[Paper Review] Gröbner-Shirshov bases and PBW theorems

L. A. Bokut, Yuqun Chen|arXiv (Cornell University)|Nov 8, 2014

Algebraic structures and combinatorial models52 references3 citations

TL;DR

This paper presents a comprehensive review of Gröbner-Shirshov (GS) bases and their applications in algebra, particularly through the Composition-Diamond lemma, which provides a systematic method to construct linear bases and normal forms for various algebras. The key contribution is the unification of PBW-type theorems, normal forms for groups and semigroups, and embeddings of algebras using GS bases, with applications to Lie, conformal, Rota-Baxter, and other non-associative algebras.

ABSTRACT

We review some applications of Gröbner-Shirshov bases, including PBW theorems, linear bases of free universal algebras, normal forms for groups and semigroups, extensions of groups and algebras, embedding of algebras.

Motivation & Objective

• To unify and systematize the application of Gröbner-Shirshov bases across diverse algebraic structures, including associative, Lie, conformal, and Rota-Baxter algebras.
• To establish a general framework based on the Composition-Diamond lemma for deriving linear bases and normal forms in free universal algebras.
• To provide new proofs and insights into classical theorems such as PBW theorems and embedding theorems using GS basis techniques.
• To extend the GS basis method to algebras over commutative algebras and non-associative varieties, including dialgebras and L-algebras.
• To demonstrate the utility of GS bases in solving word problems, constructing normal forms for groups (e.g., braid groups, HNN extensions), and enabling embeddings of countably generated algebras into 2-generated ones.

Proposed method

• Utilizes the Composition-Diamond lemma as the foundational tool, establishing equivalence between a set being a GS basis and the existence of a linear basis via irreducible words.
• Applies the lemma to various algebraic categories: associative algebras, Lie algebras, dialgebras, conformal algebras, and Rota-Baxter algebras, with specific GS bases defined for each.
• Employs two types of compositions: inclusion and intersection compositions, defined via least common multiples of leading terms, ensuring trivial reductions modulo the basis.
• Extends the method to algebras over commutative algebras by introducing double-free algebras and handling non-trivial lcm structures in the coefficient part.
• Applies the GS basis technique to derive normal forms for groups and semigroups, including braid groups, HNN extensions, and one-relator groups.
• Uses GS bases to prove embedding theorems, such as embedding any countably generated Lie or differential algebra into a simple 2-generated algebra.

Experimental results

Research questions

• RQ1 How can the Composition-Diamond lemma be generalized to various varieties of algebras, including non-associative and conformal algebras?
• RQ2 What conditions ensure that a set of polynomials forms a Gröbner-Shirshov basis, and how does this relate to the existence of a linear basis in the quotient algebra?
• RQ3 Can GS bases be used to derive PBW-type theorems for Lie conformal algebras and n-conformal associative algebras?
• RQ4 How can GS bases be applied to construct normal forms for complex groups such as braid groups and HNN extensions?
• RQ5 To what extent can GS bases enable the embedding of countably generated algebras and groups into finitely generated ones?

Key findings

• The Composition-Diamond lemma establishes that a set S is a GS basis if and only if every composition of elements in S reduces trivially, ensuring that the set of irreducible words forms a linear basis for the quotient algebra.
• For Lie conformal algebras, a 1/2 PBW theorem is proven: the universal enveloping associative conformal algebra has a linear basis formed by irreducible words from the GS basis of defining relations.
• The Hall basis and Lyndon-Shirshov basis for free Lie algebras are derived as irreducible words from anti-commutative GS bases in the associative algebra setting.
• The Loday basis for free dialgebras and the basis for free Rota-Baxter algebras are constructed via GS bases, confirming their linear independence and spanning properties.
• A normal form for HNN extensions of groups is established using an S-partially monomial order, proving the existence of a computable normal form via GS bases.
• Every countably generated Lie algebra can be embedded into a simple 2-generated Lie algebra, and every countably generated differential algebra into a simple 2-generated differential algebra, using GS basis techniques.