Kac--Moody groups and automorphic forms in low dimensional supergravity theories

Kac--Moody groups $G$ over $\mathbb{R}$ have been conjectured to occur as symmetry groups of supergravity theories dimensionally reduced to dimensions less than 3, and their integral forms $G(\mathbb{Z})$ conjecturally encode quantized symmetries. In this review paper, we briefly introduce the conjectural symmetries of Kac--Moody groups in supergravity as well as the known evidence for these conjectures. We describe constructions of Kac--Moody groups over $\R$ and $\Z$ using certain choices of fundamental modules that are considered to have physical relevance. Eisenstein series on certain finite dimensional algebraic groups are known to encode quantum corrections in the low energy limit of superstring theories. We describe briefly how the construction of Eisenstein series extends to Kac--Moody groups. The constant terms of Eisenstein series on $E_9$, $E_{10}$ and $E_{11}$ are predicted to encode perturbative string theory corrections.Comment: arXiv admin note: text overlap with arXiv:1308.619

Integral forms of Kac-Moody groups and Eisenstein series in low dimensional supergravity theories

Kac-Moody groups $G$ over $\mathbb{R}$ have been conjectured to occur as symmetry groups of supergravities in dimensions less than 3, and their integer forms $G(\mathbb{Z})$ are conjecturally U-duality groups. Mathematical descriptions of $G(\mathbb{Z})$, due to Tits, are functorial and not amenable to computation or applications. We construct Kac-Moody groups over $\mathbb{R}$ and $\mathbb{Z}$ using an analog of Chevalley's constructions in finite dimensions and Garland's constructions in the affine case. We extend a construction of Eisenstein series on finite dimensional semisimple algebraic groups using representation theory, which appeared in the context of superstring theory, to general Kac-Moody groups. This coincides with a generalization of Garland's Eisenstein series on affine Kac-Moody groups to general Kac-Moody groups and includes Eisenstein series on $E_{10}$ and $E_{11}$. For finite dimensional groups, Eisenstein series encode the quantum corrections in string theory and supergravity theories. Their Kac-Moody analogs will likely also play an important part in string theory, though their roles are not yet understood

Generators and relations for (generalised) Cartan type superalgebras

In Kac's classification of finite-dimensional Lie superalgebras, the contragredient ones can be constructed from Dynkin diagrams similar to those of the simple finite-dimensional Lie algebras, but with additional types of nodes. For example, $A(n-1,0) = \mathfrak{sl}(1|n)$ can be constructed by adding a "gray" node to the Dynkin diagram of $A_{n-1} = \mathfrak{sl}(n)$, corresponding to an odd null root. The Cartan superalgebras constitute a different class, where the simplest example is $W(n)$, the derivation algebra of the Grassmann algebra on $n$ generators. Here we present a novel construction of $W(n)$, from the same Dynkin diagram as $A(n-1,0)$, but with additional generators and relations.Comment: 6 pages, talk presented at Group32, Prague, July 2018. v2: Minor change

Generators and relations for Lie superalgebras of Cartan type

We give an analog of a Chevalley-Serre presentation for the Lie superalgebras W(n) and S(n) of Cartan type. These are part of a wider class of Lie superalgebras, the so-called tensor hierarchy algebras, denoted W(g) and S(g), where g denotes the Kac-Moody algebra A_r, D_r or E_r. Then W(A_{n-1}) and S(A_{n-1}) are the Lie superalgebras W(n) and S(n). The algebras W(g) and S(g) are constructed from the Dynkin diagram of the Borcherds-Kac-Moody superalgebras B(g) obtained by adding a single grey node (representing an odd null root) to the Dynkin diagram of g. We redefine the algebras W(A_r) and S(A_r) in terms of Chevalley generators and defining relations. We prove that all relations follow from the defining ones at level -2 and higher. The analogous definitions of the algebras in the D- and E-series are given. In the latter case the full set of defining relations is conjectured.Comment: 42 pages. v2: Minor changes. Version accepted for publication in J. Phys.

Abstract simplicity of complete Kac-Moody groups over finite fields

Let $G$ be a Kac-Moody group over a finite field corresponding to a generalized Cartan matrix $A$, as constructed by Tits. It is known that $G$ admits the structure of a BN-pair, and acts on its corresponding building. We study the complete Kac-Moody group $\hat{G}$ which is defined to be the closure of $G$ in the automorphism group of its building. Our main goal is to determine when complete Kac-Moody groups are abstractly simple, that is have no proper non-trivial normal subgroups. Abstract simplicity of $\hat{G}$ was previously known to hold when A is of affine type. We extend this result to many indefinite cases, including all hyperbolic generalized Cartan matrices $A$ of rank at least four. Our proof uses Tits' simplicity theorem for groups with a BN-pair and methods from the theory of pro-$p$ groups.Comment: Final version. The statement and the proof of Theorem 5.2 have been corrected. The main result (Theorem 1.1) now holds under slightly stronger restriction

Dimensions of Imaginary Root Spaces of Hyperbolic Kac--Moody Algebras

We discuss the known results and methods for determining root multiplicities for hyperbolic Kac--Moody algebras