DifferentialGeometry/LieAlgebras/Query/Closed UnderConjugate - Maple Help
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Query[ClosedUnderConjugation] - check if a list of vectors or matrices is closed under complex conjugation

Query[ClosedUnderTransposition] - check if a list of square matrices is closed under transposition

Query[ClosedUnderHermitianTransposition] - check if a list of square matrices is closed under Hermitian (complex conjugation) transposition

Calling Sequences

Query()

Query()

Query()

Parameters

A       - a list of column vectors, or a list of square matrices

option  - the keyword method

Description

 • Let be a list of square matrices. These query commands return true if for each  for some whereconjugation) or transposition) or Hermitian transposition).
 • With the keyword option method = "span", these commands return true if for each  span($A$).
 • These commands are useful in the study of classical semi-simple Lie algebras. For example, if a semi-simple matrix Lie algebra is given, then a Cartan decomposition can easily be computed if the algebra is closed under Hermitian transposition. As another example, to calculate the Satake diagram for a non-compact simple Lie algebra, one must use a set of positive roots closed under complex conjugation.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.

We check that a list of vectors is closed under complex-conjugation.

 > $\mathrm{A1}≔\left[⟨I,-I,1⟩,⟨0,1,0⟩,⟨-I,I,1⟩\right]$
 > $\mathrm{Query}\left(\mathrm{A1},"ClosedUnderConjugation"\right)$

Example 2.

We check that the list of matrices defining is closed under transposition.

 > $\mathrm{A2}≔\mathrm{map}\left(\mathrm{Matrix},\left[\left[\left[1,0,0,0\right],\left[0,0,0,0\right],\left[0,0,-1,0\right],\left[0,0,0,0\right]\right],\left[\left[0,1,0,0\right],\left[0,0,0,0\right],\left[0,0,0,0\right],\left[0,0,-1,0\right]\right],\left[\left[0,0,0,0\right],\left[1,0,0,0\right],\left[0,0,0,-1\right],\left[0,0,0,0\right]\right],\left[\left[0,0,0,0\right],\left[0,1,0,0\right],\left[0,0,0,0\right],\left[0,0,0,-1\right]\right],\left[\left[0,0,0,-1\right],\left[0,0,1,0\right],\left[0,0,0,0\right],\left[0,0,0,0\right]\right],\left[\left[0,0,0,0\right],\left[0,0,0,0\right],\left[0,-1,0,0\right],\left[1,0,0,0\right]\right]\right]\right)$
 > $\mathrm{Query}\left(\mathrm{A2},"ClosedUnderTransposition"\right)$

Example 3.

We check that the span of the matrices defining is closed under Hermitian transposition.

 > $\mathrm{A3}≔\mathrm{map}\left(\mathrm{Matrix},\left[\left[\left[0,-1,0,0\right],\left[1,0,0,0\right],\left[0,0,0,-1\right],\left[0,0,1,0\right]\right],\left[\left[0,-I,0,0\right],\left[I,0,0,0\right],\left[0,0,0,I\right],\left[0,0,-I,0\right]\right],\left[\left[0,0,1,0\right],\left[0,0,0,0\right],\left[-1,0,0,0\right],\left[0,0,0,0\right]\right],\left[\left[0,0,0,1\right],\left[0,0,1,0\right],\left[0,-1,0,0\right],\left[-1,0,0,0\right]\right],\left[\left[0,0,0,0\right],\left[0,0,0,1\right],\left[0,0,0,0\right],\left[0,-1,0,0\right]\right],\left[\left[0,0,0,-I\right],\left[0,0,I,0\right],\left[0,-I,0,0\right],\left[I,0,0,0\right]\right]\right]\right)$
 > $\mathrm{Query}\left(\mathrm{A3},"ClosedUnderHermitianTransposition"\right)$
 > $\mathrm{Query}\left(\mathrm{A3},\mathrm{method}="span","ClosedUnderHermitianTransposition"\right)$