Ad-nilpotent elements in algebras and superalgebras

In this thesis we will deal with ad-nilpotent elements in associative algebras and superalgebras with involution and superinvolution, and ad-nilpotent elements in Lie superalgebras. The first aim of this work fits with Herstein’s branch of theory that studies nilpotent inner derivations in algebras. There are many studies on this area, highlighting for our work the articles of W. S. Martindale and C. R. Miers [55], [56] and T. K. Lee [54]. Later, in the second part, we study how to associate some Jordan structures to a Lie superalgebra, motivated by the work of A. Fernández, E. García and M. Gómez Lozano [24].

Three objectives are addressed throughout this thesis. In the first instance, we seek to describe in detail the ad-nilpotent elements in semiprime associative algebras with involution. The second aim of this thesis is to carry over the descriptions of ad-nilpotent elements in semiprime associative algebras to prime associative superalgebras, that is, to give a detailed description of homogeneous ad-nilpotent elements belonging to prime associative superalgebras. Finally, motivated by the work of A. Fernández, E. García and M. Gómez Lozano in [24], to associate a Jordan superstructure to a Lie superalgebra with an ad-nilpotent element of a certain index.

To develop the first two goals we have worked within the framework of semiprime algebras with involution and prime associative superalgebras with superinvolution. Moreover, the extended centroid will be an important tool in this thesis. For the last objective, we have worked with nonassociative superstructures such as Lie and Jordan superalgebras, defined by the Grassmann envelope, and Jordan superpairs. We can highlight the high combinatorial content throughout the entire thesis.

We have successfully covered the three initial goals. First, we have described in detail ad-nilpotent elements belonging to a semiprime associative algebra. Moreover, we have succeeded in reducing the torsion in the classification of ad-nilpotent elements in semiprime associative algebras with involution due to the new concept of a pure ad-nilpotent element, introduced in this thesis in Chapter 2. The conditions on the scalar rings has been weakened to be free of (ns) and s torsion with s := [n+12] instead of being free of n! torsion.

On the other hand, for the skew-symmetric ad-nilpotent elements of a semiprime associative algebra R with involution *, we have given a description that depends on their ad-nilpotent index modulo 4. In this description we can emphasize: If a skew-symmetric element a is ad-nilpotent such that its index of ad-nilpotence of K := Skew(R, *) and R do not coincide, that is, adanK=0 but adanR≠0, (it can only occur for ad-nilpotent indices of K congruent to 0 or 3 modulo 4) then a certain corner of R satisfies a PI, hence R holds a GPI. These results have been developed throughout Chapter 2 which have originated an article that has been published in the journal Bulletin of the Malaysian Mathematical Sciences Society ([12]). The second aim, to describe in prime associative superalgebras with superinvolution nilpotent inner derivations, has also been positively solved during Chapter 3. This description depends on the parity of the homogeneous element: if the element is even, what has been developed in the previous chapter in algebra settings ([12]), is largely rescued. However, if the element is odd, we have worked on its square, which is an even ad-nilpotent element, and we have applied the descriptions for even ad-nilpotent elements studied above. These results has been published in the journal Linear and Multilinear Algebra ([28]). During Chapter 4, we have given examples for each of the cases appearing in the descriptions of the elements in both algebras and superalgebras, thus showing that these descriptions are not trivial. Finally, in Chapter 5, we have associated a Jordan superstructure to a Lie superalgebra L with a homogeneous ad-nilpotent element a of index 3 or 4, according to its parity. Furthermore, the Jordan superpair we have constructed following the spirit of the paper of A. Fernández, E. García and M. Gómez Lozano [24], coincides with the subquotient of a Lie superalgebra associated with an abelian inner ideal [a, [a, L]]. This last chapter has been published and can be consulted in the journal Communications in Algebra ([30]).