Commutative Power-associative Nilalgebras Research Articles

On power-associative modules

The aim of this paper is to study the structure of irreducible modules in the variety [Formula: see text] of commutative power-associative nilalgebras of nilindex [Formula: see text]. If [Formula: see text] with dimension at most 5, then we prove that [Formula: see text] is contained in the annihilator of every irreducible [Formula: see text]-module in the variety [Formula: see text]. Also, we consider the enveloping algebra of an algebra [Formula: see text] in the variety [Formula: see text] and we obtain a new example of a commutative power-associative non-nilpotent nilalgebra of dimension 9.

Power associative nilalgebras of dimension 9

In [19] authors described some properties about commutative power associative nilalgebras of nilindex 5. Here we will get new results about the structure of this class of algebras. Those results will allow us to prove that every commutative power associative algebra of dimension 9 and nilindex 5 over a field of characteristic different from 2, 3 and 5 is solvable. Consequently the famous Albert's conjecture ([1], Problem 1.1) is setted for dimension ≤9 and characteristic 0 since the case of dimension 9 and nilindex ≠5 have already been examined in [27], [11], [14] and [20].

Some results on the structure of a class of commutative power-associative nilalgebras of nilindex four

When a commutative nilalgebra has nilidex four, we can separate their study in three big cases. One of them is when the algebra \({\mathcal {A}}\) satisfies the identity \( (((yx)x)x)x= 0\). In this paper we get results on the structure of this kind of algebras. Let W be the subspace of \({\mathcal {A}}\) generated by the cubes. We prove that \(W^2 = 0\) and we use this to get others properties of W.

On Power-Associative Nilalgebras of Dimension N and Nilindex N-1

Whether or not a finite-dimensional, commutative, power-associative nilalgebra is solvable is a well-known open problem. In this paper, we describe commutative, power-associative nilalgebras of dimension n ≥ 6 and nilindex n − 1 based on the condition that n − 4 ≤ dim 𝔄3 ≤ n − 3. This paper is a continuation of [10], where we describe commutative power-associative nilalgebras of dimension and nilindex n. We observe that the Jordan case was obtained by L. Elgueta and A. Suazo in [2].

On the Classification of Commutative Right-Nilalgebras of Nilindex 5 and Dimension 4

Gerstenhaber and Myung (1975) classified all commutative power-associative nilalgebras of dimension 4. In Elduque and Labra (2007), Gerstenhaber and Myung's results are generalized by giving a classification of commutative right-nilalgebras of right-nilindex 4 and dimension at most 4, without assuming power-associativity. In this article we complete this research and give a classification of commutative right-nilalgebras of right-nilindex 5 and dimension 4, without assuming power-associativity, thus completing the classification of commutative right-nilalgebras of dimension at most 4.

Nilpotence of a class of commutative power-associative nilalgebras

Let A be a commutative algebra over a field F of characteristic ≠ 2 , 3 . In [M. Gerstenhaber, On nilalgebras and linear varieties of nilpotent matrices II, Duke Math. J. 27 (1960) 21–31], M. Gerstenhaber proved that if A is a nilalgebra of bounded index t and the characteristic of F is zero (or greater than 2 t − 3 ), then the right multiplication R x is nilpotent and R x 2 t − 3 = 0 for all x ∈ A . In this work, we prove that this result is also valid for commutative power-associative algebras of characteristic ⩾ t. In Section 3, we prove that when A is a power-associative nilalgebra of dimension ⩽6, then A is nilpotent or ( A 2 ) 2 = 0 . In Section 4, we prove that every power-associative nilalgebra A of dimension n and nilindex t ⩾ n − 1 is either nilpotent of index t or isomorphic to the Suttles' example.

Commutative Power-Associative Nilalgebras of Nilindex 5

In this paper we study the structure of commutative power-associative nilalgebras of dimension 8 and nilindex ≤ 5 over a field of characteristic different from 2, 3 and 5. We prove that every algebra in this class verifies the identities x4y = 0 and x(x(x(x(xy)))) = 0. In particular, we finish the proof of the Albert’s problem [0] in the following case: every commutative power-associative nilalgebra of dimension ≤ 8 over a field of characteristic ≠ 2, 3 and 5 is solvable. The solvability of these algebras for dimension 4, 5 and 6 were proved in [0], [0] and [0] respectively.

Nilpotent linear transformations and the solvability of power-associative nilalgebras

We prove some results about nilpotent linear transformations. As an application we solve some cases of Albert’s problem on the solvability of nilalgebras. More precisely, we prove the following results: commutative power-associative nilalgebras of dimension n and nilindex n − 1 or n − 2 are solvable; commutative power-associative nilalgebras of dimension 7 are solvable.

On the solvability of the commutative power-associative nilalgebras of dimension 6

We prove that commutative power-associative nilalgebras of dimension 6 over a field of characteristic ≠2,3,5 are solvable.

JORDAN NILALGEBRAS OF NILINDEX N AND DIMENSION N+1

ABSTRACT It is known the classification of Jordan nilalgebras of nilindex n and dimension n with . In this work we describe Jordan nilalgebras of nilindex n and dimension . We also give a description of commutative power-associative nilalgebras of dimension 5.

JORDAN NILALGEBRAS OF DIMENSION 6

It is known the classification of commutative power-associative nilalgebras of dimension ? 4 (see, [4]). In [2], we give a description of commutative power-associative nilalgebras of dimension 5. In this work we describe Jordan nilalgebras of dimension 6.

On a Class of Commutative Power-Associative Nilalgebras

We prove that commutative power associative nilalgebras of nilindexnand dimensionnare nilpotent of indexn. We find a necessary and sufficient condition for such an algebra to be a Jordan algebra and give all corresponding isomorphism classes.

On Commutative Power-Associative Nilalgebras of Low Dimension

Commutative power-associative nilalgebras of dimension 4 and characteristic $\ne 2$ are shown to be nilpotent and all their isomorphism classes are determined.

On commutative power-associative nilalgebras of low dimension

Commutative power-associative nilalgebras of dimension 4 and characteristic *2 are shown to be nilpotent and all their isomorphism classes are determined. The long-standing conjecture, originally due to A. A. Albert, that a commutative power-associative nilalgebra of finite dimension over a field is nilpotent, has recently been disproved by Suttles [41, who gave a counterexample of dimension 5. This dimension is generally the best possible, for we show here that every commutative power-associative nilalgebra of dimension 4 over a field of characteristic 4 2 is nilpotent, and we determine the isomorphism classes of all such algebras. The proof is elementary and cannot be significantly simplified by the use of the general results of [1] and [2], which require, moreover, additional assumptions about the characteristic. Throughout, A will denote a commutative power-associative nilalgebra of dimension 4 over a field F of characteristic 4 2. The subspace of A generated by elements u, v, w, ... will be denoted by (u, v, w, ). Since every x E A is nilpotent, the powers of x are linearly independent, so we must have xn = 0 for n > 5; the least n such that xn = 0 for anl x E A is called the nilindex. Now the product of any two elements of A can be written as a linear combination of squares, for xy = ,'4[(x + y)2 _ x 2 _ y 2 ]. Therefore, if the nilindex is 2, then every product vanishes. If it is 5, then A = (x, x2, x3, x4) for any x with x4 A 0. These are trivial cases, in each of which there is, up to isomorphism, a unique, associative, algebra. Only the cases of nilindex 3 and 4 are of interest. 1. Nilindex 3. Linearizing the identity (x2)x 0 yields 2[(xy)z + (yz)x + (zx)y] = 0 for all x, y, z E A. Therefore, denoting right Received by the editors January 2, 1974. AMS (MOS) subject classifications (1970). Primary 17A10.