On the structure of a relatively free Grassmann algebra
We investigate the multiplicative and T-space structure of the relatively free algebra F<sup>(3)</sup> with a unity corresponding to the identity [[x<sub>1</sub>,x<sub>2</sub>],x<sub>3]=</sub>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<sup>(3)</sup>, which form infinite chains of the inclusions. One of the main results is the decomposition of quotient T-spaces connected with F<sup>(3)</sup> into a direct sum of simple components. Also, the studied T-spaces are commutative subalgebras of F<sup>(3)</sup>; thus, the structure of F<sup>(3)</sup> and its subalgebras can be described as modules over these commutative algebras. Separately, we consider the specifics of the case p=2. In Appendix, we study nonunitary closed T-spaces and the case of a field of zero characteristic.