Ad-nilpotent elements in algebras and superalgebras

Abstract

In this thesis we will deal with ad nil potent elements in associative algebras and super algebras with involution and super involution, and ad nil potent element sin Lie super algebras. The rstaim of this work ts with Herstein's branch of theory that studies nil potent inner derivations in algebras. There are many studies on this area, high lighting for our work the articles of W.S.Martin dale and C.R.Miers [55],[56] and T.K.Lee [54]. Later, in these condpart, we study how to associate some Jordan structures to a Lie super algebra , motivated by the work of A. Fernandez, E. García and M. Gómez Lozano [24]. Objectives Three objectives are addressed through out this thesis. In therst instance ,we seek to describe in detail thead-nil potent elements in semi prime associative algebras with involution. These condaim of this thesis is to carry over the description so fad-nil potent elements in semi prime associative algebras to prime associative super algebras, that is, to give a detailed description of homogeneous ad nil potent elements belonging to prime associative super algebras. Finally, motivated by the work of A.Fernández, E. García and M. Gómez Lozano in[24], to associate a Jordan super structure to a Lie super algebra with a nad nil potent element of a certain in dex. Methodology To develop the goals we have worked with in the frame work of semi prime algebras with involution and prime associative super algebras with super involution.

Moreover ,the extended centro id will be an important tool in this thesis. For the last objective ,we have worked with non associative super structures such as Lie and Jordan super algebras, de ned by the Grass mann envelope, and Jordan super pairs. We can high light the high combinatorial content through out theen tire thesis. Results We have successful ly covered the three initial goals. First, we have described in detail ad nil potent elements belonging to a semiprime associative algebra. Moreover, we have succeeded in reducing the torsion in the classi cation of ad-nil potent elements in semi prime associative algebras with involution due to the new concept of a pure ad nil potent element , introduced in this thesis in Chapter2. The conditions on the scalar rings has been weakened to be free of on and torsion with s := [n+12 ] instead of being free of n! torsion. On the other hand, for thes kew-symmetricad-nil potent elements of a semi prime associative algebra R with involution , we have given a description that depend son their ad-nil potent index modulo 4. In this description we can emphasize: If ask ew-symmetric element a is ad-nil potent such that its index of ad-nil potence of K := Skew(R; ) and R do not coincide , that is, adn aK = 0 but adn a R 6= 0, (it can only occur for ad-nil potent in dices of K congruent to 0 or 3 modulo 4) then a certain corner of R satis es a PI, hence R holds a GPI. These results have been developed through out Chapter2 which have originated an article that has been published in the journal Bulletin of the Malaysian Mathematical Sciences Society ([12]). These condaim, to describe in prime associative super algebras with super involution nil potent inner derivations, has also been positive ly solved during Chapter 3. This description depend son 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-nil potent element ,and we have applied the descriptions for even ad-nil potent elements studied above. These results has been published in the journal Linear and Multi linear 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 super algebras ,thus showing that these descriptions are not trivial. Finally, in Chapter 5, we have associated a Jordan super structure to a Lie super algebra L with a homogeneous ad nil potent element a of index 3or4, according to its parity. Furthermore, the Jordan super pair we have constructed following the spirit of the paper of A. Fernández, E. García and M. Gómez Lozano[24], coincides with the sub quo tient of a Lie super algebra associated with anabel ian inner ideal[a; [a; L]]. This last chapter has been published and can be consulted in the journal Communication sin Algebra ([30]).