[Paper Review] Gröbner-Shirshov bases for some Lie algebras

Yuqun Chen, Li Yu+1 more|arXiv (Cornell University)|May 15, 2013

Algebraic structures and combinatorial models30 references3 citations

TL;DR

This paper establishes Gröbner-Shirshov bases for the Drinfeld-Kohno Lie algebra $<math>\mathbf{L}_n<math>$ and Kukin's Lie algebra $A_P$ associated with a semigroup $P$, proving that $\mathbf{L}_n$ is a free $\mathbb{Z}$-module with an explicit $\mathbb{Z}$-basis. It provides a new proof of Kukin's theorem: if $P$ has an undecidable word problem, so does $A_P$, using the algebraic structure of Gröbner-Shirshov bases.

ABSTRACT

We give Gröbner-Shirshov bases for Drinfeld-Kohno Lie algebra $ extbf{L}_{n}$ in \cite{[Et]} and Kukin Lie algebra $A_P$ in \cite{Kukin}, where $P$ is a semigroup. As applications, we show that as $\mathbb{Z}$-module $ extbf{L}_{n}$ is free and a $\mathbb{Z}$-basis of $ extbf{L}_{n}$ is given. We give another proof of Kukin Theorem: if semigroup $P$ has the undecidable word problem then the Lie algebra $A_P$ has the same property.

Motivation & Objective

• To construct Gröbner-Shirshov bases for the Drinfeld-Kohno Lie algebra $\mathbf{L}_n$ over $\mathbb{Z}$.
• To establish a $\mathbb{Z}$-basis for $\mathbf{L}_n$ using the Gröbner-Shirshov basis and prove it is a free $\mathbb{Z}$-module.
• To provide a new algebraic proof of Kukin's theorem: if a semigroup $P$ has an undecidable word problem, then the associated Lie algebra $A_P$ also has an undecidable word problem.
• To demonstrate that $\mathbf{L}_n$ is an iterated semidirect product of free Lie algebras via the basis structure.

Proposed method

• Applying the Composition-Diamond Lemma for Lie algebras over a field to construct a Gröbner-Shirshov basis $S$ for $\mathbf{L}_n$ using deg-lex ordering and Shirshov special bracketing.
• Verifying that all compositions of relations in $S$ reduce to zero modulo $S$, confirming $S$ is a Gröbner-Shirshov basis.
• Using the normal $S$-words in the basis $Irr(S)$ to define a $\mathbb{Z}$-basis for $\mathbf{L}_n$ as non-associative Lyndon-Shirshov words.
• Analyzing the relations of $A_P$ via Gröbner-Shirshov basis techniques to show that the word problem of $P$ is inherited by $A_P$.
• Employing the structure of the basis to show $\mathbf{L}_n$ is an iterated semidirect product of free Lie algebras.
• Extending results from $\mathbb{Z}$ to arbitrary commutative rings with identity by preserving the algebraic structure.

Experimental results

Research questions

• RQ1 Does the Drinfeld-Kohno Lie algebra $\mathbf{L}_n$ admit a Gröbner-Shirshov basis over $\mathbb{Z}$?
• RQ2 Can a $\mathbb{Z}$-basis for $\mathbf{L}_n$ be explicitly constructed using Gröbner-Shirshov basis theory?
• RQ3 Does the word problem of a semigroup $P$ determine the word problem of the associated Lie algebra $A_P$?
• RQ4 Is $\mathbf{L}_n$ isomorphic to an iterated semidirect product of free Lie algebras?
• RQ5 Can the undecidability of the word problem in $P$ be algebraically inherited by $A_P$ via Gröbner-Shirshov bases?

Key findings

• The set $S$ of defining relations for $\mathbf{L}_n$ forms a Gröbner-Shirshov basis over $\mathbb{Z}$, verified by reduction of all compositions to zero.
• The Drinfeld-Kohno Lie algebra $\mathbf{L}_n$ is a free $\mathbb{Z}$-module with a $\mathbb{Z}$-basis given by $Irr(S) = \{[t_{ik_1}t_{ik_2}\cdots t_{ik_m}] \mid t_{ik_1}t_{ik_2}\cdots t_{ik_m} \text{ is an ALSW in } T^*, m \in \mathbb{N}\}$.
• The paper provides a new proof of Kukin's theorem: if $P$ has an undecidable word problem, then $A_P$ also has an undecidable word problem, using the Gröbner-Shirshov basis structure.
• The Lie algebra $\mathbf{L}_n$ is an iterated semidirect product of free Lie algebras, as shown by the decomposition $\mathbf{L}_n = A_1 \oplus A_2 \oplus \cdots \oplus A_{n-2}$ with $A_i \triangleleft A_i + \cdots + A_{n-2}$.
• All results extend to arbitrary commutative rings with identity, not just $\mathbb{Z}$, preserving the Gröbner-Shirshov basis and basis structure.
• The composition of relations in $S$ is trivial modulo $S$ in all cases, confirming the basis is complete and minimal for $\mathbf{L}_n$.