over
sl
2
(
C
)
, it is possible to associate for any of them an irreducible module over the four-dimensional zero-algebra on the ...class of commutative power-associative algebras. This association allows to construct an embedding from the category of
sl
2
(
C
)
-modules into the category of the four-dimensional zero-algebra modules. Furthermore, in this paper it is shown that for any
n
greater than or equal to two, there exist two non-isomorphic families of irreducible modules of dimension 3
n
over the commutative power-associative algebra of dimension four and zero multiplication.
In this paper we introduce a class of noncommutative (finitely generated) monomial algebras whose Hilbert series are algebraic functions. We use the concept of graded homology and the theory of ...unambiguous context-free grammars for this purpose. We also provide examples of finitely presented graded algebras whose corresponding leading monomial algebras belong to the proposed class and hence possess algebraic Hilbert series.
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