Structurable Tori

Abstract

The classification of structurable tori with nontrivial involution, which was begun by Allison and Yoshii, is completed. New examples of structurable tori are obtained using a construction of structurable algebras from a semilinear version of cubic forms satisfying the adjoint identity. The classification uses techniques borrowed from quadratic forms over ℤ ₂ and from the geometry of generalized quadrangles. Since structurable tori are the coordinate algebras for the centerless cores of extended affine Lie algebras of type BC₁, the results of this article provide a classification and new examples for this class of Lie algebras.

Key Words:

• Cubic form with adjoint identity
• Extended affine Lie algebra
• Generalized quadrangle
• Lie algebra of type BC ₁
• Lie torus
• Structurable algebra
• Structurable torus

2000 Mathematics Subject Classification:

• Primary 17A30, 17B60
• Secondary 17A75, 17C40, 17B67

ACKNOWLEDGMENT

Bruce Allison gratefully acknowledges the support of the Natural Sciences and Engineering Research Council of Canada.

Notes

¹In Allison (1978) there is an overriding assumption that algebras are finite dimensional. However, that assumption is not used in the sections (1–5 and 8(iii)) of Allison (1978) that we use in the infinite dimensional setting.

²The motivation for the terminology comes from the case when and 𝒜 is associative. In that case, ϵ is a multiplicative version of a quadratic form (see Lemma 5.14) and the terminology is standard.

³The product on 𝒜(k) defined in Allison (1978) and Allison and Yoshii (2003) is the opposite of the product defined here. However, the involution on 𝒜(k) is an isomorphism of 𝒜(k) with 𝒜(k)^op as algebras with involution.

⁴In Allison and Yoshii (2003, Proposition 4.5), the M-torus ℬ is realized as a quantum torus with involution.

⁵Nondegeneracy of h is equivalent to the map α:u → h(, u) being an injection of 𝒲 into its dual space for ℰ. If α is a bijection, then (h, N) automatically has an adjoint.

⁶It is not difficult to give a direct argument for this without using Neher and Yoshii (2003, Proposition 4.9), a result that uses the classification of Jordan tori.

Communicated by A. Elduque.

Dedicated to the memory of Professor Issai Kantor.