Table of Lie groups
From Biocrawler, the free encyclopedia.
This article gives a table of some common Lie groups and their associated Lie algebras.
We remark on the topological properties of the group (dimension, connectedness, compactness, the nature of the fundamental group, and whether or not they are simply connected), as well as their on their algebraic properties (abelian, simple, semisimple).
For more examples of Lie groups and other related topics see List of Lie group topics.
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Real Lie groups and their algebras
| Lie group | Description | Remarks | Lie algebra | Description | dim/R |
|---|---|---|---|---|---|
| Rn | Euclidean space with addition | abelian, simply connected, not compact | Rn | the Lie bracket is zero | n |
| R× | nonzero real numbers with multiplication | abelian, not connected, not compact | R | the Lie bracket is zero | 1 |
| R+ | positive real numbers with multiplication | abelian, simply connected, not compact | R | the Lie bracket is zero | 1 |
| S1 | the circle group: complex numbers of absolute value 1, with multiplication; also known as U(1). | abelian, connected, not simply connected, compact; isomorphic to SO(2) and R/Z. | R | the Lie bracket is zero | 1 |
| H× | non-zero quaternions with multiplication | simply connected, not compact | H | quaternions, with Lie bracket the commutator | 4 |
| S3 | quaternions of absolute value 1, with multiplication; also known as Sp(1); topologically a 3-sphere | simply connected, compact, simple and semi-simple; isomorphic to SU(2) and to Spin(3) | Im(H) | quaternions with zero real part, with Lie bracket the commutator; isomorphic to real 3-vectors, with Lie bracket the cross product; also isomorphic to su(2) and to so(3) | 3 |
| GL(n,R) | general linear group: invertible n×n real matrices | not connected, not compact | M(n,R) | n×n matrices, with Lie bracket the commutator | n2 |
| GL+(n,R) | n×n real matrices with positive determinant | connected, not compact, for n≥2: not simply connected | M(n,R) | n×n matrices, with Lie bracket the commutator | n2 |
| SL(n,R) | special linear group: real matrices with determinant 1 | connected, for n≥2: not simply connected, not compact | sl(n,R) | square matrices with trace 0, with Lie bracket the commutator | n2−1 |
| O(n) | orthogonal group: real orthogonal matrices | not connected, compact | so(n) | skew-symmetric square real matrices, with Lie bracket the commutator | n(n−1)/2 |
| SO(n) | special orthogonal group: real orthogonal matrices with determinant 1 | connected, compact, for n ≥ 2: not simply connected, for n = 3 and n≥5: simple and semisimple | so(n) | skew-symmetric square real matrices, with Lie bracket the commutator | n(n−1)/2 |
| Spin(n) | spin group: double cover of SO(n) | compact, for n≥2: connected, for n≥3: simply connected | so(n) | skew-symmetric square real matrices, with Lie bracket the commutator | n(n−1)/2 |
| Sp(2n,R) | symplectic group: real symplectic matrices | connected, not simply connected, not compact, simple and semisimple | sp(2n,R) | real matrices that satisfy JA + ATJ = 0 where J is the standard skew-symmetric matrix | n(2n+1) |
| Sp(n) | compact symplectic group: quaternionic n×n unitary matrices | compact, simply connected, simple and semisimple | sp(n) | square quaternionic matrices A satisfying A = −A*, with Lie bracket the commutator | n(2n+1) |
| U(n) | unitary group: complex n×n unitary matrices | connected, not simply connected, compact. For n=1: isomorphic to S1. Note: this is not a complex Lie group/algebra | u(n) | square complex matrices A satisfying A = −A*, with Lie bracket the commutator | n2 |
| SU(n) | special unitary group: complex n×n unitary matrices with determinant 1 | simply connected, compact, for n ≥ 2: simple and semisimple. Note: this is not a complex Lie group/algebra | su(n) | square complex matrices A with trace 0 satisfying A = −A*, with Lie bracket the commutator | n2−1 |
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Complex Lie groups and their algebras
The dimensions given are dimensions over C. Note that every complex Lie group/algebra can also be viewed as a real Lie group/algebra of twice the dimension.
| Lie group | Description | Remarks | Lie algebra | Description | dim/C |
|---|---|---|---|---|---|
| Cn | group operation is addition | abelian, simply connected, not compact | Cn | the Lie bracket is zero | n |
| C× | nonzero complex numbers with multiplication | abelian, not simply connected, not compact | C | the Lie bracket is zero | 1 |
| GL(n,C) | general linear group: invertible n×n complex matrices | connected, not simply connected, not compact. For n=1: isomorphic to C× | M(n,C) | n×n matrices, with Lie bracket the commutator | n2 |
| SL(n,C) | special linear group: complex matrices with determinant 1 | simply connected, for n≥2: not compact, simple and semisimple. | sl(n,C) | square matrices with trace 0, with Lie bracket the commutator | n2−1 |
| O(n,C) | orthogonal group: complex orthogonal matrices | not connected, for n≥2: not compact | so(n,C) | skew-symmetric square complex matrices, with Lie bracket the commutator | n(n−1)/2 |
| SO(n,C) | special orthogonal group: complex orthogonal matrices with determinant 1 | connected, for n≥2: not simply connected, not compact, for n=3 and n≥5: simple and semisimple | so(n,C) | skew-symmetric square complex matrices, with Lie bracket the commutator | n(n−1)/2 |
| Sp(2n,C) | symplectic group: complex symplectic matrices | simply connected, not compact, simple and semisimple | sp(2n,C) | complex matrices that satisfy JA + ATJ = 0 where J is the standard skew-symmetric matrix | n(2n+1) |
