Noncommutative Associative Algebra Research Articles

Test Elements, Retracts and Automorphic Orbits of Free Algebras

A nonzero element a of an algebra A is called a test element if for any endomorphism φ of A it follows from φ(a)=a that φ is an automorphism of the algebra A. A subalgebra B of A is a retract if there is an ideal I of A such that A=B ⊕ I. We consider the main types of free algebras with the Nielsen–Schreier property: free nonassociative algebras, free commutative and anti-commutative nonassociative algebras, free Lie algebras and superalgebras, and free Lie p-algebras and p-superalgebras. For any free algebra F of finite rank of such type we prove that an element u is a test element if and only if it does not belong to any proper retract of F. Test elements for monomorphisms of F are exactly elements that are not contained in proper free factors of F. These results give analogs of Turner's results on test elements of free groups. We also characterize retracts of the algebra F. We prove that if some endomorphism φ preserve the automorphic orbit of some nonzero element of F, then φ is a monomorphism. For free Lie algebras and superalgebras over a field of characteristic zero and for free Lie p-(super)algebras over a field of prime characteristic p we show that in this situation φ is an automorphism. We discuss some related topics and formulate open problems.

Actions of Hopf algebras

We consider an action of a finite-dimensional Hopf algebra on a non-commutative associative algebra . Properties of the invariant subalgebra in are studied. It is shown that if is integral over its centre then in each of three cases will be integral over (the invariant subalgebra in ): 1) the coradical is cocommutative and char , 2) is pointed, has no nilpotent elements, is an affine algebra, and , 3) is cocommutative. We also consider an action of a commutative Hopf algebra on an arbitrary associative algebra, in particular, the canonical action of on the tensor algebra . A structure theorem on Hopf algebras is proved by application of the technique developed. Namely, every commutative finite-dimensional Hopf algebra whose coradical is a sub-Hopf algebra or cocommutative, where or , is cosemisimple, that is, . In particular, a commutative pointed Hopf algebra with or will be a group Hopf algebra. An example is also constructed showing that the restrictions on are essential.

Quantum space–time and classical gravity

A method has been recently proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example a version of quantized space–time is considered here. It is found that there is a natural differential calculus over this version of quantized space–time using which the only possible torsion-free, metric-compatible, linear connection has zero curvature. It is then the noncummutative version of Minkowski space–time. Perturbations of this calculus are shown to give rise to nontrivial gravitational fields.

Differential calculi and linear connections

A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a one-to-one correspondence, between the module structure of the 1-forms and the metric torsion-free connections on it. In the commutative limit the connection remains as a shadow of the algebraic structure of the 1-forms.

Variations on a theme of Schwinger and Weyl

The theme of doing quantum mechanics on all Abelian groups goes back to Schwinger and Weyl. This theme was studied earlier from the point of view of approximating quantum systems in infinite-dimensional spaces by those associated to finite Abelian groups. This Letter links this theme to deformation quantization, and explores the set of noncommutative associative algebra structures on the Schwartz-Weil algebra of any locally compact separable Abelian group. If the group is a vector space of even dimension over a non-Archimedean local fieldK, there exists a family of noncommutative (Moyal) structures parametrized by the local field and containing membersarbitrarily close to the classical one, although the classical algebra is rigid in the sense of deformation theory. The-products are defined by Fourier integral operators. The problem of constructing sucharithmetic Moyal structures on the algebra of Schwartz-Bruhat functions on manifolds that are locally likeK 2n is raised.

Q-analytic functions on quantum spaces

In this article a different interpretation is given of the quantum plane introduced by Manin and the Q-analytic functions on its two-intervals are characterized. The A1(q) modules of Q-analytic functions defined in this article are in fact noncommutative associative algebras with unity, which can be considered as the deformations of the algebras of analytic functions on the two intervals of R2. Some results obtained by this approach are briefly mentioned.

On commutative non-associative algebras associated with a group of automorphisms of a 2-design

On commutative non-associative algebras associated with a group of automorphisms of a 2-design

A string construction of a commutative non-associative algebra related to the exceptional Jordan algebra

We demonstrate how a bosonic realization of the parastatistical operator product expansion of Fateev and Zamolodchikov relates to and extends the recent representation of the Jordan algebra exhibited in terms of superstring vertex operators found by Goddard, Nahm, Olive, Ruegg and Schwimmer.

On a commutative non-associative algebra associated with a doubly transitive group

“$Ti = xjxi zz -xi xj for l<i<j<n. Putx~=--x~-xx2-...-~x,. As shown in Lemma 1 of [2], we have x0x0 = (I2 1)x,, and xoxi = x,x0 for 1 < i < 17. Therefore if G is a permutation group on R = {O, 1, 2,..., n), then G acts on A by (.xJc = xi8 for g E G and prcscrves the product of A. Hence G can be embedded in the group Aut(A) of the automorphisms of A. Here Aut(A) is defined as Aut(A)= (a& GL(A)j (xy)“=x”y”,x,yEA}. In Theorem A of 121, the author showed that Aut(A) z x,,+, , the symmetric group of degree N + 1, provided that the characteristic of K (denoted by char(K) henceforth) is zero or greater than n + 1. Recently

A natural representation of the Fischer-Griess Monster with the modular function J as character

We announce the construction of an irreducible graded module V for an "affine" commutative nonassociative algebra [unk]. This algebra is an "affinization" of a slight variant [unk] of the commutative nonassociative algebra B defined by Griess in his construction of the Monster sporadic group F(1). The character of V is given by the modular function J(q) = q(-1) + 0 + 196884q +.... We obtain a natural action of the Monster on V compatible with the action of [unk], thus conceptually explaining a major part of the numerical observations known as Monstrous Moonshine. Our construction starts from ideas in the theory of the basic representations of affine Lie algebras and develops further the calculus of vertex operators. In particular, the homogeneous and principal representations of the simplest affine Lie algebra A(1) ((l)) and the relation between them play an important role in our construction. As a corollary we deduce Griess's results, obtained previously by direct calculation, about the algebra structure of B and the action of F(1) on it. In this work, the Monster, a finite group, is defined and studied by means of a canonical infinite-dimensional representation.

On commutative nonassociative algebras associated with the doubly transitive permutation groups PSLm(q), m ≥ 3

On commutative nonassociative algebras associated with the doubly transitive permutation groups PSL_m(q), m ≥ 3

A construction of F 1 as automorphisms of a 196,883-dimensional algebra

In this note, I announce the construction of the finite simple group F(1), whose existence was predicted independently in 1973 by Bernd Fischer and by me. The group has order 2(46)3(20)5(9)7(6)11(2)13(3)17.19.23.29.31.41. 47.59.71 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 and is realized as a group of automorphisms of a 196,883-dimensional commutative nonassociative algebra over the rational numbers, which has an associative form. Equivalently, it is a group of automorphisms of a cubic form in 196,883 variables. It turns out that all the relevant arguments and calculations may be done by hand. Furthermore, existence of the group F(1) implies the existence of a number of other sporadic simple groups for which existence proofs formerly depended on work with computers. We are beginning to look upon this group as a "friendly giant."

Commutative nonassociative algebras and cubic forms

Commutative nonassociative algebras and cubic forms

Sequences in genetic algebras for overlapping generations

When Etherington (2) introduced linear commutative non-associative algebras in connection with problems in theoretical genetics, he pointed out that various sequences of elements in these algebras represented different mating systems. In all such systems it was however assumed that the generations did not overlap, and this restriction has been kept in later work in this field. In this paper we treat sequences which make it possible to find the probability distribution in successive generations in a discrete time model where the generations may be overlapping. We also consider idempotents in genetic algebras and outline how the method used on the overlapping generation sequence may be applied to other sequences.

Commutative Non-Associative Algebras and Identities of Degree Four

The main result of this paper is the following.Theorem 1. Let A be a simple, commutative, finite-dimensional algebra containing an idempotent over a field of characteristic 0, and let the algebra A' obtained from A by adjoining a unity element satisfy an identity of degree ≦ 4 not implied by commutativity. Then either A is a Jordan algebra or A is two-dimensional over an appropriate field E.

Useful Theorems on Commutative Non-associative Algebras

Recently J. M. Osborn has investigated the structure of a simple commutative non-associative algebra with unity element satisfying a polynomial identity, (4), (5) and (6). From his work it seems likely that if such an algebra is of degree three or more it is necessarily power-associative. In (4) he establishes a hierarchy of identities with the property that each identity is satisfied by an algebra satisfying no preceding identity. Following (5), (6),the next identity to consider iswhere a, c and h are elements of the ground field.

On commutative nonassociative algebras

In this paper we study a class of commutative nonassociative algebras which includes those special Jordan algebras which arise as the set of all elements fixed by an involution in a primitive ring with nonzero socle and with centralizer which is a field of characteristic not 2. Perhaps the most unusual feature of our approach is that we do not assume that our algebras satisfy any identity, but only that enough primitive idempotents exist satisfying certain properties that follow from the Jordan identity. (For an introduction to the standard theory, see [6].) We also need an axiom which insures that the discrete topology (Jacobson’s finite topology) will suffice. More specifically, we assume that the following four axioms on idempotents hold in our algebra A over the field @ (A will be assumed to be commutative and of characteristic not 2 hereafter):

Non-Associate Powers and a Functional Equation

Several writers have studied algebras in which multiplication is non-associative, that is, x yz≠xy z. It is necessary in a non-associative algebra to distinguish the possible interpretations of a power xn In a non-commutative non-associative algebra x2 is unique, x3 can mean xx2 or x2x; x4 can mean x xx2, x x2x, x2x2, xx2x or x2x x, x5 has 14 interpretations; x6 has 42; and so on. In a commutative non-associative algebra, the possible interpretations are fewer x3 is unique, x4 can mean xx3 or x2x2, x5 can mean x xx3, x x2x2 or x2x3, x6 has 6 interpretations, and so on. The problem considered here is how many meanings are there for xn (A) in a general non-commutative non-associative algebra ? (B) in a general commutative non-associative algebra ? The answer to (A) is I am not able to find any such simple formula for (B).