Algebras Defined by Monic Gröbner Bases over Rings

Abstract

Let $K⟨X⟩=K⟨X_1,...,X_n⟩$ be the free algebra of $n$ generators over a field $K$, and let $R⟨X⟩=R⟨X_1,...,X_n⟩$ be the free algebra of $n$ generators over an arbitrary commutative ring $R$. In this semi-expository paper, it is clarified that any monic Gröbner basis in $K⟨X⟩$ may give rise to a monic Gröbner basis of the same type in $R⟨X⟩$, and vice versa. This fact turns out that many important $R$-algebras have defining relations which form a monic Gröbner basis, and consequently, such $R$-algebras may be studied via a nice PBW structure theory as that developed for quotient algebras of $K⟨X⟩$ in ([LWZ], [Li2, 3]).

new version, 23 pages

References in corpus (1)

• Rosso-Yamane Theorem on PBW basis of $U_q(A_N)$

Cited by in corpus (1)

• Valuation Extensions of Algebras Defined by Monic Gröbner Bases