Abstract Nonsense

Crushing one theorem at a time

Some Natural Identifications (Pt. IV)


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August 14, 2012 Posted by | Algebra, Linear Algebra | , , , , , | Leave a comment

Some Natural Identifications (Pt. III)


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August 14, 2012 Posted by | Algebra, Linear Algebra | , , , , , , , | 1 Comment

Some Natural Identifications (Pt. II)


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August 14, 2012 Posted by | Algebra, Linear Algebra | , , , , , , , | 1 Comment

Some Natural Identifications


Point of Post: In this post we take our discussion last time concerning tensor algebras and their quotients to the case of vector spaces, and show that there are simple identifications that can be made. We also discuss the notion of orientations and volume forms.

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Motivation

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We now are able to take the very formal, very general way of defining tensor powers/exterior powers and show how they relate to much tamer objects–multilinear and alternating forms. The main reason for doing this is, as mentioned before, to help explain the somewhat confusing notation used in some differential geometry textbooks. For example, they merely denote the space of alternating k-forms on V as \Lambda^k(V^\ast). But, why? This notation makes no sense, especially considering that this notation has other meaning in mathematics (the k^{\text{th}} exterior power of the dual space of V). They should really only conflate these two if there is some sort of natural isomorphism between these two. Well, to state the obvious, they are naturally isomorphic. But, this is never stated, let alone proven, in differential geometry textbooks. This is the point of this post.

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August 14, 2012 Posted by | Algebra, Linear Algebra | , , , , , | 1 Comment

Tensor Algebra and Exterior Algebra (Pt. VII)


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May 10, 2012 Posted by | Algebra, Module Theory, Ring Theory | , , , , | 4 Comments

Tensor Algebra and Exterior Product (Pt. VI)


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May 10, 2012 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , | 1 Comment

The Tensor Algebra and Exterior Algebra (Pt. V)


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May 10, 2012 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , , , | 1 Comment

The Tensor Algebra and Exterior Algebra (Pt. IV)


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May 10, 2012 Posted by | Algebra, Module Theory, Ring Theory | , , , , , | 1 Comment

The Tensor Algebra and Exterior Algebra (Pt. III)


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May 10, 2012 Posted by | Algebra, Module Theory, Ring Theory | , , , , , | 1 Comment

The Tensor Algebra and the Exterior Algebra (Pt. II)


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May 10, 2012 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , | 1 Comment