[Paper Review] Lectures on extended affine Lie algebras

Erhard Neher|arXiv (Cornell University)|Mar 11, 2010

Algebraic structures and combinatorial models34 references3 citations

TL;DR

This paper provides a comprehensive introduction to extended affine Lie algebras (EALAs), unifying finite-dimensional semisimple, affine, and toroidal Lie algebras under a common axiomatic framework. It establishes that all EALAs arise from invariant Lie tori via central extensions and derivations, with a complete classification via the construction $\operatorname{E}(L,D,\tau)$, where $L$ is an invariant Lie torus, $D$ a graded subalgebra of special derivations, and $\tau$ an affine cocycle.

ABSTRACT

We give an introduction to the structure theory of extended affine Lie algebras, which provide a common framework for finite-dimensional semisimple, affine and toroidal Lie algebras. The notes are based on a lecture series given during the Fields Institute summer school at the University of Ottawa in June 2009.

Motivation & Objective

• To provide a structured, accessible introduction to the theory of extended affine Lie algebras (EALAs) for graduate students and researchers.
• To unify finite-dimensional semisimple, affine Kac-Moody, and toroidal Lie algebras under a common axiomatic structure.
• To establish the general construction of all EALAs using invariant Lie tori, central extensions, and derivations.
• To characterize discrete EALAs over $\mathbb{C}$ via injective evaluation maps and affine cocycles.
• To present foundational tools such as affine reflection systems, Lie tori, and centroidal derivations for structural analysis.

Proposed method

• Define EALAs via axioms (EA1)–(EA4), emphasizing root space decomposition and nondegenerate invariant bilinear forms.
• Introduce affine reflection systems as the root system framework underlying EALAs, generalizing affine root systems.
• Construct EALAs via the general formula $\operatorname{E}(L,D,\tau) = L \oplus D^{\mathrm{gr}*} \oplus D$, with Lie bracket defined by $[l_1 \oplus c_1 \oplus d_1, l_2 \oplus c_2 \oplus d_2]$ involving actions and cocycles.
• Use the centroid and centroidal derivations of Lie tori to classify and construct EALAs systematically.
• Employ degree maps and graded derivations to define the structure of the core and centreless core of an EALA.
• Verify axioms (EA1)–(EA4) for the constructed algebra $\operatorname{E}(L,D,\tau)$ using the nondegenerate, invariant bilinear form $ (\cdot|\cdot) $.

Experimental results

Research questions

• RQ1 How can finite-dimensional semisimple, affine, and toroidal Lie algebras be unified under a single algebraic framework?
• RQ2 What are the structural properties of extended affine root systems, and how do they generalize affine root systems?
• RQ3 How can all extended affine Lie algebras be systematically constructed from invariant Lie tori?
• RQ4 What conditions ensure that an EALA is discrete, particularly over $\mathbb{C}$?
• RQ5 What is the role of central extensions and derivations in the classification of EALAs?

Key findings

• All extended affine Lie algebras arise from the construction $\operatorname{E}(L,D,\tau)$, where $L$ is an invariant Lie torus, $D$ a graded subalgebra of special derivations, and $\tau$ an affine 2-cocycle.
• The core of an EALA is $L \oplus D^{\mathrm{gr}*}$, and the centreless core is $L$, which is an invariant Lie torus.
• The symmetric bilinear form $ (\cdot|\cdot) $ on $\operatorname{E}(L,D,\tau)$ is nondegenerate and invariant, ensuring the algebra satisfies the EALA axioms.
• Over $\mathbb{C}$, a discrete EALA is characterized by an injective evaluation map $\Lambda \to D^{0*}$ and an affine cocycle $\tau$.
• The structure of EALAs mirrors that of affine Kac-Moody algebras: central extension of a generalized loop algebra (invariant Lie torus) plus derivations.
• The classification of EALAs reduces to classifying invariant Lie tori and their special derivations, with concrete realizations possible via matrix algebras and Laurent polynomials.