# 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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### Exterior Derivative over Quaternions

I was wondering if it is possible to define the exterior derivative of a quaternionic valued function. I am doing the quaternionic analogue of a previously complex valued computation, namely something ...
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### Are all bivectors in three dimensions simple?

I want to show that all bivectors in three dimensions are simple. If I understand correctly, a bivector is simply an element from the two-fold exterior product $\bigwedge^2V$ of a vector space $V$, ...
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### How to calculate the wedge product of differential forms with arbitrary coefficients

I need to calculate the wedge product between some differential forms of the type:   $\omega=P_1(x_1, ..., x_n)dx_1+\cdots+P_n(x_1, ..., x_n) dx_n$ and $d\omega$, i-e, $\omega\wedge d\omega$. where ...
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### Exterior algebra: If $\alpha \land \beta =0$ for all $\beta \in \Lambda ^{n-k} V$ then $\alpha =0$.

How do I prove the following: If $V$ is $n$-dimensional, and $\alpha \in \Lambda ^{k} V$, if $\alpha \land \beta =0$ for all $\beta \in \Lambda ^{n-k}$ then $\alpha =0$. For $k=1$, then we can form ...
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### One-dimensional null space for a 2-form

Consider the following 2-form on $\mathbb R^{2n+1}$ with coordinates $x_1...x_n;y_1...y_n;t$: $$\omega^2=\sum dx_i \land dy_i-\omega^1 \land dt$$ where $\omega^1$ is any 1-form on $\mathbb R^{2n+1}$. ...
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### Showing that a 2-form on an odd dimensional space is not degenerate

On an odd-dimensional space $\mathbb R^{2n+1}$ with coordinates $x_1...x_n;y_1...y_n;t$ consider the following 2-form: $$\omega^2=\sum dx_i \land dy_i-\omega^1 \land dt$$ where $\omega^1$ is any 1-...
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### Lagrange identity for determinants

Let A $\in M_{(n-1)},n(\mathbb{R})$ and for each $1\leq j \leq n$,let $A_j$ the matrix obtained from A by removing the j-th column. Show that: $det (AA^t)= \sum\limits_{j=1}^n det(A_j)^2$ My first ...
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### Functional Derivative of Product of Two Grassmann Functionals

Say, I have a product of two Grassmann functionals: $F[\psi(x)]$ and $G[\psi(x)]$ given by $F[\psi(x)]G[\psi(x)]$. I want to take the functional derivative of this product with respect to $\psi(x)$: ...
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### Confusion over an exterior derivative product

I have an elementary question regarding the exterior derivative which confuses me. I am reading Theorem 2.1.13 of the book Aspects of Multivariate Statistical Theory by Robb J Muirhead. The part that ...
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### The wedge of an exact form with a closed form is exact.

I'm trying to prove that the wedge of a closed form $\xi$ with an exact form $\omega$ is exact. We already have that half of it is exact. Maybe we can use the equation of $\xi$ being closed to rewrite ...
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### A natural identification of the second exterior power of linear operators ?

Let $K$ be a field of characteristic zero and $V=K^n$. Let $A\in M(n,K)$, so we can think of $A$ as a linear map $A: V \to V$ be a linear map . Let $\wedge^2 V$ be the second exterior power of $V$ and ...
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### 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 ...
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### On the “exterior derivative” for not-necessarily-differential forms

Suppose that we are talking about the linear map $$e^i\wedge:\text{Alt}(\otimes^pV^*)\to\text{Alt}(\otimes^{p+1}V^*),$$ which maps exterior $p$-forms to $(p+1)$-forms. In my mind, this is the ...
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### Does this geometric rigidity condition forces the map to be the identity?

$\newcommand{\Cof}{\operatorname{cof}}$ $\newcommand{\id}{\operatorname{Id}}$ $\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$ Let $V$ be a real $d$-dimensional vector ...
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### Wedge Product on $C\ell^+(0,3,0)$ Relationship to Quaternion Cross Product

The even Clifford sub-algebra $C\ell^+(0,3,0)$ is isomorphic to the quaternion algebra. The mapping between terms is $e_0 \mapsto 1$, $e_{23} \mapsto i$, $e_{31} \mapsto j$, $e_{12} \mapsto k$. In ...
Let $\alpha, \beta$ $1$-forms in the open $U \subset \mathbb{R}^{3}$ such that $\alpha \wedge \beta \neq 0$ in every $x \in U$. If a form $\omega$, of degree $2$ in $U$ is such that $\omega \wedge \... 1answer 26 views ### (Exterior Algebra) Relation between positive oriented space and$r$-forms space. Let$S$a oriented vector space of dimension$m$and equipped with a inner product. Given$v \in S$, let$\omega = \phi(v) \in \mathcal{A}_{m-1}(S)$defined by $$\omega(v_{1},...,v_{m-1}) = \langle ... 1answer 39 views ### Write \omega in terms of the standard basis dx^i ∧dx^j at each point. At each point p ∈ \mathbb R^3, define a bilinear function \omega_p on T_p(\mathbb R^3) by$$\omega_p(\textbf{a},\textbf{b})=\omega_p(\begin{bmatrix} a^1 \\ a^2 \\ a^3 \\ ... 1answer 66 views ### MTW's Gravitation. Factor of contraction of p-vector with p-form I'm reading MTW's Gravitation. On page 92 we have the following statement$$\langle\omega^{i_1}\wedge\dots\wedge\omega^{i_p},e_{j_1}\wedge\dots\wedge e_{j_p}\rangle = \delta^{i_1\dots i_p}_{j_1\... 0answers 72 views ### Classifying the orbits of the natural$\text{GL}(V)$-action on the exterior power$\bigwedge^k V$Let$V$be a real$d$-dimensional vector space, and let$1 < k < d$. Consider the following action of$\text{GL}(V)$on$\bigwedge^k V$:$(T,\omega) \to (\bigwedge^k T) \omega$. Can we ... 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^...
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 $V$ ...