Lifting Gröbner Bases from a Class of Algebras

Abstract

Let K be a field, K⟨X⟩ = K⟨X ₁, …, X _n ⟩ the free K-algebra of n generators, K _Q [x _{i 1} , …, x _{i n} ] the multiparameter quantized coordinate ring of affine n-space with parameter matrix Q, and let π: K⟨X⟩ → K _Q [x _{i 1} , …, x _{i n} ] be the canonical algebra epimorphism. By extending the results of [5] to K _Q [x _{i 1} , …, x _{i n} ], we show that if G is a Gröbner basis in K _Q [x _{i 1} , …, x _{i n} ], then a Gröbner basis 𝒢 of the pre-image ℐ = π⁻¹(I) of the ideal I = ⟨G⟩ may be constructed subject to G (Theorem 2.3), and that the way of producing 𝒢 also simultaneously provides a criterion for the finiteness of 𝒢 (Theorem 2.6). Applications to some well-known algebras are given.

Key Words:

• Filtration
• Gradation
• Gröbner basis
• Monomial ordering

2000 Mathematics Subject Classification:

• 16W70
• 68W30

ACKNOWLEDGMENT

The original version of this article is about lifting finite Gröbner bases from the enveloping algebra of a K-Lie algebra (arXiv:math.RA/0701120). The exploration on lifting finite Gröbner bases from the algebra K _Q [x _{i 1} , …, x _{i n} ] was suggested kindly by the anonymous referee of the original version. The author would like to thank the referee for reading the (revised) manuscript patiently, and for making valuable remarks on this topic.

This project is supported by the National Natural Science Foundation of China (10571038).

Notes

Communicated by M. Cohen.