ON THE STRUCTURE OF A RELATIVELY FREE GRASSMANN ALGEBRA

摘要

We investigate the multiplicative and T-space structure of the relatively free algebra F~（（3）） with a unity corresponding to the identity [[x_1,x_2],x_3] = 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.