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

In summary, the conversation discusses the identification of the Lie Algebra of the Lie group GL(n,H) and the elements that satisfy the conditions for being a tangent vector. The discussion also briefly mentions the Lie Algebra of sl(n,H) and the necessary conditions for it.
  • #1

asm

1
0
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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  • #2
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 [tex]\gamma(t)\colon R\rightarrow GL(n,H)[/tex] such that [tex]\gamma(0)=\tilde1[/tex] and then take the derivative of that path at the identity.

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

ZM

PS: typically the requirement for [tex]\hbox{det} g=1[/tex] for [tex]g\in G[/tex] means that the element in the Lie Algebra has zero trace.
 
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  • #3
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
 
  • #4
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}
 
  • #5
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?
 
  • #6
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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  • #7
Thanks, how about sl(n,H) ?
 
  • #8
Do the same thing: it must be satisfy two conditions.
 

1. What is a Lie group?

A Lie group is a type of mathematical group that is also a smooth manifold. It is named after the mathematician Sophus Lie and is used to study transformations and symmetries in mathematics and physics.

2. How is a Lie group different from a regular group?

A regular group is a set of elements with a binary operation that satisfies certain axioms, while a Lie group has the additional structure of a smooth manifold. This allows for the study of continuous symmetries and transformations.

3. What are some examples of Lie groups?

Some examples of Lie groups include the rotation group, the special orthogonal group, and the general linear group. These groups are commonly used in physics to describe symmetries in space and time.

4. What is the significance of Lie groups in mathematics and physics?

Lie groups are important in both mathematics and physics because they provide a powerful framework for understanding symmetries and transformations. They have applications in areas such as geometry, differential equations, and quantum mechanics.

5. Can Lie groups be applied to real-world problems?

Yes, Lie groups have many practical applications in fields such as robotics, computer vision, and control theory. They can be used to model and analyze complex systems with symmetries, making them a valuable tool for solving real-world problems.

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