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.

设F为域，表示\(\mathcal {F}_L=F\langle X_1^{\pm 1},X_2^{\pm 1},\ldots \rangle \)自由生成的自由群的群代数由\(X_i^{\pm 1}\)。它的元素是几个变量的（非交换）洛朗多项式。对于结合酉代数R ，我们用U ( R )表示其单位群。当\(\mathcal {F}_L\)的元素在U ( R )的所有元素替换下在R中消失时，它就是U ( R ) （或R ）的洛朗多项式恒等式 (LPI) 。由\(y_1\)和\(y_2\)根据关系\(y_1^2=y_2^2=0\)生成的代数\(\mathcal {F} \)在理论中起着重要作用LPI；更一般地，它出现在对给定群G的群恒等式（定律）与其群代数FG满足的多项式恒等式之间关系的研究中。在本文中，我们将\(\mathcal {F}\)的LPI 与二阶矩阵代数\(M_2(F)\)的 LPI 联系起来。从我们的结果可以看出，如果基域F是无限的并且是二次封闭的，那么这两个代数的 LPI 的理想是一致的。