Finding the Real Lie Algebra of SL(n,H) in GL(n,H)

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To find the real Lie algebra of SL(n,H) within GL(n,H), one can start by considering the tangent space at the identity, which involves creating a path in GL(n,H) and differentiating it. The condition for the determinant to equal one implies that the corresponding Lie algebra elements must have zero trace. The discussion highlights that GL(n,H) consists of matrices that satisfy specific conjugation relations, leading to the conclusion that the Lie algebra is defined by matrices D that satisfy the commutation relation DJ - JD* = 0. This approach provides a foundational understanding of the structure of SL(n,H) and its Lie algebra. The conversation emphasizes the importance of differentiating conditions to derive the algebraic properties.
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Hi all,

Anybody knowes how to find, or at least knows the reference that shows, the real lie algebra of sl(n,H)?
By sl(n,H), I mean the elements in Gl(n,H) [i.e. the invertible quaternionic n by n matrices] whose real determinant is one.

Many Thanks
Asi
 
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Given that the Lie Algebra of a Lie Group is identified with the tangent space of the Lie Group at the identity, I would create an arbitrary path \gamma(t)\colon R\rightarrow GL(n,H) such that \gamma(0)=\tilde1 and then take the derivative of that path at the identity.

I haven't seen a representation of GL(n,H) so, if you find one, post it and we'll see if we can take it's derivative.

ZM

PS: typically the requirement for \hbox{det} g=1 for g\in G means that the element in the Lie Algebra has zero trace.
 
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Actually, GL(n,H) is the set of all matrices A in GL(2n,C) such that AJ=Ja where a is complex conjugate of A and J is 2n by 2n matrice with the rows
(0 -I) and (I 0), I is the n by n identity matrix.

Thanks
 
Differentiate the condition. The lie algebra is the set of matrices D satisfing Id+eD is in GL(n,H) mod e^2.

Thus (id+eD)J=J(Id+eD)^*

Thus eDJ=eJD* or DJ-JD^*=0. So the lie algebra is

gl(n,H):={ D : DJ-JD^*=0}
 
Many thanks, But how did you get the lie algebra is the set of matrices D satisfing Id+eD is in GL(n,H) mod e^2 by differentiation?
 
Because that is the definition of the lie algebra. I just wrote down the conditions necessary to be a tangent vector (and ignored convergence issues - everytihing is defined in terms of polynomials so there is no concern here).
 
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Thanks, how about sl(n,H) ?
 
Do the same thing: it must be satisfy two conditions.
 

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