Recent content by Don Aman

Unfortunately even your original comment was more misleading than it was useful, but I thank you from refraining to make additional useless comments. And thank you to others in this thread who tried to answer. At this point, I guess I have to conclude that perhaps Bartels was mistaken about...

Still there?

Well, I think so... I mean, I'm pretty sure... For example, the alternating square of a 2d vector space is 1d, whose symmetric square is also 1d. While the symmetric square of a 2d vector space is 3d, and its alternating square is 3d again. So clearly they cannot be isomorphic. But why do...

OK, yes that's right. The Hodge star can be defined in terms of a local orthonormal or holonomic frame, and then the Volume form can be defined in terms of it. But of course this is what I was hoping to avoid. Maybe you're right, this is the best that I can do. What prompted me to ask the...

The Hodge star is usually defined in terms of the volume form, so your proposed definition seems circular. Do you have a different definition of the Hodge star in mind?

The volume form on a Riemannian manifold is usually defined either in terms of a local holonomic frame or orthonormal frame. Since it's defined globally, I would like there to be a global definition, like there is with symplectic manifolds (vol = ωn). Is there one?

that any two bases have same cardinality is not equivalent to choice; it's strictly weaker.

tangent vectors are first order differential operators. the product of two of those is a second order differential operator.

Of course the distinction is clear. Every (fd) Lie algebra is isomorphic to a matrix algebra, which has a natural product. But that doesn't mean it is a matrix algebra, in exactly the same sense that not every group is a fundamental group. I mention now, for the third time, an example of a...

It seems you consider the universal enveloping algebra "contrived", but as it is a universal functor from the category of Lie algebras, it is natural in the usual sense of the word. But every Lie algebra has a multiplication, and the only way you'll find one without multiplication is if you...

And of course the same argument works with the space of derivations of any algebra. The commutator is again a derivation, but the product (which is defined by composition of maps) is not. Or what about the tangent space to any Lie group? If the Lie group is not a matrix group, there is no...

Jason: Yes, the method you suggest can be employed for calculating integral transforms (convolving is harder than multiplying). f and f^-1 may be hard, but there are tables for them. Note that these contexts are not purely group-theoretic, but the idea is the same. Matt grime: what's your...

Can you say what an exact category is? I'm not familiar with that term. I think neither kernels nor cokernels exist. For vector bundles anyway, I think the point is that the pointwise kernel and cokernel may not have constant dimension, and therefore they do not comprise a bundle. I read...

I'm pretty sure that det(1)=1

if adj means the adjugate, that is the transpose of the matrix of minors (which I guess some people also call the adjoint, but I save that for the Hermitian adjoint), then use the fact that A^-1=adj(A)/det(A)