Questions tagged [exterior-algebra]

It is a quotient - of the tensor algebra, obtained by taking graded sum over whole numbers $n$ of $n$-fold tensor products - by the ideal generated by elements of the form $a\otimes a$. We write the residue class of $a\otimes b$ in this algebra, as $a\wedge b$ and call it the wedge product.

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Trace of an exterior power of a linear map

In my lecture notes for my linear algebra course we got this question - Let $\phi : \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be a linear mapping with eigenvalues 2,1 and -1. What is the trace of the ...
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In what sense $\bigwedge(V\oplus W) \simeq \bigwedge V \otimes \bigwedge W$?

According to the Wikipedia article on the exterior algebra, there is a natural isomorphism $$\bigwedge(V\oplus W) \simeq \bigwedge V \otimes \bigwedge W,$$ where $V,W$ is a finite-dimensional $\Bbbk$-...
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Show that $\big\{A(v_{i_1}\otimes \cdots\otimes v_{i_k})\in V^{\wedge k}:i_1<\cdots<i_k \big\}$ is linearly independent

Let $V$ be vector space over $\mathbb{R}$ and $V^{\otimes k}$ be the $k$th tensor power of $V$. Denote by $S_k$ the set which contains exactly all the permutations of a set with $k$ elements. Using ...
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Different definitions on the tensor and wedge products

I'm studying differential forms and some linear algebra doubts popped up: Given $V$ a vector space, we define $A_k(V)$ as the space of alternating $k$-linear maps $V^k \to \mathbb{R}$. The tensor ...
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Prove that $\theta: M^m \rightarrow T^mM$ such that$(x_1,...,x_m) \rightarrow \frac{1}{m!} \sum_{σ∈Sm}ε(σ)x_{σ(1)} ⊗ · · · ⊗ x_{σ(m)}$ is alternating

How to prove that $\theta: M^m \rightarrow T^mM$ defined as $(x_1,...,x_m) \rightarrow \frac{1}{m!} \sum_{σ∈Sm}ε(σ)x_{σ(1)} ⊗ · · · ⊗ x_{σ(m)}$ is alternating ? Alternating (for a map) means that as ...
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substituting values in a wedge expression $\omega=zdx \wedge dy \in A^2(R^3)$ given function g

if we have $g(\phi,\theta)=\begin{bmatrix}r\cos\theta\\r\sin\theta\\\sqrt{1-r^2} \end{bmatrix}$ and we have $\omega=zdx \wedge dy \in A^2(R^3)$ i want to find the expression for $g^*\omega$ I know ...
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Computing the wedge product of some differentials

I am trying to compute the wedge product of \begin{align*} &2x_p \mathrm{d}x_p \bigwedge_{i=1,\cdots,p-1}2(x_i \mathrm{d}x_i +y_i\mathrm{d}y_i)\bigwedge_{i=1,\cdots,p-1}x_{i+1}\mathrm{d}y_i+y_i\...
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How do you show that $e_{I}^{*}(e_{J}):=\delta_{I,J}$ is a basis for $(\Lambda_{k}V)^{*}$, $\Lambda_{k}V:= V^{\otimes k}/A$?

In Tu's book "Geometry" there is the following statement on page 171: I would like to show that $span\{e_{I}^{*} \}=(\Lambda_{k}V)^{*}$. For some reason I get completely confused by all the ...
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Where does this permutation product live?

This is from Loring Tu's An introduction to manifolds. The coefficient $1/(k!ℓ!)$ in the definition of the wedge product compensates for repetitions in the sum: for every permutation $σ ∈ S_{k+ℓ}$ , ...
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Decomposition of Hodge Operator

Given a decomposition of a vector space $V \simeq U \oplus W$. Then as taking the exterior algebra preserves coproducts (it is left adjoint to the forgetful functor from graded-commutative graded ...
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One form wedge Product with $n - 1$ form.

Let $a$ be an arbitrary one-form on $\mathbb{R}^{n}$. Then prove that there exists a $n - 1$ form $b$ on $\mathbb{R}^{n} - \{\mathbf{0}\}$ such that $$a \wedge b = dx^1 \wedge \cdots \wedge dx^n.$$ ...
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The function $1/z$ is locally integrable with resepct to $dz\wedge d\bar{z}$

I was starting to read Hörmander's Introduction to Several Complex Variables, and in it he says that $1/z$ is integrable over compact sets. The form he is integrating against is $dz\wedge d\bar{z}$. I ...
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Proving that $\dim(\bigwedge^k(V^*)) = \binom{n}{k}$ without constructing an explicit basis

Text: Discussion: I find this argument kind of hard to follow because an explicit basis is never constructed; the argument seems kind of indirect. I'm relatively comfortable with the first sentence ...
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Show that $v_1\wedge\dots\wedge v_k = x_1\wedge\dots\wedge x_k \implies \text{span}\{v_1,\dots, v_k\} = \text{span}\{x_1,\dots, x_k\}$

Let $V$ be an $n$-dimensional space and $v_1,\dots, v_k \in V$ are linearly independent. It is clear that if $x_1,\dots, x_k \in V$ have the same span as $v_1\dots v_k \in V$ then there is a scalar $t$...
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Koszul homology of $k[x_1,...,x_n]/\mathfrak{m}^d$

In the book Homology of Local Rings of Gulliksen and Levin, it is mentioned that if we set $R=k[x_1,...,x_n]/\mathfrak{m}^d$ where $\mathfrak{m}$ is the homogeneous maximal ideal and consider the ...
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Is this a valid tensor representation of the Hodge star operation?

Since the Hodge star operation is a linear operation $*:\Lambda^kV \to \Lambda^{n-k} V$, then I thought that one could represent this linear map by tensor as one can can do for a linear map $A:V\to W$ ...
how do we show $\wedge^2 I \neq 0$ but $\wedge^2 A=0$? [duplicate]
The following question is from Appendix C of Matsumura's Commutative ring Theory, page $283$. Given an Ideal $I \subset A$ of a ring $A$, the author claims that it might happens that the exterior ...