List of Results Oleg N. Smirnov

This list is a short overview of some of my results. One can nd a more detailed description in my Research Statement or at 1. Simple associative algebras with nite Z-grading. Here, R is a simple algebra with a nite Z-grading over a commutative ring. A description of all such algebras was given 18]. As a corollary, it was proved that every (not necessarily nite dimensional) simple Lie algebra of characteristic 0 with a non-trivial Z-grading has a Z-grading with at most 5 summands 18]. Comment: For nite dimensional Lie algebras the result is due to Allison 2]. Using the graded associative algebras it was proved that, for every projective complex variety X, the semisimplicity of the ring of correspondences of X implies the Lefschetz standard conjecture 17]. Comment: This fact along with Jannsen's theorem 7] provides the converse to the well-known result by Grothendieck and Kleiman 8] that the Lefschetz standard conjecture implies that the homological and numerical equivalencies are the same. More recent results in this area: A construction which yields every simple algebra with a nite Z-grading is found joint work with Allison, in preparation]. Comment: As a special case it gives Jacobson's rings of all continuous linear transformations of nite rank. A description of the category of graded modules over such algebra R is obtained in prepa-ration]. Comment: It implies the strange fact, that the category of graded R-modules does not depend on the grading of R; these categories are all equivalent! 2. Simple Lie algebras with nite Z-grading. Here, (R;) is a simple associative algebra with involution over a commutative ring k, 1=2 2 k, K = fx 2 R : x = ?xg, K 0 = K; K], and K 0 = K 0 =Z(K 0). A grading means a nite Z-grading. In 19] the notion of symplectic involution was extended to innnite dimensional algebras and it was shown that Every grading of the Lie algebra K (and also of K 0 and K 0) is induced by a unique grading of R. In particular, one has either A. where m 2n, dim(R m) = 1.