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A Magnus group construction for a class of Borcherds algebras

Lisa Carbone, Elizabeth Jurisich

TL;DR

The paper addresses the problem of associating a group-theoretic analogue to a class of Borcherds algebras that include imaginary simple roots. It achieves this by exploiting a direct-sum decomposition $\mathfrak g = \mathfrak u^+ \oplus (\mathfrak g_J + \mathfrak h) \oplus \mathfrak u^-$, forming a semidirect product $G = G(S') \rtimes G_J$ where $G_J$ is a Kac–Moody (or semisimple) group and $G(S') = \exp(\widehat{L}(S'))$ is a Magnus group for the free Lie algebra $\mathfrak u^-$. The authors prove that the resulting group is independent of the choice of basis for $V'$ (up to isomorphism) and illustrate the construction across several concrete examples including the Monster Lie algebra, Fricke-type Monstrous algebras, Borcherds algebras from hyperbolic lattices, an $E_{10}$-based case with missing modules, and the gnome Lie algebra. This framework provides a hybrid between minimal and maximal Kac–Moody groups, with potential avenues for topology and pro-algebraic structure in further work. Overall, the work offers a principled group-theoretic realization for a broad class of Borcherds algebras and broadens the toolkit for exploring Moonshine-related and hyperbolic constructions in Lie theory.

Abstract

We construct a group associated to a class of Borcherds algebras that admit a direct sum decomposition into a Kac--Moody (or semi-simple) subalgebra and a pair of free Lie subalgebras. Such Borcherds algebras have no mutually orthogonal imaginary simple roots.Our group is a semi-direct product of a Kac--Moody (or semi-simple) group and a Magnus group of invertible formal power series corresponding to a basis of a certain highest weight module determined by the simple imaginary roots. We show that our group is independent of this choice of basis, up to isomorphism. We apply our construction to a number of concrete examples, such as certain Borcherds algebras formed using root lattices of hyperbolic Kac--Moody algebras, the Monster Lie algebra, Monstrous Lie algebras of Fricke type and the gnome Lie algebra.

A Magnus group construction for a class of Borcherds algebras

TL;DR

The paper addresses the problem of associating a group-theoretic analogue to a class of Borcherds algebras that include imaginary simple roots. It achieves this by exploiting a direct-sum decomposition , forming a semidirect product where is a Kac–Moody (or semisimple) group and is a Magnus group for the free Lie algebra . The authors prove that the resulting group is independent of the choice of basis for (up to isomorphism) and illustrate the construction across several concrete examples including the Monster Lie algebra, Fricke-type Monstrous algebras, Borcherds algebras from hyperbolic lattices, an -based case with missing modules, and the gnome Lie algebra. This framework provides a hybrid between minimal and maximal Kac–Moody groups, with potential avenues for topology and pro-algebraic structure in further work. Overall, the work offers a principled group-theoretic realization for a broad class of Borcherds algebras and broadens the toolkit for exploring Moonshine-related and hyperbolic constructions in Lie theory.

Abstract

We construct a group associated to a class of Borcherds algebras that admit a direct sum decomposition into a Kac--Moody (or semi-simple) subalgebra and a pair of free Lie subalgebras. Such Borcherds algebras have no mutually orthogonal imaginary simple roots.Our group is a semi-direct product of a Kac--Moody (or semi-simple) group and a Magnus group of invertible formal power series corresponding to a basis of a certain highest weight module determined by the simple imaginary roots. We show that our group is independent of this choice of basis, up to isomorphism. We apply our construction to a number of concrete examples, such as certain Borcherds algebras formed using root lattices of hyperbolic Kac--Moody algebras, the Monster Lie algebra, Monstrous Lie algebras of Fricke type and the gnome Lie algebra.
Paper Structure (19 sections, 9 theorems, 111 equations, 2 figures)

This paper contains 19 sections, 9 theorems, 111 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary Material
  3. Universal Enveloping Algebras
  4. Borcherds algebras
  5. Symmetric bilinear form
  6. Modules
  7. Our class of Borcherds algebras
  8. Magnus group for the free Lie algebra
  9. Group construction
  10. The Kac--Moody group $G_J$ associated to ${\mathfrak g}_J$ and the module $W$
  11. Defining an action of $\mathfrak{g}_J$ on $T(V^\prime)$ as formal polynomials.
  12. Action of $G_J$ on $\widehat{T}(V')$
  13. The semi-direct product $G$
  14. Examples
  15. A Magnus group for the Monster Lie algebra
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $A$ be a matrix satisfying conditions B1-B3. Let $J$ and $\mathfrak g_J$ be as above. Assume that if $i,j \in I \backslash J$ and $i \neq j$ then $a_{ij}<0$. Then where $\mathfrak u^- = L(\coprod_{i \in I\backslash J}{ U}(\mathfrak n^-_J)\cdot f_i)$ and $\mathfrak u^+= L(\coprod_{i \in I\backslash J}{ U}(\mathfrak n^+_J)\cdot e_i )$. For all $i \in I\backslash J$ the vector spaces $V^\prime_

Figures (2)

  • Figure 1: The hyperbola supporting the real roots of $H(3)$ in $\mathfrak h^*\cong {\mathbb R}^{1,1}$ is indicated by the bold line curves. The asymptotes are indicated by the dashed lines. The imaginary root $\alpha_1+\alpha_2$ lies inside the imaginary cone.
  • Figure 2: The Dynkin diagram of $E_{10}$

Theorems & Definitions (14)

  • Definition
  • Theorem 2.1
  • Corollary 2.2
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • ...and 4 more