# Graded Simple Jordan Superalgebras of Growth One

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*V. G. Kac; C. Martinez; E. Zelmanov*

The superconformal algebras with a “hidden” Jordan structure are
those of type \(K\) and the recently discovered Cheng-Kac superalgebras
\(CK(6)\). We show that Jordan superalgebras related to the type
\(K\) are Kantor Doubles of some Jordan brackets on associative
commutative superalgebras and list these brackets.

#### Table of Contents

# Table of Contents

## Graded Simple Jordan Superalgebras of Growth One

- Contents vii8 free
- Introduction 112 free
- Chapter 1. Structure of the Even Part 920
- Chapter 2. Cartan type 3344
- Chapter 3. Even Part is Direct Sum of two Loop Algebras 3748
- Chapter 4. A is a Loop Algebra 5970
- Chapter 5. J is a finite dimensional Jordan Superalgebra or a Jordan Superalgebra of a Superform 6980
- Chapter 6. The Main Case 7586
- Chapter 7. Impossible Cases 127138
- 7.1. I = (0), A = A[sup((1))] ⊕ A[sup((2))]; A[sup((1))] is a loop algebra, A[sup((2))] is one-sided graded 127138
- 7.2. A = A[sup((1))] ⊕ A[sup((2))], A[sup((1))] is a negatively graded algebra, A[sup((2))] is a positively graded algebra 128139
- 7.3. A = A[sup((1))] ⊕ A[sup((2))] with A[sup((1))] infinite dimensional Jordan algebra of a bilinear form 129140
- 7.4. I ≠ (0), A/I is an infinite dimensional Jordan algebra of a nondegenerate symmetric bilinear form 131142
- 7.5. A = A[sup((1))] ⊕ A[sup((2))], A[sup((1))] is finite dimensional; A[sup((2))] is a loop algebra 133144

- Bibliography 139150