DifferentialGeometry/LieAlgebras/SimpleLieAlgebraDataDetails

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Details for SimpleLieAlgebraData - definitions of the classical matrix algebras

Description

The following two tables describe the Lie algebras which can be intialized with the command SimpleLieAlgebraData .

The Classical Simple Real Matrix Algebras

Name	Dim	Type	Rank	Matrices	Conditions	Examples
sl(n)		A				Example 1.
		A				Example 2
version 1		A			, arbitrary, skew-symmetric symmetric	Example 3
version 2		A			arbitrary	Example 3
		A				Example 4
version 1		B		$Matrix(3, 3, {(1, 1) = A, (1, 2) = B, (1, 3) = C, (2, 1) = D, (2, 2) = -A^t, (2, 3) = E, (3, 1) = -E, (3, 2) = -C^t, (3, 3) = F})$	arbitrary ,	Example 5
version 2		B		$Matrix(2, 2, {(1, 1) = A, (1, 2) = B, (2, 1) = -B^t, (2, 2) = C})$	A arbitrary	Example 5
		C		$Matrix(2, 2, {(1, 1) = A, (1, 2) = B, (2, 1) = C, (2, 2) = -A^t})$		Example 6
		C			arbitrary = 0, , , ,	Example 7
		C				Example 7
version 1		D		$Matrix(3, 3, {(1, 1) = A, (1, 2) = B, (1, 3) = C, (2, 1) = D, (2, 2) = -A^t, (2, 3) = E, (3, 1) = -E, (3, 2) = -C^t, (3, 3) = F})$	arbitrary ,	Example 8
version 2		D		$Matrix(2, 2, {(1, 1) = A, (1, 2) = B, (2, 1) = -B^t, (2, 2) = C})$	A arbitrary	Example 9
		D		$(Matrix(2, 2, {(1, 1) = Z[1], (1, 2) = Z[2], (2, 1) = -conjugate(Z[2]), (2, 2) = conjugate(Z[1])})) = (Matrix(2, 2, {(1, 1) = A[1]+IA[2], (1, 2) = B[1]+IB[2], (2, 1) = -B^t+IB[2]^t, (2, 2) = A[1]-IA[2]}))$	,	Example 10

The following algebras can also be initialized with the command SimpleLieAlgebraData .

Other Classical Real Matrix Algebras

Name	Dim	Matrices	Conditions	Examples
			Z,	Example 11
			Z,	Example 11
			Z, Z, trace-free	Example 12
			arbitrary	Example 13
			Z, Z, skew-symmetric	Example 14
		$Matrix(2, 2, {(1, 1) = A[1]+IA[2], (1, 2) = B[1]+IB[2], (2, 1) = C[1]+IC[2], (2, 2) = -A^t-IA[2]^t})$		Example 15
			upper triangular	Example 16
			strictly upper triangular	Example 17

Examples

Example 1.

(2.1)

Example 2.

(2.2)

Example 3.

(2.3)

(2.4)

Example 4.

(2.5)

Example 5.

(2.6)

(2.7)

Example 6.

(2.8)

Example 7.

(2.9)

Example 8.

(2.10)

Example 9.

(2.11)

(2.12)

Example 10.

(2.13)

Example 11.

[Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 0, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 0, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 1, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 0, (2, 2) = 1})]

(2.14)

(2.15)

Example 12.

[Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 0, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 1, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = I, (2, 1) = 0, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = I, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 0, (2, 2) = -1}), Matrix(2, 2, {(1, 1) = I, (1, 2) = 0, (2, 1) = 0, (2, 2) = -I})]