Scalar, associative, and Jordan algebras catalogue

Core idea

109 separately identified objects and parameterized families. Real and complex scalar choices have distinct entries. An isomorphism is a recorded relationship, not a reason to collapse the entries.

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Cayley dickson algebras

Object	Dimensions	Kind of entry
$\mathbb S$	real: 16	specified object

Complex composition algebras

Object	Dimensions	Kind of entry
$\mathbb C\otimes_{\mathbb R}\mathbb C$	complex: 2; real: 4	specified object
$\mathbb H\otimes_{\mathbb R}\mathbb C$	complex: 4; real: 8	specified object
$\mathbb O\otimes_{\mathbb R}\mathbb C$	complex: 8; real: 16	specified object

Full matrix jordan algebras

Object	Dimensions	Kind of entry
$M_{1}(\mathbb C)^+$	real: 2; complex: 1	specified object
$M_{1}(\mathbb R)^+$	real: 1	specified object
$M_{2}(\mathbb C)^+$	real: 8; complex: 4	specified object
$M_{2}(\mathbb R)^+$	real: 4	specified object
$M_{3}(\mathbb C)^+$	real: 18; complex: 9	specified object
$M_{3}(\mathbb R)^+$	real: 9	specified object
$M_{n}(\mathbb C)^+$	real: 2*n^2; complex: n^2	parameterized family
$M_{n}(\mathbb R)^+$	real: 1*n^2	parameterized family

Hermitian jordan algebras

Object	Dimensions	Kind of entry
$\operatorname{Herm}_{1}(\mathbb C)$	real: 1	specified object
$\operatorname{Herm}_{1}((\mathbb C\otimes_{\mathbb R}\mathbb C))$	complex: 1; real: 2	specified object
$\operatorname{Herm}_{1}(\mathbb H)$	real: 1	specified object
$\operatorname{Herm}_{1}((\mathbb H\otimes_{\mathbb R}\mathbb C))$	complex: 1; real: 2	specified object
$\operatorname{Herm}_{1}(\mathbb O)$	real: 1	specified object
$\operatorname{Herm}_{1}((\mathbb O\otimes_{\mathbb R}\mathbb C))$	complex: 1; real: 2	specified object
$\operatorname{Herm}_{1}(\mathbb R)$	real: 1	specified object
$\operatorname{Herm}_{1}((\mathbb R\otimes_{\mathbb R}\mathbb C))$	complex: 1; real: 2	specified object
$\operatorname{Herm}_{1}(\mathbb C_s)$	real: 1	specified object
$\operatorname{Herm}_{1}(\mathbb H_s)$	real: 1	specified object
$\operatorname{Herm}_{1}(\mathbb O_s)$	real: 1	specified object
$\operatorname{Herm}_{2}(\mathbb C)$	real: 4	specified object
$\operatorname{Herm}_{2}((\mathbb C\otimes_{\mathbb R}\mathbb C))$	complex: 4; real: 8	specified object
$\operatorname{Herm}_{2}(\mathbb H)$	real: 6	specified object
$\operatorname{Herm}_{2}((\mathbb H\otimes_{\mathbb R}\mathbb C))$	complex: 6; real: 12	specified object
$\operatorname{Herm}_{2}(\mathbb O)$	real: 10	specified object
$\operatorname{Herm}_{2}((\mathbb O\otimes_{\mathbb R}\mathbb C))$	complex: 10; real: 20	specified object
$\operatorname{Herm}_{2}(\mathbb R)$	real: 3	specified object
$\operatorname{Herm}_{2}((\mathbb R\otimes_{\mathbb R}\mathbb C))$	complex: 3; real: 6	specified object
$\operatorname{Herm}_{2}(\mathbb C_s)$	real: 4	specified object
$\operatorname{Herm}_{2}(\mathbb H_s)$	real: 6	specified object
$\operatorname{Herm}_{2}(\mathbb O_s)$	real: 10	specified object
$\operatorname{Herm}_{3}(\mathbb C)$	real: 9	specified object
$\operatorname{Herm}_{3}((\mathbb C\otimes_{\mathbb R}\mathbb C))$	complex: 9; real: 18	specified object
$\operatorname{Herm}_{3}(\mathbb H)$	real: 15	specified object
$\operatorname{Herm}_{3}((\mathbb H\otimes_{\mathbb R}\mathbb C))$	complex: 15; real: 30	specified object
$\operatorname{Herm}_{3}(\mathbb O)$	real: 27	specified object
$\operatorname{Herm}_{3}((\mathbb O\otimes_{\mathbb R}\mathbb C))$	complex: 27; real: 54	specified object
$\operatorname{Herm}_{3}(\mathbb R)$	real: 6	specified object
$\operatorname{Herm}_{3}((\mathbb R\otimes_{\mathbb R}\mathbb C))$	complex: 6; real: 12	specified object
$\operatorname{Herm}_{3}(\mathbb C_s)$	real: 9	specified object
$\operatorname{Herm}_{3}(\mathbb H_s)$	real: 15	specified object
$\operatorname{Herm}_{3}(\mathbb O_s)$	real: 27	specified object
$\operatorname{Herm}_{n}(\mathbb C)$	real: n+2n(n-1)/2	parameterized family
$\operatorname{Herm}_{n}((\mathbb C\otimes_{\mathbb R}\mathbb C))$	complex: n+2n(n-1)/2; real: 2(n+2n*(n-1)/2)	parameterized family
$\operatorname{Herm}_{n}(\mathbb H)$	real: n+4n(n-1)/2	parameterized family
$\operatorname{Herm}_{n}((\mathbb H\otimes_{\mathbb R}\mathbb C))$	complex: n+4n(n-1)/2; real: 2(n+4n*(n-1)/2)	parameterized family
$\operatorname{Herm}_{n}(\mathbb O)$	real: n+8n(n-1)/2	parameterized family
$\operatorname{Herm}_{n}((\mathbb O\otimes_{\mathbb R}\mathbb C))$	complex: n+8n(n-1)/2; real: 2(n+8n*(n-1)/2)	parameterized family
$\operatorname{Herm}_{n}(\mathbb R)$	real: n+1n(n-1)/2	parameterized family
$\operatorname{Herm}_{n}((\mathbb R\otimes_{\mathbb R}\mathbb C))$	complex: n+1n(n-1)/2; real: 2(n+1n*(n-1)/2)	parameterized family
$\operatorname{Herm}_{n}(\mathbb C_s)$	real: n+2n(n-1)/2	parameterized family
$\operatorname{Herm}_{n}(\mathbb H_s)$	real: n+4n(n-1)/2	parameterized family
$\operatorname{Herm}_{n}(\mathbb O_s)$	real: n+8n(n-1)/2	parameterized family

Matrix algebras

Object	Dimensions	Kind of entry
$M_{1}(\mathbb C)$	real: 2; complex: 1	specified object
$M_{1}(\mathbb H)$	real: 4	specified object
$M_{1}(\mathbb R)$	real: 1	specified object
$M_{1}(\mathbb C_s)$	real: 2	specified object
$M_{1}(\mathbb H_s)$	real: 4	specified object
$M_{2}(\mathbb C)$	real: 8; complex: 4	specified object
$M_{2}(\mathbb H)$	real: 16	specified object
$M_{2}(\mathbb R)$	real: 4	specified object
$M_{2}(\mathbb C_s)$	real: 8	specified object
$M_{2}(\mathbb H_s)$	real: 16	specified object
$M_{3}(\mathbb C)$	real: 18; complex: 9	specified object
$M_{3}(\mathbb H)$	real: 36	specified object
$M_{3}(\mathbb R)$	real: 9	specified object
$M_{3}(\mathbb C_s)$	real: 18	specified object
$M_{3}(\mathbb H_s)$	real: 36	specified object
$M_{n}(\mathbb C)$	real: 2*n^2; complex: n^2	parameterized family
$M_{n}(\mathbb H)$	real: 4*n^2	parameterized family
$M_{n}(\mathbb R)$	real: 1*n^2	parameterized family
$M_{n}(\mathbb C_s)$	real: 2*n^2	parameterized family
$M_{n}(\mathbb H_s)$	real: 4*n^2	parameterized family
$M_{1}(\mathbb O)$	real: 8	specified object
$M_{1}(\mathbb S)$	real: 16	specified object
$M_{2}(\mathbb O)$	real: 32	specified object
$M_{2}(\mathbb S)$	real: 64	specified object
$M_{3}(\mathbb O)$	real: 72	specified object
$M_{3}(\mathbb S)$	real: 144	specified object
$M_{n}(\mathbb O)$	real: 8*n^2	parameterized family
$M_{n}(\mathbb S)$	real: 16*n^2	parameterized family

Real composition algebras

Object	Dimensions	Kind of entry
$\mathbb C$	real: 2; complex: 1	specified object
$\mathbb H$	real: 4	specified object
$\mathbb O$	real: 8	specified object
$\mathbb R$	real: 1	specified object

Real split composition algebras

Object	Dimensions	Kind of entry
$\mathbb C_s$	real: 2	specified object
$\mathbb H_s$	real: 4	specified object
$\mathbb O_s$	real: 8	specified object

Spin factors

Object	Dimensions	Kind of entry
$J(\mathbb C^{1})$	complex: 2; real: 4	specified object
$J(\mathbb R^{1})$	real: 2	specified object
$J(\mathbb C^{2})$	complex: 3; real: 6	specified object
$J(\mathbb R^{2})$	real: 3	specified object
$J(\mathbb C^{3})$	complex: 4; real: 8	specified object
$J(\mathbb R^{3})$	real: 4	specified object
$J(\mathbb R^{5})$	real: 6	specified object
$J(\mathbb R^{9})$	real: 10	specified object
$J(\mathbb C^{n})$	complex: n+1; real: 2*(n+1)	parameterized family
$J(\mathbb R^{n})$	real: n+1	parameterized family

Zero product jordan algebras

Object	Dimensions	Kind of entry
$(\mathbb C^{1},\circ=0)$	complex: 1; real: 2	specified object
$(\mathbb R^{1},\circ=0)$	real: 1	specified object
$(\mathbb C^{2},\circ=0)$	complex: 2; real: 4	specified object
$(\mathbb R^{2},\circ=0)$	real: 2	specified object
$(\mathbb C^{3},\circ=0)$	complex: 3; real: 6	specified object
$(\mathbb R^{3},\circ=0)$	real: 3	specified object
$(\mathbb C^{n},\circ=0)$	complex: n; real: 2*n	parameterized family
$(\mathbb R^{n},\circ=0)$	real: n	parameterized family