Looking for Gröbner basis theory for (almost) skew 2-nomial algebras

In this paper, we introduce (almost) skew 2-nomial algebras, establish the existence of a skew multiplicative K-basis for a skew 2-nomial algebra, and explore the existence of a Gröbner basis theory for such algebras. As a by-product, a family of Noetherian algebras is recognized by the finite Gröbner basis theory. Let K be a field, and let R be a finitely generated free K-algebra, or a path algebra defined by a finite directed graph over K. Then it is well-known that R holds an effective Gröbner basis theory ([Mor], [FFG]) which generalizes successfully the celebrated algorithmic Gröbner basis theory for commutative polynomial algebras [Bu]. Let I be a (two-sided) ideal of R. In order to study the quotient algebra A = R/I and its modules in a computational way, we are naturally concerned about the existence of a Gröbner basis theory for A. Familiar quotient algebras of R that have an effective Gröbner basis theory are exterior algebras, Clifford algebras [HT], and solvable polynomial algebras in the sense of [K-RW] (including Weyl algebras and enveloping algebras of finite dimensional K-Lie algebras). As we learnt from loc. cit., the first step to have a Gröbner basis theory for A = R/I is to have a “nice” K-basis for A, for instance, a multiplicative K-basis B in the sense that u, v ∈ B implies uv = 0 or uv ∈ B. In [Gr1–2], a Gröbner basis theory for algebras (modules) with a multiplicative K-basis (a coherent K-basis) was studied in detail. In particular, the following result was obtained in [Gr2]. Theorem ([Gr2], Theorem 2.3) Suppose that R is a K-algebra with multiplicative K-basis C. Let I be an ideal in R and π: R → R/I be the canonical surjection. Let π(C) = π(C) − {0}. Project supported by the National Natural Science Foundation of China (10571038). e-mail: [email protected]