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Table legend: S: Is this algebra simple? (Yes or No) SS: Is this algebra semi-simple? (Yes or No) Lie algebra Description S SS Remarks dim/R R the real numbers, the Lie bracket is zero 1 Rn the Lie bracket is zero n R3 the Lie bracket is the cross product 3 H quaternions, with Lie bracket the commutator 4 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,R) Y Y 3 M(n,R) n×n matrices, with Lie bracket the commutator n2 sl(n,R) square matrices with trace 0, with Lie bracket the commutator Y Y n2−1 so(n) skew-symmetric square real matrices, with Lie bracket the commutator. Y Y Exception: so(4) is semi-simple, but not simple. n(n−1)/2 sp(2n,R) real matrices that satisfy JA + ATJ = 0 where J is the standard skew-symmetric matrix Y Y n(2n+1) sp(n) square quaternionic matrices A satisfying A = −A*, with Lie bracket the commutator Y Y n(2n+1) u(n) square complex matrices A satisfying A = −A*, with Lie bracket the commutator n2 su(n) n≥2 square complex matrices A with trace 0 satisfying A = −A*, with Lie bracket the commutator Y Y n2−1 Complex Lie groups and their algebras[edit] 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 CM \pi_0 \pi_1 UC Remarks Lie algebra dim/C Cn group operation is addition N 0 0 abelian Cn n C× nonzero complex numbers with multiplication N 0 Z abelian C 1 GL(n,C) general linear group: invertible n×n complex matrices N 0 Z For n=1: isomorphic to C× M(n,C) n2 SL(n,C) special linear group: complex matrices with determinant 1 N 0 0 for n=1 this is a single point and thus compact. sl(n,C) n2−1 SL(2,C) Special case of SL(n,C) for n=2 N 0 0 Isomorphic to Spin(3,C), isomorphic to Sp(2,C) sl(2,C) 3 PSL(2,C) Projective special linear group N 0 Z2 SL(2,C) Isomorphic to the Möbius group, isomorphic to the restricted Lorentz group SO+(3,1,R), isomorphic to SO(3,C). sl(2,C) 3 O(n,C) orthogonal group: complex orthogonal matrices N Z2 – compact for n=1 so(n,C) n(n−1)/2 SO(n,C) special orthogonal group: complex orthogonal matrices with determinant 1 N 0 Z n=2 Z2 n>2 SO(2,C) is abelian and isomorphic to C×; nonabelian for n>2. SO(1,C) is a single point and thus compact and simply connected so(n,C) n(n−1)/2 Sp(2n,C) symplectic group: complex symplectic matrices N 0 0 sp(2n,C) n(2n+1) Complex Lie algebras[edit] The dimensions given are dimensions over C. Note that every complex Lie algebra can also be viewed as a real Lie algebra of twice the dimension. Lie algebra Description S SS Remarks dim/C C the complex numbers 1 Cn the Lie bracket is zero n M(n,C) n×n matrices, with Lie bracket the commutator n2 sl(n,C) square matrices with trace 0, with Lie bracket the commutator Y Y n2−1 sl(2,C) Special case of sl(n,C) with n=2 Y Y isomorphic to su(2) \otimes C 3 so(n,C) skew-symmetric square complex matrices, with Lie bracket the commutator Y Y Exception: so(4,C) is semi-simple, but not simple. n(n−1)/2 sp(2n,C) complex matrices that satisfy JA + ATJ = 0 where J is the standard skew-symmetric matrix Y Y n(2n+1) |
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