Composition-diamond lemma for modules

Abstract

In this paper we give some relationships among the Gröbner-Shirshov bases in free associative algebras, free left modules and "double-free" left modules (free modules over a free algebra). We give the Chibrikov's Composition-Diamond lemma for modules and show that Kang-Lee's 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.

Gröbner–Shirshov bases and their calculation

“…Theorem 9 (Chibrikov [90], see also [78], the CD-lemma for modules) Consider a non-empty set S ⊂ mod k X Y with all s ∈ S monic and choose an ordering < on X * Y as before. The following statements are equivalent:…”

Gröbner-Shirshov bases of irreducible modules over the quantum group $$U_q(F_4)$$ U q ( F 4 )

“…For the convenience of the reader, in this section we recall some notions and results about Gröbner-Shirshov bases of double-free modules and the quantum groups from [7,17], respectively.…”

“…, see also [17]) Let S ⊂ mod k X Y be a none-empty subset generated by some monic elements, " ≺ " the admissible ordering defined above. The following statements are equivalent:…”

“…Several years later, in [16] Chibrikov used another approach to the Gröbner-Shirshov bases for the modules, and the key idea in his approach is that a module of an algebra is viewed as a free module over a free algebra. Later, in [17] authors gave a Gröbner-Shirshov bases for some modules using Chibrikov lemma [16]. In this paper, on the bases of Gröbner-Shirshov bases for the quantum group U q (F 4 ), we construct a Gröbner-Shirshov bases for the irreducible module V q (λ) of U q (F 4 ), first, by using the method in [17] and then, by specializing a suitable version of U q (F 4 ) at q = 1, get a Gröbner-Shirshov bases for the universal enveloping algebra of the simple Lie algebra of type F 4 and finite dimensional irreducible module V (λ) of U q (F 4 ).…”

“…Later, in [17] authors gave a Gröbner-Shirshov bases for some modules using Chibrikov lemma [16]. In this paper, on the bases of Gröbner-Shirshov bases for the quantum group U q (F 4 ), we construct a Gröbner-Shirshov bases for the irreducible module V q (λ) of U q (F 4 ), first, by using the method in [17] and then, by specializing a suitable version of U q (F 4 ) at q = 1, get a Gröbner-Shirshov bases for the universal enveloping algebra of the simple Lie algebra of type F 4 and finite dimensional irreducible module V (λ) of U q (F 4 ). And by comparing our basis for the universal enveloping algebra of the simple Lie algebra of type F 4 with the one given in [8], we found that they are coincide.…”