Is Hom(Lambda^k(V),Lambda^(k+1)(V)) and element of End(Lambda(V))?

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In summary: The k-th exterior power of V is an abstract construction, and its definition does not involve any basis. So, the embedding of V into Hom(Lambda^k(V), Lambda^(k+1)(V)) is indeed canonical, in the sense that it does not depend on any choice of basis.
  • #1
precondition
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Sorry I don't know how to write in symbols so I'm using Latex codes, anyway,
the question is as in title:

Is an element of Hom(Lambda^k(V),Lambda^(k+1)(V)) and element of End(Lambda(V))?

In words, is an element of the homomorphism between k th grade exterior algebra over V and k+1 th exterior algebra over V also an element of endomorphism of exterior algebra over vector space V?

I appreciate your help
 
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  • #2
What is the domain of an element of

Hom(Lambda^k(V),Lambda^(k+1)(V))?

What is the domain of an element of

End(Lambda(V))?

Can a function have two different domains?
 
  • #3
I'm afraid I'm going to disagree. Let n be the dimension of V. Then canonically

[tex] \Hom(\Lambda(V),\Lambda(V))= \oplus\Hom(\Lambda^r(V),\Lambda^s(V)[/tex]

0<=r,s<=n.

Thus any element of the vector space Hom(\Lambda^k(V),\Lambda^{k+1}(V)) is canonically an element of End(\Lambda(V)).

The question boils down to: if x is in X, is x also an element of X\oplus Y, and it is. I think the function notion is very misleading at times.
 
  • #4
I agree with matt grime

Yes I agree with what matt grime said,
When I carefully read the text again it said,
"Let V-->Hom(Lambda^kV, Lambda^(k+1)V) be the action of V on LambdaV"
and since action of V on LambdaV is precisely the homomorphism between V and End(LambdaV) I can see that an element of Hom(Lambda^kV, Lambda^(k+1)V) is precisely an element of End(Lambda V)

Thank you for your help anyways
 
  • #5
matt grime said:
Thus any element of the vector space Hom(\Lambda^k(V),\Lambda^{k+1}(V)) is canonically an element of End(\Lambda(V)).
But being canonically an element isn't the same as actually being an element... which is what I had asumed the OP meant.

And there's a problem with this argument: it would also prove, for example, that every 1x1 real matrix is canonically an element of any group of mxn real matrices.

I had also assumed End(\Lambda V) was the ring of algebra endomorphisms, not vector space endomorphisms. The canonical embedding you describe doesn't live in the subset of algebra endomorphisms.

precondition said:
Yes I agree with what matt grime said,
When I carefully read the text again it said,
"Let V-->Hom(Lambda^kV, Lambda^(k+1)V) be the action of V on LambdaV"
and since action of V on LambdaV is precisely the homomorphism between V and End(LambdaV) I can see that an element of Hom(Lambda^kV, Lambda^(k+1)V) is precisely an element of End(Lambda V)

Thank you for your help anyways
I'm pretty sure that the text doesn't mean

For any particular k, V-->Hom(Lambda^kV, Lambda^(k+1)V) is the action of V on Lambda V​

but instead means

The collection of all V-->Hom(Lambda^kV, Lambda^(k+1)V) constitutes the action of V on Lambda V​

which makes more sense, because to have an endomorphism of V, you have to know what happens to every Lambda^j V. (The canonical embedding matt describes for a particular k assumes the zero map on every j unequal to k)
 
  • #6
Hurkyl said:
But being canonically an element isn't the same as actually being an element... which is what I had asumed the OP meant.

And there's a problem with this argument: it would also prove, for example, that every 1x1 real matrix is canonically an element of any group of mxn real matrices.

It doesn't prove that at all. There is nothing canonical about the description you just gave. Anything that invokes matrices is by definition not canonical. Although I agree it is a wooly question: if 1 is a real number is 1 an element of C?
 
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  • #7
matt grime said:
It doesn't prove that at all. There is nothing canonical about the description you just gave. Anything that invokes matrices is by definition not canonical. Although I agree it is a wooly question: if 1 is a real number is 1 an element of C?
Canonically,

[tex]
(\mathbb{R}^m, \mathbb{R}^n)
\cong \bigoplus_{\substack{1 \leq i \leq m \\ 1 \leq j \leq n}} ( \mathbb{R}, \mathbb{R} )
[/tex]

I guess my real point, though, is that it's an abuse of language to say "An element of X is an element of [itex]X \oplus Y[/itex]", since what we really mean is "we have an embedding [itex]X \mapsto X \oplus Y[/itex] which we will invoke implicitly as needed". Using an example with a repeated summand was just an indirect way to get at this.
 
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  • #8
I never have liked canonical. How are you picking out a summand, canonically? What if I change bases? I then change that decomposition. If you start writing down copies of R like that it stops being canonical in any meaningful sense. The fact is that a 2 dimensional real vector space is not canonically isomorphic to R\oplus R.The decomposition of Lambda(V) does not depend on any choice of basis in V.
 
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1. What is Hom(Lambda^k(V),Lambda^(k+1)(V))?

Hom(Lambda^k(V),Lambda^(k+1)(V)) is the set of linear transformations from the vector space Lambda^k(V) to the vector space Lambda^(k+1)(V). This set is also known as the space of k-th order tensors on the vector space V.

2. What does it mean for an element to be in End(Lambda(V))?

An element of End(Lambda(V)) is a linear transformation from the vector space Lambda(V) to itself. This is also known as an endomorphism of the vector space Lambda(V).

3. Why is it important to study Hom(Lambda^k(V),Lambda^(k+1)(V))?

Understanding the space Hom(Lambda^k(V),Lambda^(k+1)(V)) is crucial in the study of multilinear algebra and differential geometry. It allows us to define and manipulate higher order tensors, which are essential in many mathematical and scientific fields.

4. What is the relationship between Hom(Lambda^k(V),Lambda^(k+1)(V)) and End(Lambda(V))?

Hom(Lambda^k(V),Lambda^(k+1)(V)) is a subset of End(Lambda(V)), as it only includes linear transformations between specific vector spaces. However, not all elements of End(Lambda(V)) are in Hom(Lambda^k(V),Lambda^(k+1)(V)), as the latter set has specific constraints on the dimensions and order of tensors.

5. Can you give an example of an element in Hom(Lambda^k(V),Lambda^(k+1)(V))?

One example of an element in Hom(Lambda^2(R^3),Lambda^3(R^3)) is the cross product operator. This linear transformation takes two vectors in R^3 and returns a third vector that is orthogonal to the original two. It can be represented as a 3rd order tensor, making it an element of Hom(Lambda^2(R^3),Lambda^3(R^3)).

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