Preserve this form:
u^2+t^2-x^2-y^2
t^2-x^2
{t,x}
RotationMatrix[Iθ]
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{{Cosh[θ],-Sinh[θ]},{-Sinh[θ],Cosh[θ]}}.{t,x}
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{tCosh[θ]-xSinh[θ],xCosh[θ]-tSinh[θ]}
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%.{{1,0},{0,-1}}.%
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(xCosh[θ]-tSinh[θ])(-xCosh[θ]+tSinh[θ])+
2
(tCosh[θ]-xSinh[θ])
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FullSimplify[%]
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(t-x)(t+x)
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What is matrix representation
What is matrix representation
u^2+t^2-x^2-y^2
Has 2 rotation subgroups
https://math.stackexchange.com/questions/1662181/finding-the-basis-of-mathfrakso2-2-lie-algebra-of-so2-2
liealgebra={SparseArray[{{1,2}1,{2,1}-1},{4,4}],SparseArray[{{3,4}1,{4,3}-1},{4,4}],SparseArray[{{1,3}1,{3,1}1},{4,4}],SparseArray[{{1,4}1,{4,1}1},{4,4}],SparseArray[{{2,3}1,{3,2}1},{4,4}],SparseArray[{{2,4}1,{4,2}1},{4,4}]}
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DiagonalMatrix[{1,1,-1,-1}]
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{{1,0,0,0},{0,1,0,0},{0,0,-1,0},{0,0,0,-1}}
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anticommute[m_]:=m.DiagonalMatrix[{1,1,-1,-1}]-DiagonalMatrix[{1,1,-1,-1}].Transpose[m]
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anticommute/@liealgebra
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{True,True,True,True,True,True}
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Reduce[anticommute[Array[a,{4,4}]]]
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a[1,1]0&&a[1,2]-a[2,1]&&a[2,2]0&&a[1,3]a[3,1]&&a[2,3]a[3,2]&&a[3,3]0&&a[1,4]a[4,1]&&a[2,4]a[4,2]&&a[3,4]-a[4,3]&&a[4,4]0
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The group O(p, q) is defined for vector spaces over the reals. For complex spaces, all groups O(p, q; C) are isomorphic to the usual orthogonal group O(p + q; C), since the transform {\displaystyle z_{j}\mapsto iz_{j}}z_j \mapsto iz_j changes the signature of a form.
MatrixExp[t#]&/@liealgebra
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#.{u,t,x,y}&/@%202//FullSimplify
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#.DiagonalMatrix[{1,1,-1,-1}].#&/@%
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FullSimplify[%]
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{+--,+--,+--,+--,+--,+--}
2
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Counts[%]
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+--6
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