In this thesis we have studied ad-nilpotent elements belonging to semiprime associative algebras with involution or prime associative superalgebras with superinvolution, and ad-nilpotent elements in Lie superalgebras. First, we have dealt with semiprime associative algebras with involution. In these algebras we have defined the notion of pure ad-nilpotent element; this notion will be a relevant definition throughout Chapter 2 because it will allow us to give a more precise description of such elements, and to weaken the torsion conditions required to the whole algebra.
We have described the pure ad-nilpotent elements belonging to a semiprime associative algebra R with involution * and belonging to K := Skew(R, *). Indeed, if a is a pure ad-nilpotent element in R of index n, with R free of and s-torsion, with , then there exists λ in the extended centroid of R such that a — λ is nilpotent of index s. On the other hand, if a is a pure ad-nilpotent element in K of index n, the description of a depends on the equivalence class of n modulo 4, and there are three posibilities: If n ≡4 0 then the index of ad-nilpotence of a in R is greater than n and there exists a corner of R that satisfies a PI. If n ≡4 0 then the index of ad-nilpotence of a in R is n and we can conclude that there exists λ in the extended centroid such that a — λ is nilpotent. If n ≡4 3 then a can be descomposed as an orthogonal sum of an ad-nilpotent element of R of index n and another ad-nilpotent element of R of index greater than n. It is important to note that in semiprime associative algebras the extended centroid is not a field, but it is a von Neumann regular ring.
In the next chapter, we have studied homogeneous ad-nilpotent elements in prime associative algebras R = R0+R1 with superinvolution *. We have started by studying the homogeneous ad-nilpotent elements a of index n in R. If a is even, since R0 is an algebra, we can use the above description of ad-nilpotent elements in associative algebras. Although it is an almost direct implication of the previous chapter, we have to deepen into the structure of the extended centroid C(R) to ensure that a — λ is nilpotent with λ an even element in the extended centroid. On the other hand, if a ∈ R1, we have focused on a2. Thus, unlike the descriptions of even elements, two different cases appear: If n is even then a2 ∈ R0 is ad-nilpotent of R of index which implies that there exists λ ∈ C(R)0 such that a2 — λ is nilpotent of index . If n is odd then a is ad-nilpotent of R of index and hence a is nilpotent of index
Continuing in the super setting, we have described the homogeneous ad-nilpotent elements a ∈ K := Skew(R, *) of index n of K. Once we have shown that any homogeneous ad-nilpotent element of K is either an ad-nilpotent element of R of the same index or nilpotent, we can describe these elements in depth. This description depends on the parity of the element: In the even case, the proof and the description is strongly supported by the non-super case. While, in the odd case, we will focus on a2 and thus use the description of even ad-nilpotent elements. More precisely, if a ∈ K1 is an ad-nilpotent element of K of index n and R has characteristic p > n, there are seven possibilities depending on the equivalence class of n modulo 8:
(1) If n ≡8 0 then a is nilpotent of index , ad-nilpotent of R of index n + 1 and (so is a commutative trivial local superalgebra).
(2) If n ≡8 1 then , and a is nilpotent of index and ad-nilpotent of R of index n.
(3) If n ≡8 2 then there exists λ, a skew-symmetric element in the extended centroid, such that a2 — λ is nilpotent of index and a is ad-nilpotent of R of index n.
(4) If n ≡8 5 then , and a is nilpotent of index and ad-nilpotent of R of index n.
(5) If n ≡8 6 then there exists λ, a skew-symmetric element in the extended centroid, such that a2 — λ is nilpotent of index and a is ad-nilpotent of R of index n.
(6) If n ∈8 7 then a is nilpotent of index , ad-nilpotent of R of index n+2 and for every homogeneous k ∈ K (so is a commutative trivial local superalgebra).
(7) The cases n ≡8 3 and n ≡8 4 do not occur.
Afterwards, we have given examples of elements fitting these descriptions. Our examples are matrices considered in the superalgebra M(r|s) over a field with a nontrivial superinvolution. Although these examples are considered in superalgebras, restricting to the even part yields examples in the non-super setting. These examples allow us to ensure that all the cases appearing in our descriptions hold.
Finally, in Chapter 5, given any Lie superalgebra over Φ with , we have studied the even ad-nilpotent elements of index 3 and the odd ad-nilpotent elements of index 4. For the even elements it is possible to associate a Jordan superalgebra to the initial Lie superalgebra by transferring to super setting the existing result in Lie and Jordan algebras due to A. Fernández, E. García and M. Gómez Lozano in [24]. However, for odd ad-nilpotent elements of index 4 we have obtained a Jordan superpair. We have also introduced the notion of subquotient of a Lie superalgebra associated to an abelian inner ideal. Furthermore, the subquotient of a Lie superalgebra associated to an abelian inner ideal is a Jordan superpair, generalizing the structure defined above by the homogeneous ad-nilpotent elements.
We have studied homogeneous ad-nilpotent elements in prime associative superalgebras but we can also study these descriptions in semiprime associative superalgebras. We note that the main difficulty of working on semiprime superalgebras is that the extended centroid drops the property of its elements being invertible with all that this entails. Another possibility could be to study these descriptions for non-homogeneous ad-nilpotent elements.
On the other hand, the subquotients of a Lie superalgebra associated to an abelian inner ideal could be a starting point to study the concept of socle and a Wedderburn-Artin theory for Lie superalgebras following the ideas of C. Draper, A. Fernández, E. García, and M. Gómez Lozano [22]. Some other future research could be to study the relationship between Jordan superstructures and Leibniz superalgebras, as R. Velásquez and R. Felipe have done in the algebra settings [65].
[1]Anquela José A. and Cortés Teresa. Local-to-global inheritance of primitivity in Jordan algebras. Arch. Math. (Basel), 70(3):219-227, 1998.
[2]Anquela José A., Cortés Teresa, and Montaner Fernando. Local inheritance in Jordan algebras. Arch. Math. (Basel), 64(5):393-401, 1995.
[3]Baxter Willard E. and Martindale Wallace S., III. The extended centroid in *-prime rings. Comm. Algebra, 10(8):847-874, 1982.
[4]Beidar K. I., Brešar M., Chebotar M. A., and Martindale W. S., 3rd. On Herstein’s Lie map conjectures. II. J. Algebra, 238(1):239-264, 2001.
[5]Beidar K. I., Brešar M., Chebotar M. A., and Martindale W. S., 3rd. On Herstein’s Lie map conjectures. III. J. Algebra, 249(1):59-94, 2002.
[6]Beidar K. I., Brešar M., Chebotar M. A., and Martindale W. S., III. On Herstein’s Lie map conjectures. I. Trans. Amer. Math. Soc., 353(10):4235-4260, 2001.
[7]Beidar K. I., Martindale W. S., III, and Mikhalev A. V.. Rings with generalized identities, volume 196 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York, 1996.
[8]Brešar Matej, Chebotar Mikhail A., and Martindale Wallace S., III. Functional identities. Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007.
[9]Brešar Matej and Špenko Špela. Functional identities in one variable. J. Algebra, 401:234-244, 2014.
[10]Brox J., López A. Fernández, and Lozano M. Gómez. Clifford elements in Lie algebras. J. Lie Theory, 27(1):283-296, 2017.
[11]Brox J., García E., Gómez Lozano M., Alcázar R. Muñoz, and de Salas G. Vera. Ad-nilpotent elements of skew-index in semiprime associative algebras with involution. Submitted.
[12]Brox J., García E., Gómez Lozano M., Alcázar R. Muñoz, and de Salas G. Vera. A description of ad-nilpotent elements in semiprime rings with involution. Bull. Malays. Math. Sci. Soc., 44(4):2577-2602, 2021.
[13]Brox Jose, García Esther, and Lozano Miguel Gómez. Jordan algebras at Jordan elements of semiprime rings with involution. J. Algebra, 468:155-181, 2016.
[14]Chuang Chen-Lian and Lee Tsiu-Kwen. Nilpotent derivations. J. Algebra, 287(2):381-401, 2005.
[15]Chung L. O.. Nil derivations. J. Algebra, 95(1):20-30, 1985.
[16]Chung L. O. and Luh Jiang. Nilpotency of derivations. Canad. Math. Bull., 26(3):341-346, 1983.
[17]Chung L. O. and Luh Jiang. Corrigendum ti the paper: “nilpotency of derivations”. Canad. Math. Bull., 29(3):383-384, 1986.
[18]Chung Lung O., Kobayashi Yuji, and Luh Jiang. Remark on nilpotency of derivations. Proc. Japan Acad. Ser. A Math. Sci., 60(9):329-330, 1984.
[19]D’Amour Alain and McCrimmon Kevin. The local algebras of Jordan systems. J. Algebra, 177(1):199-239, 1995.
[20]D’Amour Alain and McCrimmon Kevin. The structure of quadratic Jordan systems of Clifford type. J. Algebra, 234(1):31-89, 2000.
[21]Filippis V. De, Rehman N., and Raza M. A.. Strong commutativity preserving skew derivations in semiprime rings. Bull. Malays. Math. Sci. Soc., 41(4):1819-1834, 2018.
[22]Draper Cristina, López Antonio Fernández, García Esther, and Lozano Miguel Gómez. The socle of a nondegenerate Lie algebra. J. Algebra, 319(6):2372-2394, 2008.
[23]López Antonio Fernández. Jordan structures in Lie algebras, volume 240 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2019.
[24]López Antonio Fernández, García Esther, and Lozano Miguel Gómez. The Jordan algebras of a Lie algebra. J. Algebra, 308(1):164-177, 2007.
[25]Fošner Maja. On the extended centroid of prime associative superalgebras with applications to superderivations. Comm. Algebra, 32(2):689-705, 2004.
[26]Fošner Ajda and Fošner Maja. Equations related to superderivations on prime superalgebras. Math. Scand., 115(2):303-319, 2014.
[27]García E. and Lozano M. Gómez. A characterization of the Kostrikin radical of a Lie algebra. J. Algebra, 346(1):266-283, 2011.
[28]García E., Gómez Lozano M., and de Salas G. Vera. Nilpotent superderivations in prime superalgebras. Linear and Multilinear Algebra, 2021, doi: https://doi.org/10.1080/03081087.2021.1919594.
[29]García Esther and Lozano Miguel Gómez. A note on a result of Kostrikin. Comm. Algebra, 37(7):2405-2409, 2009.
[30]García Esther, Lozano Miguel Gíomez, and de Salas Guillermo Vera. Jordan supersystems related to Lie superalgebras. Comm. Algebra, 48(3):992-1000, 2020.
[31]Ghahramani H., Ghosseiri M. N., and Safari S.. Some questions concerning superderivations on ℤ2-graded rings. Aequationes Math., 91(4):725-738, 2017.
[32]Giambruno Antonio, Ioppolo Antonio, and Mattina Daniela La. Varieties of algebras with superinvolution of almost polynomial growth. Algebr. Represent. Theory, 19(3):599-611, 2016.
[33]Giambruno Antonio, Ioppolo Antonio, and Mattina Daniela La. Superalgebras with involution or superinvolution and almost polynomial growth of the codimensions. Algebr. Represent. Theory, 22(4):961–976, 2019.
[34]Giambruno Antonio, Ioppolo Antonio, and Martino Fabrizio. Standard polynomials and matrices with superinvolutions. Linear Algebra Appl., 504:272-291, 2016.
[35]Gómez-Ambrosi Carlos, Laliena Jesús, and Shestakov Ivan P.. On the Lie structure of the skew elements of a prime superalgebra with superinvolution. Comm. Algebra, 28(7):3277-3291, 2000.
[36]Gómez-Ambrosi Carlos and Montaner Fernando. On Herstein’s constructions relating Jordan and associative superalgebras. Comm. Algebra, 28(8):3743-3762, 2000.
[37]Góomez-Ambrosi Carlos and Shestakov Ivan P.. On the Lie structure of the skew elements of a simple superalgebra with superinvolution. J. Algebra, 208(1):43-71, 1998.
[38]Grzeszczuk P.. On nilpotent derivations of semiprime rings. J. Algebra, 149(2):313-321, 1992.
[39]Hennig Johanna. Simple, locally finite dimensional Lie algebras in positive characteristic. J. Algebra, 413:270-288, 2014.
[40]Herstein I. N.. On the Lie ring of a simple ring. Proc. Nat. Acad. Sci. U.S.A., 40:305-306, 1954.
[41]Herstein I. N.. Lie and Jordan structures in simple, associative rings. Bull. Amer. Math. Soc., 67:517-531, 1961.
[42]Herstein I. N.. Sui commutatori degli anelli semplici. Rend. Sem. Mat. Fis. Milano, 33:80-86, 1963.
[43]Herstein I. N.. Topics in ring theory. The University of Chicago Press, Chicago, Ill.-London, 1969.
[44]Herstein I. N.. Topics in ring theory. The University of Chicago Press, Chicago, Ill.-London, 1969.
[45]Ioppolo Antonio. Some results concerning the multiplicities of cocharacters of superalgebras with graded involution. Linear Algebra Appl., 594:51-70, 2020.
[46]Józefiak Tadeusz. Semisimple superalgebras. In Algebra—some current trends (Varna, 1986), volume 1352 of Lecture Notes in Math., pages 96-113. Springer, Berlin, 1988.
[47]Kamiya Noriaki and Okubo Susumu. On δ-Lie supertriple systems associated with (∊, δ)-Freudenthal-Kantor supertriple systems. Proc. Edinburgh Math. Soc. (2), 43(2):243-260, 2000.
[48]Kharchenko V. K.. Differential identities of prime rings. Algebra and Logic, 17(2):155-168, 1978.
[49]Kharchenko V. K.. Differential identities of semiprime rings. Algebra and Logic, 18(1):58-80, 1979.
[50]Koç Emine and Gölbaşi Öznur. Some results on ideals of semiprime rings with multiplicative generalized derivations. Comm. Algebra, 46(11):4905-4913, 2018.
[51]Bukovšek Damjana Kokol and Omladič Matjaž. Linear spaces of symmetric nilpotent matrices. Linear Algebra Appl., 530:384-404, 2017.
[52]Laliena Jesús. The derived superalgebra of skew elements of a semiprime superalgebra with superinvolution. J. Algebra, 420:65-85, 2014.
[53]Lam T. Y.. A first course in noncommutative rings, volume 131 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2008.
[54]Lee Tsiu-Kwen. Ad-nilpotent elements of semiprime rings with involution. Canad. Math. Bull., 61(2):318-327, 2018.
[55]Martindale W. S., III and Miers C. Robert. On the iterates of derivations of prime rings. Pacific J. Math., 104(1):179-190, 1983.
[56]Martindale W. S., III and Miers C. Robert. Nilpotent inner derivations of the skew elements of prime rings with involution. Canad. J. Math., 43(5):1045-1054, 1991.
[57]Martindale Wallace S., III. Prime rings satisfying a generalized polynomial identity. J. Algebra, 12:576-584, 1969.
[58]Martindale Wallace S., III and Miers C. Robert. Herstein’s Lie theory revisited. J. Algebra, 98(1):14-37, 1986.
[59]Meyberg Kurt. Lectures on algebras and triple systems. The University of Virginia, Charlottesville, Va., 1972. Notes on a course of lectures given during the academic year 1971-1972.
[60]Montaner Fernando. On the Lie structure of associative superalgebras. Comm. Algebra, 26(7):2337-2349, 1998.
[61]Montaner Fernando. Local PI theory of Jordan systems. J. Algebra, 216(1):302–327, 1999.
[62]Montgomery S.. Constructing simple Lie superalgebras from associative graded algebras. J. Algebra, 195(2):558-579, 1997.
[63]Posner Edward C.. Derivations in prime rings. Proc. Amer. Math. Soc., 8:1093–1100, 1957.
[64]Rehman Nadeem Ur and Raza Mohd Arif. On Lie ideals with generalized derivations and non-commutative Banach algebras. Bull. Malays. Math. Sci. Soc., 40(2):747-764, 2017.
[65]Velásquez Raúl and Felipe Raúl. Quasi-Jordan algebras. Comm. Algebra, 36(4):1580-1602, 2008.
[66]Wang Yu. Lie superderivations of superalgebras. Linear Multilinear Algebra, 64(8):1518-1526, 2016.
[67]Zelmanov E. I.. Primary Jordan triple systems. Sibirsk. Mat. Zh., 24(4):23-37, 1983.
[68]Zelmanov Efim. Lie algebras and torsion groups with identity. J. Comb. Algebra, 1(3):289-340, 2017.