), Alexandre Pozhidaev (app@math.nsc.ru), Yury Volkov (wolf86 666@list.ru).Abstract. We describe degenerations of four-dimensional Zinbiel and four-dimensional nilpotent Leibniz algebras over C. In particular, we describe all irreducible components in the corresponding varieties.

We prove that all Rota-Baxter operators on a quadratic division algebra are trivial. For nonzero weight, we state that all Rota-Baxter operators on the simple odd-dimensional Jordan algebra of bilinear form are projections on a subalgebra along another one. For weight zero, we find a connection between the Rota-Baxter operators and the solutions to the alternative Yang-Baxter equation on the Cayley-Dickson algebra. We also investigate the Rota-Baxter operators on the matrix algebras of order two, the Grassmann algebra of plane, and the Kaplansky superalgebra.Mathematics Subject Classification. 16T25, 17A45, 17C50.

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We prove a coordinatization theorem for noncommutative Jordan superalgebras of degree n > 2, describing such algebras. It is shown that the symmetrized Jordan superalgebra for a simple finite-dimensional noncommutative Jordan superalgebra of characteristic 0 and degree n > 1 is simple. Modulo a "nodal" case, we classify central simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0.

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It is proved that there exist no simple finite-dimensional Filippov superalgebras of type A(n, n) over an algebraically closed field of characteristic 0.

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