Lie groups catalogue - Knowlpedia

Core idea

180 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.

Open the Lie-group table · Explore pairs and categories · Catalogue overview

Additive group

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

Affine group

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

Compact symplectic group

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

Connected special pseudo orthogonal group

Object	Dimensions	Kind of entry
$\operatorname{SO}_0(p,q)$	real: (p+q)(p+q-1)/2	parameterized family

Euclidean group

Object	Dimensions	Kind of entry
$\mathrm{E}(1)$	real: 1	specified object
$\mathrm{E}(2)$	real: 3	specified object
$\mathrm{E}(3)$	real: 6	specified object
$\mathrm{E}(n)$	real: n(n+1)/2	parameterized family

Exceptional group

Object	Dimensions	Kind of entry
${E_6}^{\mathrm{cpt}}_{\mathrm{sc}}$	real: 78	specified object
$E_6(\mathbb C)_{\mathrm{sc}}$	real: 156; complex: 78	specified object
${E_6}^{\mathrm{split}}_{\mathrm{ad},0}$	real: 78	specified object
${E_7}^{\mathrm{cpt}}_{\mathrm{sc}}$	real: 133	specified object
$E_7(\mathbb C)_{\mathrm{sc}}$	real: 266; complex: 133	specified object
${E_7}^{\mathrm{split}}_{\mathrm{ad},0}$	real: 133	specified object
${E_8}^{\mathrm{cpt}}_{\mathrm{sc}}$	real: 248	specified object
$E_8(\mathbb C)_{\mathrm{sc}}$	real: 496; complex: 248	specified object
${E_8}^{\mathrm{split}}_{\mathrm{ad},0}$	real: 248	specified object
${F_4}^{\mathrm{cpt}}_{\mathrm{sc}}$	real: 52	specified object
$F_4(\mathbb C)_{\mathrm{sc}}$	real: 104; complex: 52	specified object
${F_4}^{\mathrm{split}}_{\mathrm{ad},0}$	real: 52	specified object
${G_2}^{\mathrm{cpt}}_{\mathrm{sc}}$	real: 14	specified object
$G_2(\mathbb C)_{\mathrm{sc}}$	real: 28; complex: 14	specified object
${G_2}^{\mathrm{split}}_{\mathrm{ad},0}$	real: 14	specified object

General linear group

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

Heisenberg group

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

Multiplicative group

Object	Dimensions	Kind of entry
$\mathbb C^\times$	real: 2; complex: 1	specified object
$\mathbb R^\times$	real: 1	specified object

Orthogonal group

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

Positive multiplicative group

Object	Dimensions	Kind of entry
$\mathbb R_{>0}^{\times}$	real: 1	specified object

Product group

Object	Dimensions	Kind of entry
$\mathrm{SL}(2,\mathbb C)\times\mathrm{SL}(2,\mathbb C)$	real: 12; complex: 6	specified object
$\mathrm{SU}(2)\times\mathrm{SU}(2)$	real: 6	specified object

Projective general linear group

Object	Dimensions	Kind of entry
$\operatorname{PGL}(1,\mathbb C)$	real: 0; complex: 0	specified object
$\operatorname{PGL}(1,\mathbb R)$	real: 0	specified object
$\operatorname{PGL}(2,\mathbb C)$	real: 6; complex: 3	specified object
$\operatorname{PGL}(2,\mathbb R)$	real: 3	specified object
$\operatorname{PGL}(3,\mathbb C)$	real: 16; complex: 8	specified object
$\operatorname{PGL}(3,\mathbb R)$	real: 8	specified object
$\operatorname{PGL}(n,\mathbb C)$	real: 2*(n^2-1); complex: n^2-1	parameterized family
$\operatorname{PGL}(n,\mathbb R)$	real: n^2-1	parameterized family

Projective special linear group

Object	Dimensions	Kind of entry
$\operatorname{PSL}(1,\mathbb C)$	real: 0; complex: 0	specified object
$\operatorname{PSL}(1,\mathbb R)$	real: 0	specified object
$\operatorname{PSL}(2,\mathbb C)$	real: 6; complex: 3	specified object
$\operatorname{PSL}(2,\mathbb R)$	real: 3	specified object
$\operatorname{PSL}(3,\mathbb C)$	real: 16; complex: 8	specified object
$\operatorname{PSL}(3,\mathbb R)$	real: 8	specified object
$\operatorname{PSL}(n,\mathbb C)$	real: 2*(n^2-1); complex: n^2-1	parameterized family
$\operatorname{PSL}(n,\mathbb R)$	real: n^2-1	parameterized family

Pseudo orthogonal group

Object	Dimensions	Kind of entry
$\operatorname{O}(1,1)$	real: 1	specified object
$\operatorname{O}(1,2)$	real: 3	specified object
$\operatorname{O}(1,3)$	real: 6	specified object
$\operatorname{O}(p,q)$	real: (p+q)(p+q-1)/2	parameterized family

Pseudo symplectic group

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

Pseudo unitary group

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

Quaternionic general linear group

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

Quaternionic special linear group

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

Restricted lorentz group

Object	Dimensions	Kind of entry
$\mathrm{SO}^{+}(1,3)$	real: 6	specified object

Special euclidean group

Object	Dimensions	Kind of entry
$\mathrm{SE}(1)$	real: 1	specified object
$\mathrm{SE}(2)$	real: 3	specified object
$\mathrm{SE}(3)$	real: 6	specified object
$\mathrm{SE}(n)$	real: n(n+1)/2	parameterized family

Special linear group

Object	Dimensions	Kind of entry
$\operatorname{SL}(1,\mathbb C)$	real: 0; complex: 0	specified object
$\operatorname{SL}(1,\mathbb R)$	real: 0	specified object
$\operatorname{SL}(2,\mathbb C)$	real: 6; complex: 3	specified object
$\operatorname{SL}(2,\mathbb R)$	real: 3	specified object
$\operatorname{SL}(3,\mathbb C)$	real: 16; complex: 8	specified object
$\operatorname{SL}(3,\mathbb R)$	real: 8	specified object
$\mathrm{SL}(4,\mathbb C)$	real: 30; complex: 15	specified object
$\operatorname{SL}(n,\mathbb C)$	real: 2*(n^2-1); complex: n^2-1	parameterized family
$\operatorname{SL}(n,\mathbb R)$	real: n^2-1	parameterized family

Special orthogonal group

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

Special pseudo orthogonal group

Object	Dimensions	Kind of entry
$\operatorname{SO}(1,1)$	real: 1	specified object
$\operatorname{SO}(1,2)$	real: 3	specified object
$\operatorname{SO}(1,3)$	real: 6	specified object
$\operatorname{SO}(p,q)$	real: (p+q)(p+q-1)/2	parameterized family

Special pseudo unitary group

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

Special unitary group

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

Spin group

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

Symplectic group

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

Torus group

Object	Dimensions	Kind of entry
$\mathbb T^{1}$	real: 1	specified object
$\mathbb T^{2}$	real: 2	specified object
$\mathbb T^{3}$	real: 3	specified object
$\mathbb T^{n}$	real: n	parameterized family

Unitary group

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