February 3, Graduate Center: Alexander A. Mikhalev (Moscow State University), Free Akivis Algebras
Abstract: A vector space A over a field F is called an Akivis algebra if it is endowed with an anticommutative bilinear operation [x,y] (a commutator) and a trilinear operation (x,y,z) (an associator) that satisfy the identity [[x,y],z] + [[y,z],x] + [[z,x],y] = (x,y,z) + (y,z,x) + (z,x,y) - (y,x,z) - (x,z,y) - (z,y,x). These algebras were introduced by M.A.Akivis as tangent algebras of local analytic loops. If B is an algebra over a field and [x,y]=xy-yx, (x,y,z)=(xy)z-x(yz), then the algebra B with these operations is an Akivis algebra (we denote it by Ak(B)). Let Ak(X) be the free Akivis algebra over a field F with the set X of free generators, F(X) the free nonassociative algebra over the field F with the same set X of free generators. Then the algebra Ak(X) is isomorphic to the subalgebra of Ak(F(X)) generated by the set X.
I.P.Shestakov and U.U.Umirbaev proved that subalgebras of free Akivis algebras are free, i.e. the variety of all Akivis algebras over a field F is a Schreier variety. An element u of Ak(X)) is said to be a primitive element (a coordinate polynomial) if it is an element of some set of free generators of the algebra Ak(X).
In this talk we consider the problem of recognizing automorphisms of free Akivis algebras. We prove the Freiheitssatz for free Akivis algebras. We also show that an element u of Ak(X) is a primitive element if and only if the factor algebra of Ak(X) by the ideal generated by the element u is a free Akivis algebra. We also consider some properties of primitive elements. The talk is based on a joint work with I.P.Shestakov.

February 10, Graduate Center: Alexander V. Mikhalev (Moscow State University), Nonassociative cryptography
Abstract: The main goal of the talk is to show how to use nonassociative algebraic structures in cryptography: cryptosystems over quasigroup rings; Moufang loops, Page loops; alternative rings.

February 24, Graduate Center: Alexander Ushakov (Stevens Institute of Technology), TBA
Abstract:

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