Abstract. A digraph is attached to any evolution algebra. This graph leads to some new purely algebraic results on this class of algebras and allows for some new natural proofs of known results. Nilpotency of an evolution algebra will be proved to be equivalent to the nonexistence of oriented cycles in the graph. Besides, the automorphism group of any evolution algebra E with E = E 2 will be shown to be always finite.
The fine abelian group gradings on the simple classical Lie algebras (including D 4 ) over algebraically closed fields of characteristic 0 are determined up to equivalence. This is achieved by assigning certain invariant to such gradings that involve central graded division algebras and suitable sesquilinear forms on free modules over them.
Symplectic (respectively orthogonal) triple systems provide constructions of Lie algebras (respectively superalgebras). However, in characteristic 3, it is shown that this role can be interchanged and that Lie superalgebras (respectively algebras) can be built out of symplectic triple systems (respectively orthogonal triple systems) with a different construction. As a consequence, new simple finite dimensional Lie superalgebras, as well as new models of some nonclassical simple Lie algebras, over fields of characteristic 3, will be obtained.
The new construction given by Barton and Sudbery of the Freudenthal-Tits magic square, which includes the exceptional classical simple Lie algebras, will be interpreted and extended by using a pair of symmetric composition algebras, instead of the standard unital composition algebras.
Abstract. Given a grading by an abelian group G on a semisimple Lie algebra L over an algebraically closed field of characteristic 0, we classify up to isomorphism the simple objects in the category of finite-dimensional G-graded L-modules. The invariants appearing in this classification are computed in the case when L is simple classical (except for type D 4 , where a partial result is given). In particular, we obtain criteria to determine when a finite-dimensional simple L-module admits a G-grading making it a graded L-module.
Freudenthal's Magic Square, which in characteristic 0 contains the exceptional Lie algebras other than G 2 , is extended over fields of characteristic 3, through the use of symmetric composition superalgebras, to a larger square that contains both Lie algebras and superalgebras. With one exception, the simple Lie superalgebras that appear have no counterpart in characteristic 0.
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