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.

Filter by
Sorted by
Tagged with
4
votes
0answers
602 views

Koszul sign convention and symmetric group action on the graded n-th tensor product

Let $V_\bullet = (V_k)_{k \in \mathbb{Z}}$ and $W_\bullet = (W_k)_{k \in \mathbb{Z}}$ be two graded vector spaces on 0 caracteristic field. We define the tensor product of $V_\bullet$ by $W_\bullet$ ...
3
votes
2answers
730 views

How to prove this Gram determinant

Let $E$ be an Euclidian oriented vector space of dimension $3$ and $x,y,u,w \in E$. How do we prove (without coodinates) $$ \det \begin{pmatrix} \langle x,u \rangle & \langle x,w \rangle \\ ...
3
votes
1answer
263 views

Prove the exterior derivative of the following (n-1) form is zero

Let $\omega(x)=\frac{1}{{\parallel x \parallel}^n}\displaystyle\sum_{i=1}^{n}(-1)^{i-1}x_{i} dx_{1} \wedge \dots \wedge \widehat{dx_{i}} \wedge \dots \wedge dx_{n}$ be a differential $(n-1)$ form on $\...
3
votes
2answers
1k views

Interior product and exterior product

I have seen around the internet that this should hold: $$\iota_X(\alpha\wedge\beta)=\iota_X\alpha\wedge\beta+(-1)^k\alpha\wedge\iota_X\beta,$$ where $X$ is a vector field, $\alpha$ a $k$-form, $\...
3
votes
1answer
208 views

Understanding integration of k-forms

I try to wrap my head around the idea of integrating a k-form over a manifold. Loosely speaking, my intuition is, that for each point of the manifold, we do a projection onto the tangent space and ...
2
votes
1answer
189 views

Inner Products on Exterior Powers

Let $H$ is a real, $n$-dimensional vector space. Define $\varphi \colon \operatorname{GL}(H) \rightarrow \operatorname{GL}(\wedge^{k}H)$ by $A \mapsto \wedge^{k}A$ and $\psi_{\langle \cdot, \cdot \...
5
votes
2answers
139 views

Why $\bigwedge^{d-1}A=\bigwedge^{d-1}B \Rightarrow A= \pm B$

Let $V,W$ be $d$-dimensional vector spaces, and let $A,B \in \text{Hom}(V,W)$. Consider the induced maps on the exterior algebras: $\bigwedge^{d-1}A,\bigwedge^{d-1}B :\Lambda_{d-1}(V) \to \Lambda_{d-...
5
votes
1answer
285 views

Are projective modules over exterior algebras of vector spaces necessarily free?

Let $E(V)$ be the exterior algebra of a vector space $V$ (I've also seen this denoted $\Lambda(V)$).Is it true that any projective $E(V)$-module is necessarily free? If it's any easier, is it at least ...
5
votes
1answer
449 views

Poincare's lemma for 1-form

Let $\omega=f(x,y,z)dx+g(x,y,z)dy+h(x,y,z)dz$ be a differentiable 1-form in $\mathbb{R}^{3}$ such that $d\omega=0$. Define $\hat{f}:\mathbb{R}^{3}\to\mathbb{R}$ by $$\hat{f}(x,y,z)=\int_{0}^{1}{(f(...
5
votes
1answer
128 views

Eigenvalues of the second exterior power of a linear operator

Let $K$ be a field of characteristic zero and $V=K^n$. Let $T: V \to V$ be a linear map with eigenvalues $\lambda_1,...,\lambda_n \in K$ , not necessarily all distinct. Let $\wedge^2 V$ be the second ...
4
votes
3answers
718 views

Determinant of transpose intuitive proof

We are using Artin's Algebra book for our Linear Algebra course. In Artin, det(A^T) = det(A) is proved using elementary matrices and invertibility. All of us feel that there should be a 'deeper' or a ...
4
votes
1answer
71 views

Is a pointwise decomposable differential form smoothly decomposable?

Let $\omega$ be a smooth differential form on a smooth manifold $M$. Suppose $\omega$ is pointwise decomposable, that is for every $p \in M$, $\omega_p=e^1_p \wedge e^2_p \wedge \dots \wedge e^k_p$ ...
3
votes
1answer
179 views

Exterior product generates the infinitesimal rotations — what is the geometric significance?

Question: What is the geometric significance of the fact that the exterior products of the unit basis vectors in $\mathbb{R}^3$ generate a basis for the Lie algebra of the pure rotation group $SO(3)$? ...
3
votes
1answer
920 views

The exterior derivate and pullback commute

The above question is from a past exam. I am having trouble with the fine details, ie what $F*dw$ and $dF*w$ actually look like. Can anybody show me how this question is solved? I have solved it using ...
3
votes
1answer
45 views

Show that the wedge product $ dX \wedge dX = 0 $ and $dY \wedge dY = 0$

So first I want to give you some background information: begin of the background information I'm currently reading an abstact about the Lotka Volterra differential equations: $$ x^{'} = x -xy $$ $$ ...
3
votes
1answer
167 views

Does every connection admit a parallel volume form?

Let $E$ be a smooth orientable vector bundle of rank $k$ over a manifold $M$. Let $\nabla$ be a connection on $E$. Does there always exist a non-zero parallel section of $\Lambda_k(E)$? What about $E=...
2
votes
1answer
128 views

Some wedge product computation

I want to calculate $w\wedge w$ for $w=\sum a_{ij} e_i\wedge e_j$ where the sum is over all $1\leqslant i<j\leqslant 4$. Is there a neat way to do it without writing all the terms out? What about ...
2
votes
1answer
65 views

Image and preimage of a decomposable element under hodge star operator is decomposable.

Let $V$ be a $m$ dimensional vector space over a field $\mathbb{F}$ and $1\leq l<m.$ Then consider the wedge product $\bigwedge^lV$ and $\bigwedge^{m-l}V.$ Fix a basis $\{e_1,\ldots,e_m\}$ of $V$ ...
2
votes
1answer
382 views

Computation of the hom-set of a comodule over a coalgebra: $Ext_{E(x)}(k, E(x)) = P(y)$.

First of all, since every other book somehow mentions that this is trivial, I apologize if it turns out that I am just misunderstanding something in the definitions. So here goes: The motivation for ...
2
votes
3answers
73 views

Economically computing $d\beta$

$\displaystyle \beta = z\frac{x dy \wedge dz + y dz \wedge dx + z dx \wedge dy}{(x^2+y^2+z^2)^{2}}$ Show that $d\beta=0$. So, let $r=x^2+y^2+z^2$, $\begin{align} \displaystyle d\beta &= d(z\...
2
votes
3answers
491 views

index free proof of dot product of two wedge products

I am learning geometric algebra, and meet an identy of (edited according to Andrey's comments below) $$ (a\wedge b)\cdot(c\wedge d) = (a \cdot d)(b\cdot c) - (a \cdot c)(b \cdot d)$$ as in wiki ...
2
votes
2answers
145 views
+100

Equivalence of wedge and tensor product with Levi-Civita symbol

In this answer the following is stated: \begin{eqnarray} v\land w & = & \frac{1}{2!}(v\land w-w\land v) \\ & = & \frac{1}{2!}\epsilon_{\mu\nu}v^{\mu}\land w^{\nu} \\ & = & \...
1
vote
1answer
134 views

Uniqueness in structure theorem for f.g. module over Dedekind domain

I've been trying to wrap my head around the proof that if $M$ is a finitely generated torsion-free $R$-module over a Dedekind domain $R$, then $M\cong R^n\oplus I$, where $I$ is an ideal of $R$. I'm ...
6
votes
1answer
180 views

Why is the exterior power $\bigwedge^kV$ an irreducible representation of $GL(V)$?

$\newcommand{\GL}{\operatorname{GL}}$ Let $V$ be a real $n$-dimensional vector space. For $1<k<n$ we have a natural representation of $\GL(V)$ via the $k$ exterior power: $\rho:\GL(V) \to \GL(\...
5
votes
1answer
712 views

Trace of the $k$-th Exterior Power of a Linear Operator

Let $V$ be an $n$ dimensional vector space over a field $F$ and $T$ be a linear operator over $V$. Assume that the characteristic of $F$ is not $2$. Definition. Consider the map $f_1:V^n\to \Lambda^...
4
votes
1answer
108 views

Use abstract index notation to prove Leibniz rule for exterior derivative

I want to use abstract index notation to prove Leibniz rule for exterior derivative of wedge product: For $\omega\in \Omega^k(U),\eta\in\Omega^l(U)$, d$(\omega\wedge\eta)=\text{d}\omega\wedge\eta +(...
3
votes
1answer
62 views

Exterior Algebra: Find a linear operator

I have the following problem: Let $J: \wedge^{2} \mathbb{R}^{3} \to \wedge^{2} \mathbb{R}^{3}$ isomorphism linear. Find $A: \mathbb{R}^{3} \to \mathbb{R}^{3}$ linear operator such that $\wedge^{2} A =...
3
votes
1answer
372 views

Transformation rule for a wedge product

Suppose two sets of covectors on a vector space $V, \beta^1,...\beta^k $ and $\gamma^1,...,\gamma^k,$ are related by $$\beta^i=\sum_{j=1}^k a^i_j \gamma ^j$$ where $i=1,...,k$, for a $k\times k$ ...
3
votes
1answer
33 views

Does the kernel of every alternating form contain a decomposable element?

Let $V$ be a real $n$-dimensional vector space, and let $1 < k < n$. Let $\alpha \in \bigwedge^k (V^*) \cong (\bigwedge^k V)^*$. Thinking of $\alpha$ as a linear functional $\bigwedge^k V \to \...
3
votes
0answers
903 views

Inner product exterior algebra

I have to prove that if $V$ is a real vector space ($\dim V=n$) with inner product $(.,.)$ then if we define $$ (v_{1}\wedge v_{2}\wedge\cdots\wedge v_{k},w_{1}\wedge w_{2}\wedge\cdots\wedge w_{k}) =...
3
votes
1answer
225 views

What is a more formal name for the wedgey group?

The rule $(v_1,w_1)⋅(v_2,w_2)=(v_1+v_2,w_1+w_2+(v_1∧v_2))$ defines a group structure on the vector space $V⊕(V∧V)$ whenever $V$ is itself a vector space over some field $F$. What is a more common ...
2
votes
2answers
60 views

Sobolev approximation lifts to $L^p$ convergence of the exterior powers

I am reading the book "Geometric Function Theory and Non-linear Analysis", where the following claim is used: Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set. Let $f \in W^{1,s}(\Omega,\...
2
votes
1answer
31 views

The difference between exterior maps is bounded by the norm of the difference?

Let $V,W$ be $d$-dimensional real inner product spaces, and let $A,B:V \to W$ be linear maps. Let $\bigwedge^k A,\bigwedge^k B:\Lambda_k(V) \to \Lambda_k(W)$ be the induced maps on exterior powers. ...
2
votes
2answers
206 views

Derivate of the cofactor and the determinant

$\newcommand{\Cof}{\operatorname{Cof}}$ $\newcommand{\Det}{\operatorname{Det}}$ $\newcommand{\id}{\operatorname{Id}}$ $\newcommand{\tr}{\operatorname{Tr}}$ Let $A(t)$ be a smooth path in $M_d(\mathbb{...
2
votes
0answers
69 views

Is there a natural way to view $\bigwedge^k_{\mathbb{C}}\mathbb{C}^d$ as a subspace of $\bigwedge^k_{\mathbb{R}}\mathbb{C}^d$

Does there exists an $\mathbb R$-linear embedding $\bigwedge^k_{\mathbb{C}}\mathbb{C}^d \to \bigwedge^k_{\mathbb{R}}\mathbb{C}^d$ which maps decomposable tensors to decomposable tensors? (The ...
2
votes
1answer
34 views

Connection between ranks of an endomorphism and its linear image on the exterior power

Let $V$ be an $n$-dimensional real vector space, and let $1<k<n$. Let $\psi:\text{End}(V) \to \text{End}(\bigwedge^kV)$ be the exterior power map, $\psi(A)=\bigwedge^k A$. For $B \in \text{End}...
2
votes
1answer
49 views

Exterior Algebra: General Case, find a linear operator

Is true that for all $J:\wedge^{k} \mathbb{R}^{n} \to\wedge^{k} \mathbb{R}^{n}$ isomorphism linear there exists $A:\mathbb{R}^{n} \to \mathbb{R}^{n}$ linear operator such that $\wedge^{k} A =J$? When ...
2
votes
0answers
86 views

Determining the isomorphism classes of these symmetry groups (exterior algebra)

Let $V$ be a real $d$-dimensional vector space. Let $\omega \in \bigwedge^kV$ be a fixed non-zero multivector for some $1 < k < d$. Define $ G_{\omega}=\{ T \in \text{Aut}(V) \, | \, (\bigwedge^...
2
votes
1answer
77 views

If $\omega \in \bigwedge^k\mathbb{R}^d$ is decomposable over $\mathbb C$, is it decomposable over $\mathbb R$?

Let $k,d$ be positive integers, and let $\omega \in \bigwedge^k\mathbb{R}^d$ be decomposable in $ \bigwedge^k\mathbb{C}^d$. Is $\omega$ decomposable in $\bigwedge^k\mathbb{R}^d$? Edit: Let me be ...
2
votes
2answers
215 views

Identification between wedge product and its dual

Let $\mathbb{F}$ be a field, and let $(e_i)$ be the usual elementary basis of $\mathbb{F}^n$. Let $\varphi_{ij}: \mathbb{F}^n \wedge \mathbb{F}^n \to \mathbb{F}$ be such that $v \wedge w \mapsto ...
1
vote
2answers
81 views

Confusion about a particular differential form being exact?

Consider the differential form: $$\omega=-\frac y{x^2 + y^2}dx + \frac x{x^2 + y^2}dy$$ I believe that this form is not exact. i.e: There does not exist a $\psi$ such that $d\psi=\omega$. In ...
1
vote
1answer
100 views

Questions on Cartan's magic formula $\mathcal{L}_X=i_X \circ d + d\circ i_X$

Algebra $A$ is called graded algebra if it has a direct sum decomposition $A=\bigoplus_{k\in\Bbb Z} A^k$ s.t. product satisfies $(A^k)(A^l)\subseteq(A^{k+l}) \text{ for each } k, l.$ A ...
1
vote
2answers
93 views

If $\phi_i$s are linearly dependent, $\det [\phi_i(v_j)] = 0$ - is the proof legit?

Given $v_1, \ldots, v_k \in V$ and $\phi_1, \ldots, \phi_k \in V^*$. If $\phi_1, \ldots, \phi_k \in V^*$ are linearly dependent, prove that $\det[\phi_i(v_j)] = 0.$ Here $k$ is the dimension of $V$, ...
1
vote
2answers
255 views

Compute the $n$th exterior power of a differential $2$-form

Suppose you have a differential form $\omega$ written in local coordinates as $$\omega=\sum_{i=1}^ndx_i\wedge dy_i.$$ Can anyone help me showing the following equality: $$\omega^n=n!(dx_1\wedge dy_1\...
1
vote
2answers
2k views

Area of projection of cube in $\mathbb{Z}^3$ onto a hyperplane

A cube with vertices $(\pm 1, \pm 1, \pm 1)$ gets projected into the plane perpendicular to vector $\mathbf{n}\in S^2$. The projection is a hexagon, how do I find the area? I think I can just ...
1
vote
0answers
183 views

Does the Hodge dual (star) operator make the exterior algebra an involutive (*-) algebra?

Does the Hodge dual (star) operator make the exterior algebra an involutive (*-) algebra? https://en.m.wikipedia.org/wiki/Hodge_dual https://en.m.wikipedia.org/wiki/*-algebra This would seem to be a ...
1
vote
1answer
71 views

Can we always span a decomposable form via constant coefficients?

Let $M$ be a smooth manifold of dimension $d$. Let $1 < k <d$ be an integer. Let $\omega^i$ be a local frame of the exterior power bundle $\Lambda_{k}(T^*M)$. Does there exists numbers $a_i ...
1
vote
1answer
28 views

Is “being decomposable” preserved under taking a subspace?

Let $V$ be a vector space over some field, and $W \le V$ a vector subspace. Let $1<k<\dim V$ be an integer. Suppose $\omega \in \bigwedge^k W$ is decomposable as an element in $\bigwedge^k V$....
1
vote
1answer
106 views

Exercise about wedge product and multilinear forms

I'm considering $\omega\in \Lambda^{2q+1}(V^\ast)$, i.e. a multilinear skew-symmetric form. I want to prove that $\omega\wedge\omega=0$. How shall I proceed? Any suggestions? Do I have to write $\...
1
vote
1answer
32 views

Basic question about coefficients in external algebra

The exterior algebra formed from a vector space $V$ with vector basis $\{e_1,e_2,e_3\}$ will have basis $$\begin{align} &\Delta^0 V=\langle 1 \rangle\\ &\Delta^1 V=\langle e_1,e_2,e_3 \rangle\...