Gröbner bases and syzygies on bimodules over PBW algebras

A new method for the computation of two-sided Gröbner bases of ideals and bimodules shifting the problem to the enveloping algebra is proposed. This alternative method appears to be more efficient than the one in [Kandri-Rody, A., Weispfenning, V., 1990. Non-commutative Gröbner bases in algebras of solvable type. J. Symbolic Comput. 9, 1-26] since it calls the left Buchberger algorithm once. We introduce the notion, arising from the ideas that this method involves, of two-sided syzygy, which is revealed to be useful in the computation of, e.g., the intersection of bimodules. Further applications are left for a later work.