[Paper Review] Subalgebras of Hyperbolic Kac-Moody Algebras

Alex J. Feingold, Hermann Nicolai|ArXiv.org|Mar 14, 2003

Algebraic structures and combinatorial models18 references4 citations

TL;DR

This paper establishes a systematic method to construct indefinite Kac-Moody and Borcherds subalgebras within hyperbolic Kac-Moody algebras by selecting subsets of positive real roots whose pairwise differences are not roots. It demonstrates that the rank 3 hyperbolic algebra 𝔽 contains all simply-laced rank 2 hyperbolic algebras and an infinite family of rank 3 indefinite Kac-Moody subalgebras, while also showing that hyperplane sections of the root system yield Borcherds algebras, including the simplest case with roots aligned on a line.

ABSTRACT

The hyperbolic (and more generally, Lorentzian) Kac-Moody (KM) Lie algebras $\cA$ of rank $r+2 > 2$ are shown to have a rich structure of indefinite KM subalgebras which can be described by specifying a subset of positive real roots of $\cA$ such that the difference of any two is not a root of $\cA$. Taking these as the simple roots of the subalgebra gives a Cartan matrix, generators and relations for the subalgebra. Applying this to the canonical example of a rank 3 hyperbolic KM algebra, $\cF$, we find that $\cF$ contains all of the simply laced rank 2 hyperbolics, as well as an infinite series of indefinite KM subalgebras of rank 3. It is shown that $\cA$ also contains Borcherds algebras, obtained by taking all of the root spaces of $\cA$ whose roots are in a hyperplane (or any proper subspace). This applies as well to the case of rank 2 hyperbolics, where the Borcherds algebras have all their roots on a line, giving the simplest possible examples.

Motivation & Objective

• To explore the rich internal structure of hyperbolic Kac-Moody algebras, which remain poorly understood despite their importance in mathematical physics.
• To identify and classify non-isomorphic infinite-dimensional subalgebras within hyperbolic KM algebras, particularly those that are indefinite Kac-Moody or Borcherds algebras.
• To provide a constructive method for generating such subalgebras using root systems and Weyl group actions, especially in the canonical rank 3 example 𝔽.
• To demonstrate that Borcherds algebras arise naturally as subalgebras by restricting to root spaces lying in a hyperplane, generalizing known constructions.

Proposed method

• A key theorem identifies subalgebras by selecting a set of positive real roots such that the difference of any two is not a root in the original algebra, ensuring the resulting Cartan matrix defines a valid subalgebra.
• The construction uses these selected roots as new simple roots, generating a subalgebra via standard Kac-Moody relations with a new Cartan matrix.
• For the rank 3 hyperbolic algebra 𝔽, the method yields all simply-laced rank 2 hyperbolic KM algebras and an infinite series of inequivalent rank 3 indefinite KM subalgebras.
• The Weyl group action of 𝔽, realized as the hyperbolic triangle group T(2,3,∞) on the Poincaré disk, is used to generate additional non-W-equivalent subalgebras.
• Borcherds algebras are constructed by restricting the root system to a hyperplane, and their structure is verified using Borcherds’ theorem on contravariant forms and grading.
• The decomposition 𝔠 = 𝔪₋ ⊕ 𝔥 ⊕ 𝔪₊ is used to analyze the structure of these subalgebras, with 𝔪₊ and 𝔪₋ shown to be free Lie algebras.

Experimental results

Research questions

• RQ1 How can one systematically construct indefinite Kac-Moody subalgebras inside hyperbolic Kac-Moody algebras?
• RQ2 What is the role of the Weyl group in generating inequivalent subalgebras, and how does it relate to geometric structures like conic sections?
• RQ3 Can Borcherds algebras be embedded as subalgebras in hyperbolic KM algebras via root system hyperplane sections?
• RQ4 What is the structure of the subalgebras formed by restricting to root spaces lying in a hyperplane, and how do they relate to the original algebra?
• RQ5 How do the root multiplicities and grading of these subalgebras behave, particularly in the simplest case with roots aligned on a line?

Key findings

• The rank 3 hyperbolic Kac-Moody algebra 𝔽 contains all simply-laced rank 2 hyperbolic KM algebras as subalgebras.
• An infinite family of inequivalent rank 3 indefinite KM subalgebras is constructed inside 𝔽, parameterized by an integer m ≥ 2.
• For any Lorentzian KM algebra of rank r+2, two infinite series of inequivalent indefinite KM subalgebras are found: one of rank r+1 and one of rank r+2.
• Hyperplane sections of the root system yield Borcherds algebras, with the simplest example having all roots aligned on a line, corresponding to the case of rank 2 hyperbolic algebras.
• The subalgebra 𝔠(m) = 𝔥ₘ ⊕ ⨁_{β∈Φ(𝒜_{r+1}^{indef}(m))} 𝒜_β is shown to be a Borcherds algebra for all m ≥ 2, containing the indefinite KM algebra as a proper subalgebra.
• The decomposition 𝔠 = 𝔪₋ ⊕ 𝔥 ⊕ 𝔪₊ holds, with 𝔪₊ and 𝔪₋ identified as free Lie algebras, and their structure as 𝔥-modules is suggested as a direction for further study.