[Paper Review] Magic squares and matrix models of Lie algebras

Christine H Barton, Anthony Sudbery|ArXiv.org|Mar 1, 2002

Advanced Topics in Algebra13 references4 citations

TL;DR

This paper constructs matrix models of exceptional Lie algebras using octonions and alternative algebras, generalizing classical Lie algebras like su(3), sl(3), and sp(6) to include real forms of E6, E7, and F4. It extends the Tits magic square construction via split composition algebras to yield symmetric, matrix-based descriptions of these algebras, unifying exceptional and classical Lie algebras through octonionic analogues.

ABSTRACT

This paper is concerned with the description of exceptional simple Lie algebras as octonionic analogues of the classical matrix Lie algebras. We review the Tits-Freudenthal construction of the magic square, which includes the exceptional Lie algebras as the octonionic case of a construction in terms of a Jordan algebra of hermitian 3x3 matrices (Tits) or various plane and other geometries (Freudenthal). We present alternative constructions of the magic square which explain its symmetry, and show explicitly how the use of split composition algebras leads to analogues of the matrix Lie algebras su(3), sl(3) and sp(6). We adapt the magic square construction to include analogues of su(2), sl(2) and sp(4) for all real division algebras.

Motivation & Objective

• To provide matrix-based descriptions of exceptional Lie algebras, assimilating them to classical matrix Lie algebras.
• To extend the Tits-Freudenthal magic square construction to include non-compact real forms using split composition algebras.
• To define Lie algebras sa(3,K), sl(3,K), and sp(6,K) for alternative algebras K, recovering classical algebras for K=C and exceptional real forms for K=O.
• To generalize the construction to 2×2 matrices, yielding analogues of su(2), sl(2,C), and sp(4,C) for all real division algebras.
• To establish manifestly symmetric constructions of the magic square using Vinberg’s tensor product method and Ramond’s triality algebra.

Proposed method

• Constructs Lie algebras sa(3,K), sl(3,K), and sp(6,K) as matrix algebras over alternative algebras K, with K=O yielding real forms of F4, E6, and E7.
• Adapts the Tits magic square T(K,J) to use split forms of K, producing non-symmetric magic squares with rows corresponding to the matrix Lie algebras.
• Applies Vinberg’s construction using tensor products of algebras to achieve manifest symmetry in the magic square.
• Uses Ramond’s triality algebra to provide an alternative symmetric construction of the magic square.
• Employs generalized matrix identities and associator relations in alternative algebras to verify closure and Jacobi identities.
• Derives structure constants and commutator identities using trace and Jordan product properties in H3(K) and A3(K).

Experimental results

Research questions

• RQ1 How can exceptional Lie algebras be described as octonionic analogues of classical matrix Lie algebras?
• RQ2 What is the role of split composition algebras in extending the Tits magic square to non-compact real forms?
• RQ3 How do the matrix Lie algebras sa(3,K), sl(3,K), and sp(6,K) generalize classical algebras for K=C and yield real forms of E6, E7, and F4 for K=O?
• RQ4 Can the magic square construction be made manifestly symmetric using alternative algebraic frameworks such as tensor products or triality?
• RQ5 What is the relationship between the 2×2 and 3×3 matrix constructions in the context of alternative algebras and their associated Lie algebras?

Key findings

• The Lie algebra sa(3,O) is isomorphic to the compact real form of F4, extending su(3) to the octonionic case.
• The Lie algebra sl(3,O) is a non-compact real form of E6, generalizing sl(3,C) to the octonionic setting.
• The Lie algebra sp(6,O) is a non-compact real form of E7, analogous to sp(6,C) in the complex case.
• For split algebras, L3(K,R) = sa(3,K), L3(K,C~) = sl(3,K), and L3(K,H~) = sp(6,K), forming a non-symmetric magic square.
• The 2×2 matrix construction yields sa(2,K), sl(2,K), and sp(4,K), which reduce to su(2), sl(2,C), and sp(4,C) for K=C and are isomorphic to pseudo-orthogonal algebras.
• The paper proves that the commutator identities for matrix algebras over alternative algebras satisfy the Jacobi identity via associator identities and trace conditions, confirming Lie algebra structure.