[Paper Review] An extended Freudenthal Magic Square in characteristic 3

Isabel Cunha, Alberto Elduque|ArXiv.org|May 15, 2006

Advanced Topics in Algebra4 references4 citations

TL;DR

This paper extends Freudenthal's Magic Square to characteristic 3 using symmetric composition superalgebras, constructing a supersymmetric square that includes both Lie algebras and new simple Lie superalgebras not present in characteristic 0. The key contribution is the systematic classification of these superalgebras, most of which are exotic to characteristic 0 and arise from nontrivial composition superalgebras unique to characteristic 3.

ABSTRACT

Freudenthal's Magic Square, which in characteristic 0 contains the exceptional Lie algebras other than G2, is extended over fields of characteristic 3, through the use of symmetric composition superalgebras, to a larger square that contains both Lie algebras and superalgebras. With one exception, the simple Lie superalgebras that appear have no counterpart in characteristic 0.

Motivation & Objective

• To extend Freudenthal's Magic Square to fields of characteristic 3, where classical constructions fail due to degeneracies in Lie algebra structures.
• To incorporate Lie superalgebras into the Magic Square framework by utilizing symmetric composition superalgebras, which only exist nontrivially in characteristic 3.
• To classify and describe the structure of the new Lie superalgebras that emerge in the extended square, particularly those without counterparts in characteristic 0.
• To establish a unified construction of these superalgebras via the Lie algebra construction ${\mathfrak{g}}(S, S')$ based on symmetric composition superalgebras $S$ and $S'$.

Proposed method

• The authors use symmetric composition superalgebras—classified as nontrivial only in characteristic 3—of dimensions 3 and 6 as input for the construction ${\mathfrak{g}}(S, S')$.
• They apply the Lie algebra construction from Elduque (2004), adapted to work over fields of characteristic 3, using triality Lie algebras of symmetric composition superalgebras.
• The construction yields a symmetric square of Lie superalgebras, with entries labeled by the dimensions of the input superalgebras, and the output is analyzed via root space decomposition and Cartan matrix techniques.
• The even part of each superalgebra is decomposed into direct sums of classical Lie algebras (e.g., $\mathfrak{pgl}_3$, $\mathfrak{sl}_2$), while the odd part is described as tensor products of irreducible representations.
• The authors use representation theory to verify irreducibility of the odd part and compute dimensions of the even and odd components.
• They identify the derived algebra of each superalgebra and show that, except for one case, the derived algebra is simple and of codimension 1 in the full algebra.

Experimental results

Research questions

• RQ1 How can Freudenthal's Magic Square be extended to fields of characteristic 3, where standard Lie algebra constructions break down?
• RQ2 What new Lie superalgebras arise in this extended square, and how do they differ from those in characteristic 0?
• RQ3 Which symmetric composition superalgebras are relevant in characteristic 3, and how do they contribute to the construction of new Lie superalgebras?
• RQ4 Are the resulting Lie superalgebras simple, and if not, what is the structure of their derived algebras?
• RQ5 Can the new superalgebras be classified in terms of known Lie superalgebras or structures like triple systems?

Key findings

• The extended Freudenthal Magic Square in characteristic 3 contains Lie superalgebras not present in characteristic 0, with only one exception: the pair $(6,8)$, which corresponds to $\mathfrak{psl}_{2,2}$.
• For all entries except $(6,8)$, the derived superalgebra is a simple ideal of codimension 1, indicating that the full superalgebra is not simple but has a simple core.
• The even part of each Lie superalgebra is isomorphic to a direct sum of classical Lie algebras such as $\mathfrak{pgl}_3$, $\mathfrak{sl}_2$, $\mathfrak{sp}_6$, $\mathfrak{so}_{12}$, and $\mathfrak{f}_4$, depending on the input superalgebras.
• The odd part of each superalgebra is isomorphic to a tensor product of irreducible representations: for example, $\mathfrak{psl}_3 \otimes (2)$ for $\mathfrak{pgl}_3 \oplus \mathfrak{sl}_2$.
• The superalgebra ${\mathfrak{g}}(S_{1,2}, S_{1,2})$ is isomorphic to the Lie superalgebra defined in Elduque (2006b, Theorem 4.23(ii)), confirming consistency with prior constructions.
• The superalgebra ${\mathfrak{g}}(S_{4,2}, S_{4,2})$ matches the construction in Elduque (2006a, Theorem 3.1(ii)), showing compatibility with earlier results on symplectic and orthogonal triple systems.