[Paper Review] On embedding of Lie conformal algebras into associative conformal algebras

Michael Roitman|ArXiv.org|Apr 28, 2004

Algebraic structures and combinatorial models10 references3 citations

TL;DR

This paper establishes that a Lie conformal algebra with a uniformly bounded locality function can be embedded into an associative conformal algebra, preserving the locality bound. The construction uses a universal enveloping algebra with a carefully defined ideal to enforce relations, proving that such Lie conformal algebras—especially nilpotent ones—admit nilpotent associative conformal enveloping algebras of the same index.

ABSTRACT

We prove that a Lie conformal algebra L with bounded locality function is embeddable into an associative conformal algebra A with the same bound on the locality function. If L is nilpotent, then so is A, and the nilpotency index remains the same. We also give a list of open questions concerning the embedding of Lie conformal algebras into associative conformal.

Motivation & Objective

• To determine sufficient conditions under which a Lie conformal algebra can be embedded into an associative conformal algebra.
• To address the open question of whether linear growth of the locality function suffices for embeddability.
• To generalize the classical notion of enveloping algebras to the conformal setting, particularly for nilpotent and bounded-locality Lie conformal algebras.
• To provide a constructive method for building associative conformal enveloping algebras with controlled locality properties.
• To explore connections to finite-type Lie conformal algebras and their potential for finite-type enveloping algebras.

Proposed method

• Constructs a universal associative conformal algebra ${\mathfrak{U}}$ from a Lie conformal algebra ${\mathfrak{L}}$ using formal series and divided powers.
• Defines a filtration on ${\mathfrak{U}}$ by total degree, ensuring compatibility with the conformal products and derivation $D$.
• Introduces an ideal ${\mathfrak{I}}$ generated by all words of total degree $\geq r$, where $r$ is the locality bound, to enforce the locality condition.
• Uses a ${\mathds{k}}[D]$-linear basis ${\mathcal{W}}$ of ${\mathfrak{U}}/{\mathfrak{N}}$ to prove linear independence and control the structure of the quotient.
• Shows that the quotient ${\mathfrak{A}} = {\mathfrak{U}}/{\mathfrak{I}}$ is an associative conformal algebra with the desired locality and embedding properties.
• Verifies that the quotient algebra satisfies conformal associativity and locality via explicit computation of products using the relations (4) and (6).

Experimental results

Research questions

• RQ1 Under what conditions can a Lie conformal algebra be embedded into an associative conformal algebra?
• RQ2 Is a uniformly bounded locality function sufficient for such an embedding?
• RQ3 Can nilpotent Lie conformal algebras be embedded into nilpotent associative conformal algebras of the same index?
• RQ4 Does the locality function of a central extension of a finite-type Lie conformal algebra remain bounded if the original has bounded locality?
• RQ5 Is linear growth of the locality function sufficient for embeddability, or are stronger conditions required?

Key findings

• Any Lie conformal algebra with a uniformly bounded locality function $S_{\mathfrak{L},{\mathcal{G}}}(l) \leq K$ for all $l$ admits an embedding into an associative conformal algebra with the same bound on the locality function.
• The construction yields a nilpotent associative conformal enveloping algebra of the same index for any nilpotent Lie conformal algebra.
• The universal construction via ${\mathfrak{U}}/{\mathfrak{I}}$ ensures that all words of total degree $\geq r$ vanish, enforcing the locality bound.
• The ideal ${\mathfrak{I}}$ is shown to be proper and disjoint from ${\mathfrak{L}}$, ensuring the embedding of ${\mathfrak{L}}$ into ${\mathfrak{A}} = {\mathfrak{U}}/{\mathfrak{I}}$.
• The method proves that the locality function of the enveloping algebra matches that of the original Lie conformal algebra.
• The result supports the conjecture that finite-type Lie conformal algebras with bounded locality functions may admit finite-type associative conformal enveloping algebras.