On the structure of a relatively free Grassmann algebra

We investigate the multiplicative and T-space structure of the relatively free algebra F(3) with a unity corresponding to the identity [[x1, x2], x3] = 0 over an infinite field of characteristic p > 0. The highest emphasis is placed on unitary closed T-spaces over a field of characteristic p > 2. We construct a diagram containing all basic T-spaces of the algebra F(3), which form infinite chains of the inclusions. One of the main results is the decomposition of quotient T-spaces connected with F(3) into a direct sum of simple components. Also, the studied T-spaces are commutative subalgebras of F(3); thus, the structure of F(3) and its subalgebras can be described as modules over these commutative algebras. Separately, we consider the specifics of the case p = 2. In the Appendix, we study nonunitary closed T-spaces and the case of a field of zero characteristic. © 2010 Springer Science+Business Media, Inc.