In this paper, we establish the Composition-Diamond lemma for free differential algebras. As applications, we give Groebner-Shirshov bases for free Lie-differential algebra and free commutative-differential algebra, respectively.
In this paper, we establish Composition-Diamond lemma for tensor product $k< X> otimes k< Y>$ of two free algebras over a field. As an application, we construct a Groebner-Shirshov basis in $k< X> otimes k< Y>$ by lifting a Groebner-Shirshov basis in
$k[X] otimes k< Y>$, where $k[X]$ is a commutative algebra.
In this paper we give some relationships among the Groebner-Shirshov bases in free associative algebras, free left modules and double-free left modules (free modules over a free algebra). We give the Chibrikovs Composition-Diamond lemma for modules a
nd show that Kang-Lees Composition-Diamond lemma follows from this lemma. As applications, we also deal with highest weight module over the Lie algebra $sl_2$, Verma module over a Kac-Moody algebra, Verma module over Lie algebra of coefficients of a free conformal algebra and the universal enveloping module for a Sabinin algebra.
