In this section we explain the basic theory of real forms of finite-dimensional Lie algebras. This material is somewhat technical and may therefore be skipped at a first reading. The theory of real forms of Lie algebras is required for a complete understanding of Section 7, which deals with the general case of Kac–Moody billiards for non-split real forms. However, for the benefit of the reader who wishes to proceed directly to the physical applications, we present a brief summary of the main points in the beginning of Section 7.

Our intention with the following presentation is to provide an accessible reference on the subject, directed towards physicists. We therefore consider this section to be somewhat of an entity of its own, which can be read independently of the rest of the paper. Consequently, we introduce Lie algebras in a rather different manner compared to the presentation of Kac–Moody algebras in Section 4, emphasizing here more involved features of the general structure theory of real Lie algebras rather than relying entirely on the Chevalley–Serre basis and its properties. In the subsequent section, the reader will then see these two approaches merged, and used simultaneously to describe the billiard structure of theories whose U-duality algebras in three dimensions are given by arbitrary real forms.

We have adopted a rather detailed and explicit presentation. We do not provide all proofs, however, referring the reader to [93, 129, 133, 94] for more information (including definitions of basic Lie algebra theory concepts).

There are two main approaches to the classification of real forms of finite-dimensional Lie algebras. One focuses on the maximal compact Cartan subalgebra and leads to Vogan diagrams. The other focuses on the maximal noncompact Cartan subalgebra and leads to Tits–Satake diagrams. It is this second approach that is of direct use in the billiard analysis. However, we have chosen to present here both approaches as they mutually enlighten each other.

6.1 Definitions

6.2 A preliminary example:

6.2.1 Real forms of

6.2.2 Cartan subalgebras

6.2.3 The Killing form

6.2.4 The compact real form

6.2.5 and compared and contrasted – The Cartan involution

6.2.6 Concluding remarks

6.3 The compact and split real forms of a semi-simple Lie algebra

6.4 Classical decompositions

6.4.1 Real forms and conjugations

6.4.2 The compact real form aligned with a given real form

6.4.3 Cartan involution and Cartan decomposition

6.4.4 Restricted roots

6.4.5 Iwasawa and decompositions

6.4.6 -stable Cartan subalgebras

6.4.7 Real roots – Compact and non-compact imaginary roots

6.4.8 Jumps between Cartan subalgebras – Cayley transformations

6.5 Vogan diagrams

6.5.1 Illustration – The case

and

The other real forms

Vogan diagrams

6.5.2 The Borel and de Siebenthal theorem

6.5.3 Cayley transformations in

6.5.4 Reconstruction

6.5.5 Illustrations: versus

6.5.6 A pictorial summary – All real simple Lie algebras (Vogan diagrams)

6.6 Tits–Satake diagrams

6.6.1 Example 1:

Diagonal description

Cartan involution and roots

Restricted roots

6.6.2 Example 2:

Diagonal description

Cartan involution and roots

Restricted roots

6.6.3 Tits–Satake diagrams: Definition

6.6.4 Formal considerations

6.6.5 Illustration:

6.6.6 Some more formal considerations

6.7 The real semi-simple algebras

6.7.1 Dimensions

6.7.2 Dimensions

6.7.3 Dimensions

6.8 Summary – Tits–Satake diagrams for non-compact real forms

6.2 A preliminary example:

6.2.1 Real forms of

6.2.2 Cartan subalgebras

6.2.3 The Killing form

6.2.4 The compact real form

6.2.5 and compared and contrasted – The Cartan involution

6.2.6 Concluding remarks

6.3 The compact and split real forms of a semi-simple Lie algebra

6.4 Classical decompositions

6.4.1 Real forms and conjugations

6.4.2 The compact real form aligned with a given real form

6.4.3 Cartan involution and Cartan decomposition

6.4.4 Restricted roots

6.4.5 Iwasawa and decompositions

6.4.6 -stable Cartan subalgebras

6.4.7 Real roots – Compact and non-compact imaginary roots

6.4.8 Jumps between Cartan subalgebras – Cayley transformations

6.5 Vogan diagrams

6.5.1 Illustration – The case

and

The other real forms

Vogan diagrams

6.5.2 The Borel and de Siebenthal theorem

6.5.3 Cayley transformations in

6.5.4 Reconstruction

6.5.5 Illustrations: versus

6.5.6 A pictorial summary – All real simple Lie algebras (Vogan diagrams)

6.6 Tits–Satake diagrams

6.6.1 Example 1:

Diagonal description

Cartan involution and roots

Restricted roots

6.6.2 Example 2:

Diagonal description

Cartan involution and roots

Restricted roots

6.6.3 Tits–Satake diagrams: Definition

6.6.4 Formal considerations

6.6.5 Illustration:

6.6.6 Some more formal considerations

6.7 The real semi-simple algebras

6.7.1 Dimensions

6.7.2 Dimensions

6.7.3 Dimensions

6.8 Summary – Tits–Satake diagrams for non-compact real forms

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