scholarly journals On H-simple not necessarily associative algebras

2019 ◽  
Vol 18 (09) ◽  
pp. 1950162
Author(s):  
A. S. Gordienko
Keyword(s):  
Associative Algebra ◽  
Hopf Algebras ◽  
Simple Algebra ◽  
Polynomial Identities ◽  
Associative Algebras ◽  
Graded Algebras ◽  
Finite Dimensional ◽  
Graded Polynomial Identities ◽  
Group Gradings ◽  
Polynomial Formula

An algebra [Formula: see text] with a generalized [Formula: see text]-action is a generalization of an [Formula: see text]-module algebra where [Formula: see text] is just an associative algebra with [Formula: see text] and a relaxed compatibility condition between the multiplication in [Formula: see text] and the [Formula: see text]-action on [Formula: see text] holds. At first glance, this notion may appear too general, however, it enables to work with algebras endowed with various kinds of additional structures (e.g. comodule algebras over Hopf algebras, graded algebras, algebras with an action of a semigroup by anti-endomorphisms). This approach proves to be especially fruitful in the theory of polynomial identities. We show that if [Formula: see text] is a finite dimensional (not necessarily associative) algebra over a field of characteristic [Formula: see text] and [Formula: see text] is simple with respect to a generalized [Formula: see text]-action, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of polynomial [Formula: see text]-identities of [Formula: see text]. In particular, if [Formula: see text] is a finite dimensional (not necessarily group graded) graded-simple algebra, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of graded polynomial identities of [Formula: see text]. In addition, we study the free-forgetful adjunctions corresponding to (not necessarily group) gradings and generalized [Formula: see text]-actions.

scholarly journals Co-stability of Radicals and Its Applications to PI-Theory

2016 ◽  
Vol 23 (03) ◽  
pp. 481-492 ◽  
Author(s):  
A. S. Gordienko
Keyword(s):  
Hopf Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Jacobson Radical ◽  
Associative Algebras ◽  
Finite Dimensional ◽  
Comodule Algebra ◽  
Graded Polynomial Identities ◽  
Finite Dimensional Lie Algebras ◽  
Algebra Over A Field

We prove that if A is a finite-dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite-dimensional, where either char F = 0 or char F > dim A, then the Jacobson radical J(A) is an H-subcomodule of A. In particular, if A is a finite-dimensional associative algebra over such a field F, graded by any group, then the Jacobson radical J(A) is a graded ideal of A. Analogous results hold for nilpotent and solvable radicals of finite-dimensional Lie algebras over a field of characteristic 0. We use the results obtained to prove the analog of Amitsur's conjecture for graded polynomial identities of finite-dimensional associative algebras over a field of characteristic 0, graded by any group. In addition, we provide a criterion for graded simplicity of an associative algebra in terms of graded codimensions.

scholarly journals Lattice isomorphisms of associative algebras

1966 ◽  
Vol 6 (1) ◽  
pp. 106-121 ◽  
Author(s):  
D. W. Barnes
Keyword(s):  
Associative Algebra ◽  
Simple Algebra ◽  
Lattice Isomorphism ◽  
Associative Algebras ◽  
Finite Dimensional ◽  
Image Position

Let A be an associative algebra over the field F. We denote by ℒ(A) the lattice of all subalgebras of A. By an ℒ-isomorphism (lattice isomorphism) of the algebra A onto an algebra B over the same field, we mean an isomorphism of ℒ(A) onto ℒ(B). We investigate the extent to which the algebra B is determined by the assumption that it is ℒ-isomorphic to a given algebra A. In this paper, we are mainly concerned with the case in which A is a finite- dimensional semi-simple algebra.

Cayley Symmetries in Associative Algebras

1963 ◽  
Vol 15 ◽  
pp. 285-290 ◽  
Author(s):  
Earl J. Taft
Keyword(s):  
Associative Algebra ◽  
Associative Algebras ◽  
Wedderburn Decomposition ◽  
Finite Dimensional ◽  
Principal Theorem ◽  
Algebra Over A Field

Let A be a finite-dimensional associative algebra over a field F. Let R denote the radical of A. Assume that A/R is separable. Then it is well known (the Wedderburn principal theorem) that A possesses a Wedderburn decomposition A = S + R (semi-direct), where S is a separable subalgebra isomorphic with A/R. We call S a Wedderburn factor of A.

Weyl Type Non-Associative Algebras Using Additive Groups I

2007 ◽  
Vol 14 (03) ◽  
pp. 479-488 ◽  
Author(s):  
Seul Hee Choi ◽  
Ki-Bong Nam
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Simple Algebra ◽  
Associative Algebras

A Weyl type algebra is defined in the book [4]. A Weyl type non-associative algebra [Formula: see text] and its restricted subalgebra [Formula: see text] are defined in various papers (see [1, 3, 11, 12]). Several authors find all the derivations of an associative (a Lie, a non-associative) algebra (see [1, 2, 4, 6, 11, 12]). We define the non-associative simple algebra [Formula: see text] and the semi-Lie algebra [Formula: see text], where [Formula: see text]. We prove that the algebra is simple and find all its non-associative algebra derivations.

A Note on Algebras that are Sums of Two Subalgebras

2016 ◽  
Vol 59 (2) ◽  
pp. 340-345 ◽  
Author(s):  
Marek Kępczyk
Keyword(s):  
Associative Algebra ◽  
Polynomial Identity ◽  
Arbitrary Field ◽  
Polynomial Identities ◽  
Finite Dimensional ◽  
Finite Dimensional Algebras

AbstractWe study an associative algebra A over an arbitrary field that is a sum of two subalgebras B and C (i.e., A = B+C). We show that if B is a right or left Artinian PI algebra and C is a PI algebra, then A is a PI algebra. Additionally, we generalize this result for semiprime algebras A. Consider the class of all semisimple finite dimensional algebras A = B + C for some subalgebras B and C that satisfy given polynomial identities f = 0 and g = 0, respectively. We prove that all algebras in this class satisfy a common polynomial identity.

Power-Associative Algebras in which Every Subalgebra is a Left Ideal

1970 ◽  
Vol 13 (2) ◽  
pp. 239-243
Author(s):  
D. J. Rodabaugh
Keyword(s):  
Left Ideal ◽  
Associative Algebra ◽  
Nonassociative Algebra ◽  
Single Element ◽  
Associative Algebras ◽  
Finite Dimensional ◽  
Algebra Over A Field

By an L-algebra we mean a power-associative nonassociative algebra (not necessarily finite-dimensional) over a field F in which every subalgebra generated by a single element is a left ideal. An H-algebra is a power-associative algebra in which every subalgebra is an ideal. The H-algebras were characterized by D. L. Outcalt in [2]. Let Sα be the semigroup with cardinality α such that if x, y ∊ Sα then xy = y. Consider the algebra over a field F with basis Sα. Such an algebra is an L-algebra that is not an H-algebra unless Sα contains only one element. In this paper we will prove that an algebra A over a field F with char. ≠ 2 is an L-algebra if and only if it is either an H-algebra or has a basis Sα where α is the dimension of A.

scholarly journals Identities and central polynomials for real graded division algebras

2017 ◽  
Vol 27 (07) ◽  
pp. 935-952
Author(s):  
Diogo Diniz ◽  
Claudemir Fidelis ◽  
Sérgio Mota
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Finite Basis ◽  
Division Algebras ◽  
Polynomial Identities ◽  
Finite Dimensional ◽  
Real Algebra ◽  
Graded Polynomial Identities

Let [Formula: see text] be a finite dimensional simple real algebra with a division grading by a finite abelian group [Formula: see text]. In this paper, we provide a finite basis for the [Formula: see text]-ideal of graded polynomial identities for [Formula: see text] and a finite basis for the [Formula: see text]-space of graded central polynomials for [Formula: see text].

scholarly journals Group gradings and actions of pointed Hopf algebras

2020 ◽  
pp. 2140011
Author(s):  
Yuri Bahturin ◽  
Susan Montgomery
Keyword(s):  
Hopf Algebras ◽  
Associative Algebras ◽  
Matrix Algebras ◽  
Pointed Hopf Algebras ◽  
Group Gradings

We study actions of pointed Hopf algebras on matrix algebras. Our approach is based on known facts about group gradings of matrix algebras [Y. Bahturin, S. Sehgal and M. Zaicev, Group gradings on associative algebras, J. Algebra 241 (2001) 667–698] and other sources.

GRADED POLYNOMIAL IDENTITIES OF VERBALLY PRIME ALGEBRAS

2007 ◽  
Vol 06 (03) ◽  
pp. 385-401 ◽  
Author(s):  
ONOFRIO M. DI VINCENZO ◽  
VINCENZO NARDOZZA
Keyword(s):  
Tensor Product ◽  
Vector Space ◽  
Characteristic Zero ◽  
Grassmann Algebra ◽  
Polynomial Identities ◽  
Graded Algebras ◽  
Infinite Dimensional ◽  
Graded Polynomial Identities ◽  
New Type

Let F be a field and let E be the Grassmann algebra of an infinite dimensional F-vector space. For any p,q ∈ ℕ, the algebra Mp,q(E) can be turned into a ℤp+q × ℤ2-algebra by combining an elementary ℤp+q-grading with the natural ℤ2-grading on E. The tensor product Mp,q(E) ⊗ Mr,s(E) can be turned into a ℤ(p+q)(r+s) × ℤ2-algebra in a similar way. In this paper, we assume that F has characteristic zero and describe a system of generators for the graded polynomial identities of the algebras Mp,q(E) and Mp,q(E) ⊗ Mr,s(E) with respect to these new gradings. We show that this tensor product is graded PI-equivalent to Mpr+qs,ps+qr(E). This provides a new proof of the well known Kemer's PI-equivalence between these algebras. Then we classify all the graded algebras Mp,q(E) having no non-trivial monomial identities, and finally calculate how many non-isomorphic gradings of this new type are available for Mp,q(E).

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