FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2009, VOLUME 15, NUMBER 8, PAGES 3-93

On the structure of a relatively free Grassmann algebra

A. V. Grishin
L. M. Tsybulya

Abstract

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We investigate the multiplicative and T-space structure of the relatively free algebra $F(3)$ with a unity corresponding to the identity $[[x$₁,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 Appendix, we study nonunitary closed T-spaces and the case of a field of zero characteristic.

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Last modified: September 14, 2010