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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Find the kernel of a strange map of quotients of vector space

Let $V$ be a vector space of dimension $n$ and let $W \subset V$ be a $k$-dimensional subspace. Consider the map $$ \varphi: (V/W) \otimes V \to \wedge^2(V/W) $$ which sends for $a,b \in V$ $$ (a+W) \...
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Explicit isomorphism $\Lambda^2\Lambda^2V\oplus\Lambda^4V\cong V\otimes\Lambda^3V$

I remember some years ago reading about "lambda rings" learning there are some relations between exterior powers and their combinations. The smallest example of such a relation was $\Lambda^...
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Equivalence of definition of wedge product on $\mathrm{End}(E)$.

Consider a smooth vector bundle $E$. I would like to define a wedge-product of the form $$\wedge:\Omega^{k}(\mathcal{M},\mathrm{End}(E))\times\Omega^{l}(\mathcal{M},E)\to\Omega^{k+l}(\mathcal{M},E)$$ ...
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Hodge star and non-orthonormal basis

I am working with the Hodge star and a non-orthonormal basis, and I can't see where exactly something is going wrong. Let $V$ be an oriented $n$-dimensional vector space, and let $\Lambda V$ be the ...
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Every $(n-1)$-form is decomposable

Let $\dim(V) = n \geq 2$. Show that every alternating multilinear $n-1$ form is decomposable, that is it can be written as $\omega = \alpha \wedge \beta$ with $\alpha \in \Lambda^1V^*$, $\beta \in \...
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What is meant by the wedge product?

I have encountered two definitions of the wedge product. The first is the exterior product as seen on the wikipedia page for "exterior algebra". It is a map $\wedge:\Lambda(V) \times \Lambda(...
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Intuitive understanding of oriented volume and trivectors

I get that the way a vector's arrowhead points corresponds to its orientation for a given direction (line). We can also understand vectors within $V$ as isomorphic to a set of endomorphic translations....
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Embedding from exterior product to tensor product space

I came across the following: For a basis $\phi_1,\dots,\phi_n$ of $V$ there is a natural embedding $V^{\wedge n}\hookrightarrow V^{\otimes n}$ defined as $$(\phi_1\wedge \cdots \wedge \phi_n) \...
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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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Induced $\frak{g}$-action on exterior power and symmetric power?

What to show. Let $\frak{g}$ be a Lie algebra and $V$ a $\frak{g}$-representation. I am supposed to show that for $r \geq 0$ there exists a unique action of $\frak{g}$ on the exterior power $\bigwedge ...
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$\Lambda^k(A)$ is a trace class operator

Let $A$ be a trace class operator. I am trying to understand the proof of ($\|\cdot\|_1$ means the trace class norm) $$\|\Lambda^k(A)\|_1\leq \frac{\|A\|_1^k}{k!}$$ I have a proof but I think there ...
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Determinant and symmetric function of eigenvalues

Let $H$ be an $n$-dimensional vector space and $T$ a linear operator on it with eigenvalues $\lambda_{i_1},\dots,\lambda_{i_n}$. Let $I$ be the identity operator and $z\in\mathbb{C}$. How does one ...
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(Non) Associativity of Exterior Product on Modules of $\text{dim} \geq 2$

Associativity probably isn't the right word here, but close enough. Suppose $M$ is a vector space over a field. Suppose $2 \leq \text{dim}(M) \leq \infty$. Suppose $N = \bigwedge^2 M$. Then I want to ...
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$M$ is a vector space, and $N = \bigwedge^2 M$ . If $2 \leq \dim(M) < \infty$ then is $\bigwedge^2 N \not \cong \bigwedge^4 M$

Suppose $F$ is some base field and $M$ is a $F$-vector space. Then let $N = \bigwedge^2 M$. Now suppose that $2 \leq \dim(M) < \infty$- then I want to see that $\bigwedge^2 N \not \cong \bigwedge^4 ...
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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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Antiderivation which preserves exact forms

It's easily shown that the wedge product descends to cohomology. So for a smooth manifold $M$, given a closed 1-form $\alpha \in \Omega^1(M)$, one can define the derivation of degree 1: $$ \Gamma : \...
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Anti commutativity of multicovectors

Let $V$ be a finite dimensional vector space. Let $ \omega \in \Lambda^k(V^*) $ and $\eta \in \Lambda^l(V^*)$. Then $\omega \wedge \eta = (-1)^{kl} \eta \wedge \omega$ I am trying to understand the ...
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Prove vector area formula by stokes's theorem

wiki says the following two formulas are equivalent. ${\mathbf {S}}=\int d{\mathbf {S}}$ ${\displaystyle \mathbf {S} ={\frac {1}{2}}\oint _{\partial S}{\vec {r}}\times d{\vec {r}}}$ I am learning ...
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How do we denote a surface embedded in four dimensions?

In three dimensions we can denote a surface in the $xy$-plane with area $1$ as $\vec{S} = 1\hat{z}$. This has many useful applications since we can now take a dot product between a vector field and a ...
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Proving $\bigwedge^k(U\cap V) = \left(\bigwedge^kU\right) \bigcap \left(\bigwedge^k V\right)$ via the universal mapping property

Let $V$ be a (finite dimensional) vector space, its exterior algebra of order $k$ is the vector space $\bigwedge^k V$ consisting of the formal sums of terms of the form $v_1 \wedge v_2 \wedge \dots \...
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Annihilator for $\mu\in\Lambda^3(V)$

Let $V$ a 4-dimensional real vector space and let $\mu\in \Lambda^3(V)$. I have to characterize the elements in the annihilator of $\mu$, i.e., $\text{Ann }\mu=\left\lbrace u\in V \vert u\wedge \mu=0 ...
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Computing hyper-volumes with wedge/exterior products

This is is just a simple question to make sure that I don't hold any misconceptions regarding the relationship between the wedge product and hyper-volumes. In particular, is it correct that if given $...
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Prove that $\det R = \pm 1$ iff $R \in \operatorname{Isom}(V)$, where $V$ is a finite dimensional real inner product space [duplicate]

Let $V$ be an $n$-dimensional real inner product space. An isometry on $V$ is an operator $R$ with $\langle Rx,Ry \rangle$ for all $x,y \in V$. The determinant of an Endomorphism can be defined i many ...
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$p^{n+1}+p^{n}-p$ = count 2x2 $F_p$-matrices satisfying Grassman algebra relations - true? (And the same for symmetric 3x3 Grassmans)

It seems number of 2x2 matrices over finite field $F_p$ satisfying Grassman algebra relations ( $\psi_i ^2=0, \psi_i \psi_j + \psi_j \psi_i = 0 , i=1...n$) is equal to $p^{n+1}+p^{n}-p$. (Guessed by ...
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How is the exterior product of multilinear forms defined with exterior algebra

Similar topic but different concern with this In the Exterior_algebra of Wikipedia, there is: Under this identification, the exterior product takes a concrete form: it produces a new anti-symmetric ...
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Is the (un-)even part of the exterior algebra an (anti-)commutative sub-algebra?

It is well known that the exterior algebra of an $n$-dimensional vector space $V$ has the following decomposition: $$\Lambda V=\bigoplus_{k=0}^n\Lambda^kV=\bigoplus_{\text{even }k}\Lambda^kV\oplus \...
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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 $A:V\to W$ ...
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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 ...
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calculating the wedge product for 2 4D vectors

I can't understand whether the wedge product is supposed to generate a matrix, a vector, or even an area. it seems like every source I read says something different. In one example, it seems that $v \...
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Confusion in the order of the differentials in integral using differential forms

My doubt is very general but I'm going to give an example so I can explain why I'm struggling with this. I have a 2-form $\alpha$ written in Cartesian coordinates $$ \alpha=\alpha_{ij}\,dx^i\wedge dx^...
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Indices of the suspension of a complex

Let $R$ be a commutative ring and $x$ is an $R$-regular element. It's a fact that the Koszul complex in this case is self-dual, i.e. $Hom_R(K^R(x),R)=K^R(x)[-1]$. This is true for a regular sequence ...
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Clarification on notation regarding fields, forms, and exterior algebra

Sorry if I've missed something quite obvious, but I can't seem to find a clear source for notation. $\Omega^p(T^*M)$ is the common notation I've seen for the space of $p$-forms on the cotangent ...
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Exercise 3.15 from Introduction to Manifolds by Tu

I am wondering about exercise 3.15 on page 25. If $f$ is a 3-linear function on a vector space $V$ and $v_1,v_2,v_3 \in V$, what is $(Af)(v_1,v_2,v_3)$? I have by far that $$ (Af)(v_1,v_2,v_3) = -f(...
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32 views

Alternating forms and isometries of the underlying vector space

Does a linear isometry of $\mathbb{R}^n$ yields a linear isometry of $\mathbb{R}^n \wedge \mathbb{R}^n$ ? If yes, why ? How to prove it elegantly, without explicit coordinate computations ? Using, for ...
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Why is it useful to have only $2$ orientations for 3D oriented volumes and not $3!$? (And why not $n!$ for $n$D?)

Say we have three (non-coplanar) vectors $\vec{a}, \vec{b}, \vec{c} \in \mathbb{R}^3$. Geometrically they can be used to define $3$ distinct edges of a unique non-oriented 3D parallelipiped. We define ...
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$\mathbf{x}\wedge \mathbf{w} =0$ iff $\mathbf{w}=\mathbf{x}\wedge \mathbf{w}' $

It' pretty much as stated in the title. Looking for a proof of the statement that $\mathbf{x}\wedge \mathbf{w} =0$ iff $\mathbf{w}=\mathbf{x}\wedge \mathbf{w}'$ where $\mathbf{x}, \mathbf{w}' \in V$ ...
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How to show that the integral of this volume-form over $S^n$ isn't 0?

Let $S^n$ be the n-dimensional unit sphere in $\mathbb{R}^{n+1}$. Let $\omega$ be the volume form that's defined for all $p\in M$ by $\omega_{\vec{p}} = x^1\wedge...\wedge x^n$ s.t. $x^1,...,x^n$ is ...

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