# Orthogonal Projections in Lie Theory

I have been studying a finite element method where rigid & elastic spatial motions are separated using an orthogonal projection (actually two: one for translations/stretches, the other for rotations/spins). Since all rigid & elastic modes are orthogonal with respect to the standard Euclidean inner product, I understand the projection's action in terms of linear algebra. However, I'd like to think about it from a Lie theory perspective. So far, I haven't found any literature discussing this topic. I'm wondering if the topic makes any sense &, if so, where I might explore it. Thanks, John.

2010/10/03 After more reading, I have found the following which may help: i) If I and J are two ideals in a Lie algebra g with zero intersection, then I and J are orthogonal subspaces with respect to the Killing form.; ii) If a subspace I of Lie algebra g satisfies a stronger condition that[g, I] is a subspace of I, then I is called an ideal in the Lie algebra g.; & iii) The Killing form for R^3 using the cross product bracket = 2 x standard Euclidean inner product.

If this is the right idea, then I need to show that rigid motion (e.g. I) & elastic deformation (e.g. J) are ideals with zero intersection.

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What is the difference between "in terms of linear algebra" and "from a Lie theory perspective"? – Deane Yang Oct 4 '10 at 0:42
Deane, Thanks for your comment. By "in terms of linear algebra", I mean the traditional definition of a projector P (P = P^2) & its decomposing a vector space into two orthogonal subspaces (i.e. rigid & elastic deformation in my case). By "from a Lie theory perspective", I'm asking if the vector space & its subgroups from the projection can be discussed using Lie groups, algebras &/or their adjoint representations. For example, Lie groups & algebras for rigid motion are SE(3)& se(3) -and- for elastic deformations are GL+(3,R) & gl(3, R). Any thoughts very welcome. Thanks, John. – John Craighead Oct 4 '10 at 1:47
John, I could be wrong, but I don't see how Lie theory can help you. My view of Lie theory is that it develops general abstract ideas that apply to any Lie group and not any specific one. However, much of Lie theory is motivated and arose from what people already knew about specific Lie groups such as $GL(n)$ and $O(n)$, but the theory does not tell you that much more about these groups themselves. Almost any concept in Lie theory corresponds to something more familiar and concrete when applied to a matrix group. – Deane Yang Oct 4 '10 at 13:51
Deane, Thanks for commenting so thoroughly & frankly. After more reading, I think that I should be focusing on decomposing the Lie algebra. I added an explanation dated 2010/10/03 to my original post. Any thoughts welcome if/as convenient. Thanks, John. – John Craighead Oct 4 '10 at 20:31