In this paper, we study 4-dimensional nilpotent complex associative algebras. This is a continuation of the study of the moduli space of 4-dimensional algebras. The nonnilpotent algebras were ...analyzed in an earlier paper. Even though there are only 15 families of nilpotent 4-dimensional algebras, the complexity of their behavior warranted a separate study, which we give here.
Classes of algebras and closure operations Gutierrez, I. S.; Torresblanca-Badillo, Anselmo; Towers, David A.
Communications in algebra,
01/2021, Volume:
49, Issue:
6
Journal Article
Peer reviewed
Open access
The calculus of classes and closure operations has proved to be a useful tool in group theory and has led to a deep theory in the study of finite soluble groups. More recently, parallel theories have ...started to be developed in various varieties of algebras, such as Lie, Leibniz and Malcev algebras. This paper seeks to investigate the extent to which these later theories can be generalized to the variety of all non-associative algebras.
The representation of elements of free non-associative algebras as a set of multidimensional tables of coefficients is defined. An operation for finding partial derivatives for elements of free ...non-associative algebras in the same form is considered. Using this representation, a criterion of primitivity for elements of lengths 2 and 3 in terms of matrix ranks, as well as a primitivity test for elements of arbitrary length, is derived. This test makes it possible to estimate the number of primitive elements in free non-associative algebras with two generators over a finite field. The proposed representation allows us to optimize algorithms for symbolic computations with primitive elements. Using these algorithms, we find the number of primitive elements of length 4 in a free non-associative algebra of rank 2 over a finite field.
In this paper, we classify filiform associative algebras of degree
p
over a field of characteristic zero. Moreover, over an algebraically closed field of characteristic zero, we also classify ...filiform nilpotent associative algebras and naturally graded quasi-filiform nilpotent associative algebras, described through the characteristic sequence
C
(
A
)
=
(
n
-
2
,
1
,
1
)
or
C
(
A
)
=
(
n
-
2
,
2
)
.
We prove a hom-associative version of Hilbert’s basis theorem, which includes as special cases both a non-associative version and the classical Hilbert’s basis theorem for associative Ore extensions. ...Along the way, we develop hom-module theory. We conclude with some examples of both non-associative and hom-associative Ore extensions which are all noetherian by our theorem.
Algebras and Varieties Green, Edward L.; Hille, Lutz; Schroll, Sibylle
Algebras and representation theory,
04/2021, Volume:
24, Issue:
2
Journal Article
Peer reviewed
Open access
In this paper we introduce new affine algebraic varieties whose points correspond to associative algebras. We show that the algebras within a variety share many important homological properties. In ...particular, any two algebras in the same variety have the same dimension. The cases of finite dimensional algebras as well as that of graded algebras arise as subvarieties of the varieties we define. As an application we show that for algebras of global dimension two over the complex numbers, any algebra in the variety continuously deforms to a monomial algebra.
The definitions of scattering matrix and inclusive scattering matrix in the framework of formulation of quantum field theory in terms of associative algebras with involution are presented. The ...scattering matrix is expressed in terms of Green functions on shell (LSZ formula), and the inclusive scattering matrix is expressed in terms of generalized Green functions on shell. The expression for inclusive scattering matrix can be used also for quasi-particles (for elementary excitations of any translation-invariant stationary state, for example, for elementary excitations of equilibrium state). An interesting novelty is the consideration of associative algebras over real numbers.
We investigate subalgebras in free Lie algebras, the main tool being relative growth and cogrowth functions. Our study reveals drastic differences in the behavior of proper finitely generated ...subalgebras and nonzero subideals. For instance, the growth of a proper finitely generated subalgebra H of a free Lie algebra L, with respect to any fixed free basis X, is exponentially small compared to the growth of the whole of L. Quite opposite, the cogrowth of any nonzero subideal S is exponentially small compared to the growth of L.
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