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},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 Appendix, we study nonunitary closed T-spaces and the case of
a field of zero characteristic.

Location: http://mech.math.msu.su/~fpm/eng/k09/k098/k09801h.htm

Last modified: September 14, 2010