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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The Hodge star operator and the wedge product: $\alpha \wedge (\star \beta)$

According to Wikipedia, The Hodge star operator on a vector space $V$ with an inner product is a linear operator on the exterior algebra of $V$, mapping $k$-vectors to $(n-k)$-vectors where $n=\...
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Identifying a wedge-to-metric formula

In this question, the original poster wrote: On every Riemannian manifold $M$, we can consider the Hodge $*$-operator, which is characterised by the following formula: $$a\wedge *b = (a,b)\nu.$$...
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Why is this implying a normal vector field?

Suppose $\omega$ is a $n-1$ form on a $n-1$ dimensional manifold and $(a_1(x)dx_1 + ... + a_n(x)dx_n )\wedge \omega = c\Omega $, with $c \neq 0$ and $\Omega =dx_1\wedge...\wedge dx_n$. Moreover $(a_1(...
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Proof explanation: Calculate a spectrum of a pair of commuting operators

According to the following paper of Taylor: J. L. Taylor, A joint spectrum for several commuting operators, J. Functional Anal. 6(1970), 172-191. we have Let $A= \begin{pmatrix}0&1\\1&0\...
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Equivalence of definition of symplectic form

Suppose that $V$ is a vector space of dimension $2n$, and let $\omega \in \Lambda^2(V)$. Prove that the following two statements are equivalent. (1) $\tilde{\omega} : V \rightarrow V^*$ defined ...
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Trouble in finding the Galois group of the covering spaces of $S^1 \vee S^1$.

I am studying covering spaces and deck transformations from the book Algebraic Topology written by Allen Hatcher. While reading deck transformations I came across the concept of Galois group of ...
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Exterior algebra as quotient algebra

This is my first time posting so I apologise for the stupid question. I am trying to understand the idea of exterior algebra as a quotient algebra. Let $T(V)$ be the tensor algebra of a vector space $...
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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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Product manifolds and exterior derivative with interior product

While studying differential geometry, I read this part of a proof and I didn't understand it. Given a $2$-manifold $\Omega$ and an interval $I=(-\epsilon, \epsilon)$, consider the cartesian product $M=...
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Wedge product and differential forms

I am a bit confused when it comes to wedge product and differential forms. I know the following property: $\omega\wedge\eta=(−1)^{kl}\eta\wedge\omega$ Also I know that when $k$ is odd and I am ...
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Basis for the Dual Space Involving Wedge Products

I'm pretty much stuck on the following problem. Let $V$ be an $n$-dimensional vector space, and let $\omega\in\Lambda^{2}(V^{*})$. Show that there is a basis $\{e^{1},e^{2},\ldots,e^{n}\}$ of $V^{*...
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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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Grassmann algebra

In studying associative algebras' theory I was introduced to the notion of Grassmann algebra, but I don't know if I well understood how to construct this algebraic structure. Let $F$ a field and $X=\...
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Two related(?) definitions on the wedge product

I read from a textbook that one defines the wedge product of basis elements as, for example, $$e^{i_1}\wedge...\wedge e^{i_k}\equiv k!\text{Alt}(e^{i_1}\otimes...\otimes e^{i_k})$$ and of two forms $$...
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Extending a derivative to its exterior algebra: Rotman, incorrect?

This is page 770, Lemma 9.165(ii) If $\varphi:M \rightarrow M$ is a $k$_map, there exists a unique derivation $d_\varphi: \wedge(M) \rightarrow \wedge (M)$ which is graded with $d|_\varphi M=\...
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On the idealizer of the set of elementary wedge products of two vectors in $K^4$, for a field $K$

Let $K$ be a field of characteristic zero. Consider $V=K^4$ with standard basis vectors $e_1,e_2,e_3,e_4$. We can consider the second exterior product $\bigwedge^2 V $ of $V=K^4$ with a basis given ...
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Determinant of a special type of skew symmetric matrix with complex entries

Let $a_1,...,a_{2n} \in \mathbb C$ and $A=[b_{ij}]\in M(2n,\mathbb C)$ such that $A^T=-A$ and $b_{ij}=a_ia_j,\forall i<j$. Can we find a nice expression for determinant of $A$?
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Clifford algebra is isomorphic to exterior algebra, proof in Lawson's

This Proposition 1.2, pg 10 of Lawson's Spin Geometry. Context: We have a homomorphism of graded algebra $$ \wedge^*(V) \rightarrow G^*$$ induced on each component from the map $$\wedge^r(V) \...
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Why do we need the multiple $\frac{1}{k! l!}$ in the definition of wedge product?

In the book of Analysis On Manifolds by Munkres, at page 238, it is claimed that in the definition of wedge product For alternating k-tensor $f$ and alternating l-tensor $g$ on $V$, $$f \wedge g =...
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on the matrix representation of a canonical linear map on the vector space of $4\times 4$ skew-symmetric matrices

For $t_1,t_2,...,t_6\in \mathbb R$, let $P_{(t_1,t_2,...,t_6)}=\begin{pmatrix} 0&t_1&t_2&t_3\\ -t_1&0&t_4&t_5\\ -t_2&-t_4 &0&t_6\\-t_3&-t_5&-t_6&0\end{...
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Axiomatization of Exterior Algebras ? Rotman

Rotman makes the following axiomatization Definition: If $V$ is a free $k$-module of rank $n$, then a Grassmann algebra on $V$ is a $k$ algebra $G(V)$ with identity element, $e_0$, such that ...
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differential on Koszul algebras

Let $V$ be a finite dimensional vector space over a field $k$. It is known that the symmetric algebra $S = S(V)$ is "dual" to the exterior algebra $E = \bigwedge(V^*)$. Now choose a basis $\{x_1,\...
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Is there a non-zero algebra homomorphism $\text{End}(V) \to \text{End}(\bigwedge^kV)$?

This question is totally out of curiosity. Let $V$ be a real $d$-dimensional vector space. Let $1<k<d$ be fixed. Is there a non-zero algebra homomorphism $\text{End}(V) \to \text{End}(\...
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Commutator of Lie derivative and Hodge star operator

I want to derive and expression for the commutator $[\mathcal{L}_Z,\star]\omega$. I found this post of mathoverflw that answers this question, but I have a few questions about Willie Wong's proof. How ...
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Star operator, exterior product

Let $\ast$ be the star operator $$\ast:\Lambda^p(V)\to \Lambda^{d-p}(V)$$ so that we have $$\ast(e_{i_1}\wedge...\wedge e_{i_p})=e_{j_1}...\wedge e_{j_{d-p}}$$ where $$e_{i_1},...,e_{i_p},e_{j_1},...,...
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A Hodge dual computation on a $4$-dimensional Riemannian manifold

Let $(M,g)$ be a $4$-dimensional smooth Riemannian manifold. I am trying to understand the following exterior algebra computation: Let $x^1,x^2,x^3,x^4$ be local coordinates on $M$ such that the ...
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Can the “symmetric algebra” over $\mathbb R^n$ be defined from an infinite-dimensional exterior algebra?

https://en.wikipedia.org/wiki/Symmetric_algebra If I understand that article correctly, the symmetric algebra $S(\mathbb R^n)$ is (isomorphic to) the algebra of polynomials with $n$ variables. As a ...
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Why is $\epsilon^{p,q}(X):=\Gamma(X,\bigwedge^p\Omega^1_X\otimes\overline{\bigwedge^p\Omega^1_X})$?

Here $\Omega^1_X=(T^*X)^{1,0}$, from : this notes we have $\epsilon^{p,q}(X):=\Gamma(X,\bigwedge^p\Omega^1_X\otimes\overline{\bigwedge^p\Omega^1_X})$ But I wonder why it's tensor not wedge? Shouldn'...
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Can we stay invertible while approximating linear maps in Sobolev spaces?

Let $\Omega \subseteq \mathbb{R}^d$ be an open bounded domain. Fix an integer $1<k<d$. Let $f \in W^{1,k}(\Omega;\mathbb{R}^d)$ be a continuous map with $\det df > 0$ a.e. Consider the map ...
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Expressing $*(z_1\wedge \bar z_2)$ without Hodge star operator

I have been stuck on a computation for hours, and still cannot figure out where is the mistake: For $2$-dimensional complex vector space, let $z_1,z_2$ be the basis. I want to compute $*(z_1\wedge \...
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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 ...
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A form $\omega$, of degree $2$ in $U$ such that $\omega \wedge \alpha = \omega \wedge \beta = 0$, prove that $\omega = f\cdot \alpha \wedge \beta$

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 \...
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(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 ...
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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 \\ ...
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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\...
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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 ...
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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^...
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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 $V$ ...