Composition-diamond lemma for tensor product of free algebras. (Q974391)

A. I. Shirshov established the theory of one-relator Lie algebras. This theory is in full analogy by statements but not by method with Magnus's theory of one-relator groups. He provided the algorithmic decidality of the word problem for any one-relator Lie algebra. In this paper, the authors establish a Composition-Diamond lemma for the tensor product of two free algebras over a field. Some applications are given in Section 4. As an application, they construct a Gröbner-Shirshov basis in \(k\langle X\rangle\otimes k\langle Y\rangle\) by lifting a given Gröbner-Shirshov basis in the tensor product \(k[X]\otimes k\langle Y\rangle\), in which \(k[X]\) is the polynomial algebra.