On Laurent polynomial identities

Abstract

Let F be a field and denote $\mathcal {F}_L=F\langle X_1^{\pm 1},X_2^{\pm 1},\ldots \rangle$ the group algebra of the free group freely generated by the $X_i^{\pm 1}$. Its elements are the (non-commutative) Laurent polynomials in several variables. For an associative unitary algebra R, we denote by U(R) the group of its units. An element of $\mathcal {F}_L$ is a Laurent polynomial identity (LPI) for U(R) (or for R) whenever it vanishes in R under all substitutions of elements of U(R). The algebra $\mathcal {F}$, generated by $y_1$ and $y_2$ subject to the relations $y_1^2=y_2^2=0$, plays an important role in the theory of LPI; more generally, it appears in the study of the relationship between the group identities (laws) of a given group G and the polynomial identities satisfied by its group algebra FG. In this note, we relate the LPI of $\mathcal {F}$ to the LPI of the matrix algebra of order two, $M_2(F)$. It follows from our results that if the base field F is infinite and quadratically closed, then the ideals of LPI for these two algebras coincide.