Moufang sets and structurable division algebras

Abstract

Chapter 1. Moufang sets 1.1. Definitions and basic properties 1.2. Moufang sets from linear algebraic groups 1.3. Moufang sets from Jordan algebras 1.4. Moufang sets from skew-hermitian forms Chapter 2. Structurable algebras 2.1. Definitions and basic properties 2.2. Conjugate invertibility in structurable algebras 2.3. Examples of structurable algebras 2.4. Construction of Lie algebras from structurable algebras 2.5. Isotopies of structurable algebras Chapter 3. One-invertibility for structurable algebras 3.1. Algebraicity of 5-graded Lie algebras 3.2. One-invertibility in A × S 3.3. One-invertibility for structurable division algebras Chapter 4. Simple structurable algebras and simple algebraic groups 4.1. Simple algebraic groups from simple structurable algebras 4.2. Deducing algebraicity 4.3. Structurable division algebras from simple algebraic groups of k-rank 1 Chapter 5. Moufang sets and structurable division algebras 5.1. Moufang sets from structurable division algebras 5.2. Structurable division algebras from algebraic Moufang sets Chapter 6. Examples 6.1. Associative algebras with involution 6.2. Jordan algebras 6.3. Hermitian structurable algebras 6.4. Structurable algebras of skew-dimension one 6.5. Forms of the tensor product of two composition algebras 6.6. Classification theorem for structurable division algebras Bibliography iii

On structure and TKK algebras for Jordan superalgebras

“…The construction of Ko(V + , V -) starting from Ko(V + , V -) in Proposition 5.8 fits into a more general construction. In [BDS,Section 4.1], the authors start from an arbitrary (2n + 1)-graded Lie superalgebra L = i∈Z L i (strictly speaking only Lie algebras are considered, but the procedure carries over naturally to the super case). Then [BDS,Construction 4.1.2] constructs an extension L over L, which is again a (2n + 1)-graded Lie superalgebra which satisfies…”

“…An interesting consequence of [BDS,Lemma 4.1.3] is then Out( Ko(V + , V -)) = 0, for arbitrary Jordan superpairs (V + , V -), so also for arbitrary (unital or non-unital) Jordan superalgebras.…”