BUSCADOR RECOLECTA

S-structures on Lie algebras, introduced by Vinberg, represent a broad generalization of the notion of gradings by abelian groups. Gradings by, not necessarily reduced, root systems provide many examples of natural S-structures. Here we deal with a situation not covered by these gradings: the short (SL2xSL2)-structures, where the reductive group is the simplest semisimple but not simple reductive group. The algebraic objects that coordinatize these structures are the J-ternary algebras of Allison, endowed with a nontrivial idempotent.

We study Jordan 3-graded Lie algebras satisfying 3-graded polynomial identities. Taking advantage of the Tits-Kantor-Koecher construction, we interpret the PI-condition in terms of their associated Jordan pairs, which allows us to formulate an analogue of Posner-Rowen Theorem for strongly prime PI Jordan 3-graded Lie algebras. Arbitrary PI Jordan 3-graded Lie algebras are also described by introducing the Kostrikin radical of the Lie algebras.

The nontrivial unital composition superalgebras, of dimension 3 and 6, which exist only in characteristic 3, are obtained from the split Cayley algebra and its order 3 automorphisms, by means of the process of semisimplification of the symmetric tensor category of representations of the cyclic group of order 3. Connections with the extended Freudenthal magic square in characteristic 3, that contains some exceptional Lie superalgebras specific of this characteristic are discussed too. In the process, precise recipes to go from (nonassociative) algebras in this tensor category to the corresponding superalgebras are given.

Kac’s ten-dimensional simple Jordan superalgebra over analgebraically closed field of characteristic 5 is obtained froma process of semisimplification, via tensor categories, fromthe exceptional simple Jordan algebra (or Albert algebra),together with a suitable order 5 automorphism.This explains McCrimmon’s ‘bizarre result’ asserting that,in characteristic 5, Kac’s superalgebra is a sort of ‘degree 3Jordan superalgebra’.As an outcome, the exceptional simple Lie superalgebrael(5; 5), specific of characteristic 5, is obtained from the simpleLie algebra of type E8 and an order 5 automorphism.In the process, precise recipes to obtain superalgebras fromalgebras in Rep Cp (or Rep αp), p > 2, are given

We consider the problem of classifying gradings by groups on a finite-dimensional algebra A (with any number of multilinear operations) over an algebraically closed field. We introduce a class of gradings, which we call almost fine, such that every G-grading on A is obtained from an almost fine grading on A in an essentially unique way, which is not the case with fine gradings. For abelian G, we give a method of obtaining all almost fine gradings if fine gradings are known. We apply these ideas to the case of semisimple Lie algebras in characteristic 0: to any abelian group grading with nonzero identity component, we attach a (possibly nonreduced) root system Φ and, in the simple case, construct an adapted Φ-grading.

We study Jordan 3-graded Lie algebras satisfying 3-graded polynomial identities. Taking advantage of the Tits-Kantor-Koecher construction, we interpret the PI-condition in terms of their associated Jordan pairs, which allows us to formulate an analogue of Posner-Rowen Theorem for strongly prime PI Jordan 3-graded Lie algebras. Arbitrary PI Jordan 3-graded Lie algebras are also described by introducing the Kostrikin radical of the Lie algebras., Authors are partially supported by grant PID2021-123461NB-C21, funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” and grant E22-23 Álgebra y Geometría, Gobierno de Aragón.

We study the sets of elements of Jordan pairs whose local Jordan algebras are Lesieur-Croisot algebras, that is, classical orders in nondegenerate Jordan algebras with finite capacity. It is then proved that, if the Jordan pair is nondegenerate, the set of its Lesieur-Croisot elements is an ideal of the Jordan pair., Fernando Montaner was partially supported by grants MTM2017-83506-C2-1-P (AEI/FEDER, UE) and PID2021-123461NB-C21, funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”, and by Diputación General de Aragón (Grupo de Investigación Álgebra y Geometría). Irene Paniello was partially supported by grants MTM2017-83506-C2-1-P (AEI/FEDER, UE) and PID2021-123461NB-C21, funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”.

We formulate the notion of continuous evolution algebra in terms of differentiable matrix-valued functions, to then study those such algebras arising as solutions of ODE problems. Given their dependence on natural bases, matrix Lie groups provide a suitable framework where considering time-variant evolution algebras. We conclude by broadening our approach by considering continuous evolution algebras stemming as flow lines on matrix Lie groups., Supported by grants MTM2017-83506-C2-1-P (AEI/FEDER, UE) and PID2021-123461NB-C21, funded by MCIN/AEI/10.13039/501100011033 and by ERDF A way of making Europe. The first author is also supported by Diputación General de Aragón (Grupo de Investigación Álgebra y Geometría)