Lie algebras catalogue

Core idea

152 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.

Explore pairs and categories · Catalogue overview

Abelian lie algebra

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

Affine transformation lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{aff}(1,\mathbb C)$	complex: 2; real: 4	specified object
$\mathfrak{aff}(1,\mathbb R)$	real: 2	specified object
$\mathfrak{aff}(2,\mathbb C)$	complex: 6; real: 12	specified object
$\mathfrak{aff}(2,\mathbb R)$	real: 6	specified object
$\mathfrak{aff}(3,\mathbb C)$	complex: 12; real: 24	specified object
$\mathfrak{aff}(3,\mathbb R)$	real: 12	specified object
$\mathfrak{aff}(n,\mathbb C)$	complex: n^2+n; real: 2*(n^2+n)	parameterized family
$\mathfrak{aff}(n,\mathbb R)$	real: n^2+n	parameterized family

Compact sp lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{sp}(1)$	real: 3	specified object
$\mathfrak{sp}(2)$	real: 10	specified object
$\mathfrak{sp}(3)$	real: 21	specified object
$\mathfrak{sp}(n)$	real: n(2n+1)	parameterized family

Compact su lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{su}(1)$	real: 0	specified object
$\mathfrak{su}(2)$	real: 3	specified object
$\mathfrak{su}(3)$	real: 8	specified object
$\mathfrak{su}(4)$	real: 15	specified object
$\mathfrak{su}(6)$	real: 35	specified object
$\mathfrak{su}(n)$	real: n^2-1	parameterized family

Compact u lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{u}(1)$	real: 1	specified object
$\mathfrak{u}(2)$	real: 4	specified object
$\mathfrak{u}(3)$	real: 9	specified object
$\mathfrak{u}(n)$	real: n^2	parameterized family

Direct sum lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{sl}(2,\mathbb C)\oplus \mathfrak{sl}(2,\mathbb C)$	complex: 6; real: 12	specified object
$\mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{sl}(2,\mathbb R)$	real: 6	specified object
$\mathfrak{su}(2)\oplus \mathfrak{su}(2)$	real: 6	specified object

Euclidean motion lie algebra

Object	Dimensions	Kind of entry
$\mathfrak e(1)$	real: 1	specified object
$\mathfrak e(2)$	real: 3	specified object
$\mathfrak e(3)$	real: 6	specified object
$\mathfrak e(n)$	real: n*(n+1)/2	parameterized family

Exceptional lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{e}_{6}^{\mathrm{compact}}$	real: 78	specified object
$\mathfrak{e}_{6}(\mathbb C)$	complex: 78; real: 156	specified object
$\mathfrak{e}_{6(-26)}$	real: 78	specified object
$\mathfrak{e}_{6(2)}$	real: 78	specified object
$\mathfrak{e}_{6}^{\mathrm{split}}$	real: 78	specified object
$\mathfrak{e}_{7}^{\mathrm{compact}}$	real: 133	specified object
$\mathfrak{e}_{7}(\mathbb C)$	complex: 133; real: 266	specified object
$\mathfrak{e}_{7(-25)}$	real: 133	specified object
$\mathfrak{e}_{7(-5)}$	real: 133	specified object
$\mathfrak{e}_{7}^{\mathrm{split}}$	real: 133	specified object
$\mathfrak{e}_{8}^{\mathrm{compact}}$	real: 248	specified object
$\mathfrak{e}_{8}(\mathbb C)$	complex: 248; real: 496	specified object
$\mathfrak{e}_{8(-24)}$	real: 248	specified object
$\mathfrak{e}_{8}^{\mathrm{split}}$	real: 248	specified object
$\mathfrak{f}_{4}^{\mathrm{compact}}$	real: 52	specified object
$\mathfrak{f}_{4}(\mathbb C)$	complex: 52; real: 104	specified object
$\mathfrak{f}_{4}^{\mathrm{split}}$	real: 52	specified object
$\mathfrak{g}_{2}^{\mathrm{compact}}$	real: 14	specified object
$\mathfrak{g}_{2}(\mathbb C)$	complex: 14; real: 28	specified object
$\mathfrak{g}_{2}^{\mathrm{split}}$	real: 14	specified object

General linear lie algebra

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

Heisenberg lie algebra

Object	Dimensions	Kind of entry
$\mathfrak h_{3}(\mathbb C)$	complex: 3; real: 6	specified object
$\mathfrak h_{3}(\mathbb R)$	real: 3	specified object
$\mathfrak h_{5}(\mathbb C)$	complex: 5; real: 10	specified object
$\mathfrak h_{5}(\mathbb R)$	real: 5	specified object
$\mathfrak h_{7}(\mathbb C)$	complex: 7; real: 14	specified object
$\mathfrak h_{7}(\mathbb R)$	real: 7	specified object
$\mathfrak h_{2n+1}(\mathbb C)$	complex: 2n+1; real: 2(2*n+1)	parameterized family
$\mathfrak h_{2n+1}(\mathbb R)$	real: 2*n+1	parameterized family

Indefinite orthogonal lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{so}(1,1)$	real: 1	specified object
$\mathfrak{so}(2,1)$	real: 3	specified object
$\mathfrak{so}(2,2)$	real: 6	specified object
$\mathfrak{so}(3,1)$	real: 6	specified object
$\mathfrak{so}(3,3)$	real: 15	specified object
$\mathfrak{so}(4,1)$	real: 10	specified object
$\mathfrak{so}(4,4)$	real: 28	specified object
$\mathfrak{so}(n,n)$	real: n(2n-1)	parameterized family

Indefinite so lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{so}(p,q)$	real: (p+q)*(p+q-1)/2	parameterized family

Indefinite sp lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{sp}(1,1)$	real: 10	specified object
$\mathfrak{sp}(1,2)$	real: 21	specified object
$\mathfrak{sp}(1,3)$	real: 36	specified object
$\mathfrak{sp}(p,q)$	real: (p+q)(2(p+q)+1)	parameterized family

Indefinite special unitary lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{su}(3,3)$	real: 35	specified object

Indefinite su lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{su}(1,1)$	real: 3	specified object
$\mathfrak{su}(1,2)$	real: 8	specified object
$\mathfrak{su}(1,3)$	real: 15	specified object
$\mathfrak{su}(p,q)$	real: (p+q)^2-1	parameterized family

Indefinite u lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{u}(1,1)$	real: 4	specified object
$\mathfrak{u}(1,2)$	real: 9	specified object
$\mathfrak{u}(1,3)$	real: 16	specified object
$\mathfrak{u}(p,q)$	real: (p+q)^2	parameterized family

Orthogonal lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{so}(1,\mathbb C)$	complex: 0; real: 0	specified object
$\mathfrak{so}(1,\mathbb R)$	real: 0	specified object
$\mathfrak{so}(10,\mathbb C)$	complex: 45; real: 90	specified object
$\mathfrak{so}(12,\mathbb C)$	complex: 66; real: 132	specified object
$\mathfrak{so}(12,\mathbb R)$	real: 66	specified object
$\mathfrak{so}(2,\mathbb C)$	complex: 1; real: 2	specified object
$\mathfrak{so}(2,\mathbb R)$	real: 1	specified object
$\mathfrak{so}(3,\mathbb C)$	complex: 3; real: 6	specified object
$\mathfrak{so}(3,\mathbb R)$	real: 3	specified object
$\mathfrak{so}(4,\mathbb C)$	complex: 6; real: 12	specified object
$\mathfrak{so}(4,\mathbb R)$	real: 6	specified object
$\mathfrak{so}(5,\mathbb C)$	complex: 10; real: 20	specified object
$\mathfrak{so}(5,\mathbb R)$	real: 10	specified object
$\mathfrak{so}(6,\mathbb C)$	complex: 15; real: 30	specified object
$\mathfrak{so}(6,\mathbb R)$	real: 15	specified object
$\mathfrak{so}(8,\mathbb C)$	complex: 28; real: 56	specified object
$\mathfrak{so}(8,\mathbb R)$	real: 28	specified object
$\mathfrak{so}(n,\mathbb C)$	complex: n(n-1)/2; real: 2(n*(n-1)/2)	parameterized family
$\mathfrak{so}(n,\mathbb R)$	real: n*(n-1)/2	parameterized family

Quaternionic gl lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{gl}(1,\mathbb H)$	real: 4	specified object
$\mathfrak{gl}(2,\mathbb H)$	real: 16	specified object
$\mathfrak{gl}(3,\mathbb H)$	real: 36	specified object
$\mathfrak{gl}(n,\mathbb H)$	real: 4*n^2	parameterized family

Quaternionic orthogonal real form

Object	Dimensions	Kind of entry
$\mathfrak{so}^*(12)$	real: 66	specified object

Quaternionic sl lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{sl}(1,\mathbb H)$	real: 3	specified object
$\mathfrak{sl}(2,\mathbb H)$	real: 15	specified object
$\mathfrak{sl}(3,\mathbb H)$	real: 35	specified object
$\mathfrak{sl}(n,\mathbb H)$	real: 4*n^2-1	parameterized family

Special linear lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{sl}(1,\mathbb C)$	complex: 0; real: 0	specified object
$\mathfrak{sl}(1,\mathbb R)$	real: 0	specified object
$\mathfrak{sl}(2,\mathbb C)$	complex: 3; real: 6	specified object
$\mathfrak{sl}(2,\mathbb R)$	real: 3	specified object
$\mathfrak{sl}(3,\mathbb C)$	complex: 8; real: 16	specified object
$\mathfrak{sl}(3,\mathbb R)$	real: 8	specified object
$\mathfrak{sl}(4,\mathbb C)$	complex: 15; real: 30	specified object
$\mathfrak{sl}(5,\mathbb C)$	complex: 24; real: 48	specified object
$\mathfrak{sl}(6,\mathbb C)$	complex: 35; real: 70	specified object
$\mathfrak{sl}(n,\mathbb C)$	complex: n^2-1; real: 2*(n^2-1)	parameterized family
$\mathfrak{sl}(n,\mathbb R)$	real: n^2-1	parameterized family

Strictly upper triangular lie algebra

Object	Dimensions	Kind of entry
$\mathfrak n_{1}(\mathbb C)$	complex: 0; real: 0	specified object
$\mathfrak n_{1}(\mathbb R)$	real: 0	specified object
$\mathfrak n_{2}(\mathbb C)$	complex: 1; real: 2	specified object
$\mathfrak n_{2}(\mathbb R)$	real: 1	specified object
$\mathfrak n_{3}(\mathbb C)$	complex: 3; real: 6	specified object
$\mathfrak n_{3}(\mathbb R)$	real: 3	specified object
$\mathfrak n_{n}(\mathbb C)$	complex: n(n-1)/2; real: 2(n*(n-1)/2)	parameterized family
$\mathfrak n_{n}(\mathbb R)$	real: n*(n-1)/2	parameterized family

Symplectic lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{sp}(2,\mathbb C)$	complex: 3; real: 6	specified object
$\mathfrak{sp}(2,\mathbb R)$	real: 3	specified object
$\mathfrak{sp}(2n,\mathbb C)$	complex: n(2n+1); real: 2(n(2*n+1))	parameterized family
$\mathfrak{sp}(2n,\mathbb R)$	real: n(2n+1)	parameterized family
$\mathfrak{sp}(4,\mathbb C)$	complex: 10; real: 20	specified object
$\mathfrak{sp}(4,\mathbb R)$	real: 10	specified object
$\mathfrak{sp}(6,\mathbb C)$	complex: 21; real: 42	specified object
$\mathfrak{sp}(6,\mathbb R)$	real: 21	specified object

Underlying real lie algebra

Object	Dimensions	Kind of entry
$\mathfrak{sl}(2,\mathbb C)_{\mathbb R}$	real: 6	specified object
$\mathfrak{sl}(3,\mathbb C)_{\mathbb R}$	real: 16	specified object