Embedding Theorems for L-Algebras and Anti-commutative Algebras

2019 ◽  
Vol 26 (01) ◽  
pp. 51-64
Author(s):  
Qiuhui Mo
Keyword(s):  
Associative Algebra ◽  
Commutative Algebra ◽  
Embedding Theorems ◽  
Commutative Algebras

Bokut, Chen and Huang proved that every countably generated L-algebra over a countable field can be embedded into a simple two-generated L-algebra. In this paper, we prove that every countably generated L-algebra can be embedded into a simple two-generated L-algebra. We also prove that every anti-commutative algebra can be embedded into a simple anti-commutative algebra, and that every countably generated anti-commutative algebra can be embedded into a simple two-generated anti-commutative algebra. Finally, we prove that every anti-commutative algebra can be embedded into its universal enveloping non-associative algebra.

scholarly journals Extended letterplace correspondence for nongraded noncommutative ideals and related algorithms

2014 ◽  
Vol 24 (08) ◽  
pp. 1157-1182 ◽  
Author(s):  
Roberto La Scala
Keyword(s):  
Associative Algebra ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
Free Associative Algebra ◽  
Commutative Algebras ◽  
Skew Polynomial Rings ◽  
Experimental Implementation ◽  
Buchberger Algorithm ◽  
Skew Polynomial

Let K〈xi〉 be the free associative algebra generated by a finite or a countable number of variables xi. The notion of "letterplace correspondence" introduced in [R. La Scala and V. Levandovskyy, Letterplace ideals and non-commutative Gröbner bases, J. Symbolic Comput. 44(10) (2009) 1374–1393; R. La Scala and V. Levandovskyy, Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra, J. Symbolic Comput. 48 (2013) 110–131] for the graded (two-sided) ideals of K〈xi〉 is extended in this paper also to the nongraded case. This amounts to the possibility of modelizing nongraded noncommutative presented algebras by means of a class of graded commutative algebras that are invariant under the action of the monoid ℕ of natural numbers. For such purpose we develop the notion of saturation for the graded ideals of K〈xi,t〉, where t is an extra variable and for their letterplace analogues in the commutative polynomial algebra K[xij, tj], where j ranges in ℕ. In particular, one obtains an alternative algorithm for computing inhomogeneous noncommutative Gröbner bases using just homogeneous commutative polynomials. The feasibility of the proposed methods is shown by an experimental implementation developed in the computer algebra system Maple and by using standard routines for the Buchberger algorithm contained in Singular.

Commutative non-power-associative algebras

2019 ◽  
Vol 29 (08) ◽  
pp. 1527-1539
Author(s):  
Manuel Arenas ◽  
Irvin Roy Hentzel ◽  
Alicia Labra
Keyword(s):  
Associative Algebra ◽  
Associative Algebras ◽  
Commutative Algebras ◽  
Dimension Formula

We study commutative algebras satisfying the identity [Formula: see text] It is known that for [Formula: see text] and for characteristic not [Formula: see text] or [Formula: see text], the algebra is a commutative power-associative algebra. These algebras have been widely studied by Albert, Gerstenhaber and Schafer. For [Formula: see text] Guzzo and Behn in 2014 proved that commutative algebras of dimension [Formula: see text] satisfying [Formula: see text] are solvable. We consider the remaining values of [Formula: see text] We prove that commutative algebras satisfying [Formula: see text] with [Formula: see text] and generated by one element are nilpotent of nilindex [Formula: see text] (we assume characteristic of the field [Formula: see text]).

On Reductive Lie Admissible Algebras

1971 ◽  
Vol 23 (2) ◽  
pp. 325-331 ◽  
Author(s):  
Arthur A. Sagle
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Associative Algebra ◽  
Commutative Algebra ◽  
Jacobi Identity ◽  
Direct Sum ◽  
Derivation Algebra ◽  
Image Position

A Lie admissible algebra is a non-associative algebra A such that A− is a Lie algebra where A− denotes the anti-commutative algebra with vector space A and with commutation [X, Y] = XY – YX as multiplication; see [1; 2; 5]. Next let L−(X): A− → A−: Y → [X, Y] and H = {L−(X): X ∊ A−}; then, since A− is a Lie algebra, we see that H is contained in the derivation algebra of A− and consequently the direct sum g = A − ⊕ H can be naturally made into a Lie algebra with multiplication [PQ] given by: P = X + L−(U), Q = Y + L−(V) ∊ g, thenand note that for any P, [PP] = 0 so that [PQ] = −[QP] and the Jacobi identity for g follows from the fact that A− is Lie.

scholarly journals GL-EQUIVARIANT MODULES OVER POLYNOMIAL RINGS IN INFINITELY MANY VARIABLES. II

2019 ◽  
Vol 7 ◽  
Author(s):  
STEVEN V SAM ◽  
ANDREW SNOWDEN
Keyword(s):  
Commutative Algebra ◽  
Polynomial Rings ◽  
Commutative Algebras ◽  
Representation Stability ◽  
Veronese Embeddings ◽  
Twisted Commutative Algebras

Twisted commutative algebras (tca’s) have played an important role in the nascent field of representation stability. Let $A_{d}$ be the tca freely generated by $d$ indeterminates of degree 1. In a previous paper, we determined the structure of the category of $A_{1}$-modules (which is equivalent to the category of $\mathbf{FI}$-modules). In this paper, we establish analogous results for the category of $A_{d}$-modules, for any $d$. Modules over $A_{d}$ are closely related to the structures used by the authors in previous works studying syzygies of Segre and Veronese embeddings, and we hope the results of this paper will eventually lead to improvements on those works. Our results also have implications in asymptotic commutative algebra.

scholarly journals Nonnoetherian geometry

2016 ◽  
Vol 15 (09) ◽  
pp. 1650176 ◽  
Author(s):  
Charlie Beil
Keyword(s):  
Commutative Algebra ◽  
Projective Dimension ◽  
Krull Dimension ◽  
Maximal Dimension ◽  
Algebraic Varieties ◽  
Finitely Generated ◽  
One Dimensional ◽  
Simple Modules ◽  
Commutative Algebras

We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension. The resulting geometries are algebraic varieties with positive-dimensional points, and are thus inherently nonlocal. These notions also give rise to new equivalent characterizations of noetherianity that are primarily geometric. We then consider an application to quiver algebras whose simple modules of maximal dimension are one dimensional at each vertex. We show that the vertex corner rings of [Formula: see text] are all isomorphic if and only if [Formula: see text] is noetherian, if and only if the center [Formula: see text] of [Formula: see text] is noetherian, if and only if [Formula: see text] is a finitely generated [Formula: see text]-module. Furthermore, we show that [Formula: see text] is depicted by a commutative algebra generated by the cycles in its quiver. We conclude with an example of a quiver algebra where projective dimension and geometric codimension, rather than height, coincide.

scholarly journals On some infinitely presented associative algebras

1973 ◽  
Vol 16 (3) ◽  
pp. 290-293 ◽  
Author(s):  
Jacques Lewin
Keyword(s):  
Associative Algebra ◽  
Commutative Algebra ◽  
Polynomial Identity ◽  
Finite Dimension ◽  
Negative Answer ◽  
Free Associative Algebra ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Commutator Ideal ◽  
Finitely Presented

We prove here that if F is a finitely generated free associative algebra over the field and R is an ideal of F, then F/R2 is finitely presented if and only if F/R has finite dimension. Amitsur, [1, p. 136] asked whether a finitely generated algebra which is embeddable in matrices over a commutative f algebra is necessarily finitely presented. Let R = F′, the commutator ideal of F, then [4, theorem 6], F/F′2 is embeddable and thus provides a negative answer to his question. Another such example can be found in Small [6]. We also show that there are uncountably many two generator I algebras which satisfy a polynomial identity yet are not embeddable in any algebra of n xn matrices over a commutative algebra.

Born geometry on ρ-commutative algebra

2020 ◽  
Vol 17 (14) ◽  
pp. 2050210
Author(s):  
Zahra Bagheri ◽  
Esmaeil Peyghan
Keyword(s):  
Commutative Algebra ◽  
Riemannian Geometry ◽  
Fundamental Theorem ◽  
Existence And Uniqueness ◽  
Commutative Algebras

The aim of this paper is to establish a generalization of the Born geometry to [Formula: see text]-commutative algebras. We introduce the notion of Born [Formula: see text]-commutative algebras and study the existence and uniqueness of a torsion connection which preserves the Born structure. Also, an analogue of the fundamental theorem of Riemannian geometry will be proved for these algebras.

ANTI-COMMUTATIVE ALGEBRAS WITH SKEW-SYMMETRIC IDENTITIES

2009 ◽  
Vol 08 (02) ◽  
pp. 157-180 ◽  
Author(s):  
A. S. DZHUMADIL'DAEV
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Algebra ◽  
Symmetric Polynomial ◽  
Commutative Algebras ◽  
Associative Commutative Algebra ◽  
Symmetric Identities ◽  
Generalized Lie Algebras

Generalizing Lie algebras, we consider anti-commutative algebras with skew-symmetric identities of degree > 3. Given a skew-symmetric polynomial f, we call an anti-commutative algebra f-Lie if it satisfies the identity f = 0. If sn is a standard skew-symmetric polynomial of degree n, then any s4-Lie algebra is f-Lie if deg f ≥ 4. We describe a free anti-commutative super-algebra with one odd generator. We exhibit various constructions of generalized Lie algebras, for example: given any derivations D, F of an associative commutative algebra U, the algebras (U, D ∧ F) and (U, id ∧ D2) are s4-Lie. An algebra (U, id ∧ D3 - 2D ∧ D2) is s'5-Lie, where s'5 is a non-standard skew-symmetric polynomial of degree 5.

XXV.—Properties of I-extensions of Anti-commutative Algebras

1961 ◽  
Vol 65 (4) ◽  
pp. 345-357
Author(s):  
E. W. Wallace
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Algebra ◽  
Characteristic Zero ◽  
Cartesian Product ◽  
Paper Properties ◽  
Commutative Algebras ◽  
The Given

SYNOPSISS. T. Tsou and A. G. Walker have defined the I-extension of a given Lie algebra as a certain Lie algebra on the Cartesian product of the given algebra and one of its ideals (Tsou 1955). I-extensions have been studied also in connection with metrisable Lie groups and metrisable Lie algebras. The definition can be applied immediately to any anti-commutative algebra, and in this paper properties of such I-extensions are established. A list of all proper I-extensions of dimension not greater than four over a field of characteristic zero is also given together with a set of characters.

scholarly journals Wronskian Envelope of a Lie Algebra

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Laurent Poinsot
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Commutative Algebra ◽  
Differential Algebra ◽  
Free Lie Algebra ◽  
Sufficient Condition ◽  
Enveloping Algebra ◽  
Lie Bracket ◽  
Differential Algebras

The famous Poincaré-Birkhoff-Witt theorem states that a Lie algebra, free as a module, embeds into its associative envelope—its universal enveloping algebra—as a sub-Lie algebra for the usual commutator Lie bracket. However, there is another functorial way—less known—to associate a Lie algebra to an associative algebra and inversely. Any commutative algebra equipped with a derivation , that is, a commutative differential algebra, admits a Wronskian bracket under which it becomes a Lie algebra. Conversely, to any Lie algebra a commutative differential algebra is universally associated, its Wronskian envelope, in a way similar to the associative envelope. This contribution is the beginning of an investigation of these relations between Lie algebras and differential algebras which is parallel to the classical theory. In particular, we give a sufficient condition under which a Lie algebra may be embedded into its Wronskian envelope, and we present the construction of the free Lie algebra with this property.

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