The main topic of this thesis is the study of ad-nilpotent elements belonging to Lie algebras and superalgebras. This work could be splitted in two parts: the first part sticks to the branch of Herstein’s theory which studies nilpotent inner derivations in algebras; at the same time, this part can be splitted again into two, the study of nilpotent inner derivations in associative algebras, and the study of nilpotent inner derivations in the super setting. The second part studies Jordan superstructures attached to an ad-nilpotent element of a Lie superalgebra and the subquotients associated to abelian inner ideals of Lie superalgebras.
On one hand, Herstein’s theory, which started in 1954 in [40] (see also the influential works [41] and [63]), is the study of nonassociative objects in associative prime and semiprime rings perhaps with involution, or in rings with well-behaved idempotents that provide a context rich enough for the theory to be satisfactorily developed. Among the main contributors, apart from Herstein itself, we can also cite works of K. I. Beidar, M. Bresar, M. A. Chetobar and W. S. Martindale [6], P. Grzeszczuk [38], T. K. Lee, [54], W. S. Martindale and C. R. Miers [56] and E. C. Posner [63]. Herstein’s theory developed into several similar but different branches: the study of sets with an additional nonassociative structure, as Lie and Jordan ideals (e.g. [58]), culminating in the development of GPI theory ([7]); the study of special conditions (e.g. commutativity) on certain maps (e.g. generalized derivations) over some sets (e.g. Jordan ideals), in which strong knowledge is gained through the a priori weaker properties (e.g. [9], [50], [21], [64]); and the determination of the structure of nonassociative maps, as Lie homomorphisms and derivations (e.g. [4], [5], [6]), culminating in the development of the theory of functional identities ([8]). It is to this last branch of Herstein’s theory that the first part of this thesis is about, centering on the structure of nilpotent derivations, which have been broadly studied since the 1960’. In 1963, Herstein proved that for any ad-nilpotent element a of index n in a simple ring of characteristic zero or greater than n there exists some λ in the center of R such that a − λ is nilpotent. Furthermore, he showed that the index of nilpotence of such element is not greater than , see [42, Theorem page 84]. Herstein’s result was extended by Martindale and Miers in 1983 ([55, Corollary 1]) to prime rings of characteristic greater than n making use of the extended centroid of R. In 1978, Kharchenko obtained in [48] an important result: all algebraic derivations of prime rings of characteristic zero are inner for certain elements in an overring; he extended this result to torsion-free semiprime rings in 1979, see [49]. In 1983, Chung and Luh stated that the index of nilpotence of a nilpotent derivation on a semiprime ring of zero characteristic is always odd (see [16] and [17]), and in 1984 Chung, Kobayashi and Luh ([18]) proved that if R is semiprime and char R = p > 2 then the index of nilpotence of a nilpotent derivation is of the form n = asps +as+1ps+1 +…+alpl where 0 ≤ s ≤ l, the ai are non negative integers less than p, as is odd, and as+1, …, al are even. Moreover, Chung in 1985 proved, for prime rings R of characteristic zero, that a nilpotent derivation is inner and induced by a nilpotent element of an overring, see [15]. In 1992, with different techniques, Grzeszczuk showed that any nilpotent derivation in a semiprime ring with minimal restrictions on its characteristic is an inner derivation in a semiprime subring of the right Martindale ring of quotients of R and is induced by a nilpotent element in such subring, see [38, Corollary 8] and its generalization by Chuang and T. K. Lee in [14, §3].
Some examples of Lie algebras appear when working with rings R with involution *: the Lie algebras of skew-symmetric elements K := Skew(R, *) and K/Z(R) and the derived Lie algebras [K, K] and [K, K]/([K, K] ∩ Z(R)). The nilpotent derivations of the skew-symmetric elements of prime rings with involution were studied by Martindale and Miers in the 1990’s. In this case, if R has zero characteristic and is not an order in a 4-dimensional central simple algebra, for every inner derivation with there exists an element λ in the extended centroid of R such that either or the involution is the identity in the extended centroid of R and , see [56, Main Theorem]. This result was partially extended to semiprime rings by T.K. Lee in 2018. In his main result he proved that if R is semiprime with involution and has no n!-torsion, then for any a ϵ K with there exist λ and a symmetric idempotent ϵ in the extended centroid of R such that , see [54, Theorem 1.5].
In chapter 2 we study ad-nilpotent elements in Lie algebras arising from semiprime associative algebras R free of 2-torsion. With the idea of keeping under control the torsion of R we introduce a more restrictive notion of ad-nilpotent element, pure ad-nilpotent element, which is a technical condition since every ad-nilpotent element can be expressed as an orthogonal sum of pure ad-nilpotent elements of decreasing indices. This allows us to be more precise when setting the torsion inside the algebra R in order to describe its ad-nilpotent elements. If R is a semiprime associative algebra, C(R) its extented centroid and a ϵ R is a pure ad-nilpotent element of R of index n with R free of t and -torsion for , then n is odd and there exists λ ϵ C(R) such that a − λ is nilpotent of index t. If R is a semiprime associative algebra with involution * and a is a pure ad-nilpotent element of Skew(R, *) free of t and -torsion for , then either a is an ad-nilpotent element of R of the same index n (this may occur if n ≡4 1, 3) or R is a nilpotent element of R of index t + 1 and R satisfies a nontrivial GPI (this may occur if n ≡4 0, 3). The case n ≡4 2 is not possible.
On the other hand, an associative superalgebra is a ℤ2-graded associative algebra R = R0 + R1. The elements of R0 ∪ R1 are called homogeneous elements and we say that the degree of a ϵ R0 ∪ R1 is i (denoted |a| = i) when a ϵ Ri, i ϵ {0, 1}. Given an associative superalgebra R, we obtain a Lie superalgebra if the associative product is replaced by the superbracket [a, b] = ab − (−1)|a| |b|ba for homogeneous a, b ϵ R. The Lie structure of prime/simple associative superalgebras was investigated by F. Montaner in [60] and S. Montgomery in [62].
We say that a ℤ2-linear map * : R → R is a superinvolution when (a*)* = a and (ab)* = (−1)|a| |b|b*a* for homogeneous a, b ϵ R0 ∪ R1. The set of skew-symmetric elements of an associative superalgebra is a Lie superalgebra and it will be denoted by K throughout this paper. Moreover, the study of the Lie structure of K of a simple associative superalgebra with superinvolution was iniciated by C. Gómez-Ambrosi and I. Shestakov in 1997 in [37], and their results were extended to prime superalgebras in [35]. The study of superinvolutions in associative superalgebras has been of great interest. We highlight the work of J. Laliena [52] about the description of the derived superalgebra [K, K] of a semiprime superalgebra with superinvolution, and the recent works of A. Giambruno, A. Ioppolo, D. La Mattina and F. Martino ([32], [33], [34], [45]) on superinvolutions in superalgebras related to polynomial identities and related to the growth of certain substructures of the superalgebras.
Another interesting and very active topic in superalgebras is the study of superderivations (see for example the works of A. Fošner and M. Fošner [26], H. Ghahramani, M. N. Ghosseiri and S. Safari [31] or Y. Wang [66]). A linear map d = d0 + d1 in R is called a superderivation if each di, i ϵ {0, 1}, satisfies di(Rj) ⊂ Ri+j and di(ab) = di(a)b + (−1)i|a|adi(b), for homogeneous a, b ϵ R0 ∪ R1. For instance, if a ϵ R0 ∪ R1, the map ada : R → R given by ada(x) = [a, x] is a superderivation (of degree |a|). Such a superderivation is called an inner derivation. In [31] the authors describe the structure of superderivations on some ℤ2-graded rings and study when superderivations are inner.
In chapter 3 we give an in-deph analysis of the nilpotency index of nilpotent homogeneous inner superderivations in associative prime superalgebras with and without superinvolution.
Chapter 4 is devoted to giving examples for all of the types of elements studied in the chapters 2 and 3. Since the even part of an associative superalgebra is an associative algebra and a superinvolution restricted to the even part of an associative superalgebra is an involution, the examples of even ad-nilpotent elements of an associative superalgebra with superinvolution will also provide examples of ad-nilpotent elements of an associative algebra with involution.
Finally, local algebras of Jordan systems were introduced by Meyberg [59], used by Zelmanov and revisited by D’Amour and McCrimmon in their classification of linear and quadratic Jordan systems [67], [19], [20]. Ever since their introduction, they have played a prominent role in the structure theory of Jordan systems, mainly due to the fact that nice properties flow between the system and their local algebras (see for example [1], [2] or [61]).
In [24] E. García, A. Fernández López and M. Gómez Lozano attached a Jordan algebra to any Lie algebra L with an ad-nilpotent element x of index less than or equal to three. Their construction extended the fact that every Lie algebra with an 𝔰l2-triple (e, [e, f], f) is automatically 5-graded relative to the eigenspaces of ad[e,f] and is a unital Jordan algebra. Although their object imitates the construction of a “local” algebra of a Lie algebra, they did not get a Lie algebra again but a Jordan algebra, so this object was called the Jordan algebra of L at x. Furthermore, any ℤ-graded Lie algebra L = L−n ⊕ … ⊕ L0 ⊕ … ⊕ Ln comes together with a Jordan pair V = (L−n, Ln) and any element x of Ln is ad-nilpotent of index less than or equal to three, so one can construct the local algebra of V at x (in the sense of Meyberg [59]) and this Jordan algebra coincides with the Jordan algebra of L at x.
The Jordan algebras of Lie algebras, together with their extension to subquotients (Jordan pairs) associated to abelian inner ideals of Lie algebras, have provided a new way of connecting the Lie and the Jordan settings. For example, they were used by E. Zelmanov in his proof of the Lie version of the Kurosh problem [68, §2], and by J. Hennig in her classification of ad-integrable simple, locally finite Lie algebras over algebraically closed fields of characteristic > 3 [39, Theorem 2]. This construction was also mimicked in [65] to construct a quasi-Jordan algebra from a Leibniz algebra and an ad-nilpotent element of index less than or equal to three.
In chapter 5, given a Lie superalgebra and an even ad-nilpotent element of index less or equal to 3, we can obtain a Jordan superalgebra attached to that element by using the Grassmann envelope; inspired by that construction we build a Jordan superpair attached to an odd ad-nilpotent element of index less or equal to 4. We introduce inner ideals for Lie superalgebras, and we prove that the associated subquotients are Jordan superpairs.
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