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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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What is the dimension of the subgroup of $\text{GL}(\bigwedge^k V)$ which preserves pure tensors?

Let $V$ be a $d$-dimensional real vector space, and let $1<k<d$. Set $$H=\{B\in\text{GL}(\bigwedge^k V) \, | \, B \,\text{ preserves pure tensors }\}$$ (i.e. $B \in H$ if it maps ...
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Exterior derivative of the wedge product

The book I'm reading at the moment would like to show that the following coordinate definition of the exterior derivative satisfies all the axioms assumed for the operator '$\text{d}$'. The axiom I'm ...
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Functorial proof of Cayley-Hamilton using exterior powers

Let $V$ be a rank $n$ free module over a commutative ring $R$. Let $\dagger$ denote the adjoint with respect to the natural perfect pairing given by the wedge product $$\Lambda ^k\otimes \Lambda ^{n-k}...
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Decomposing $k$th exterior powers $\Lambda^kV(\omega_1)$

Let $\Phi$ be a $G_2$ root system, $\omega_1$ the fundamental weight corresponding to the shorter root and consider the unique irreducible highest weight module $V = V(\omega_1)$ of highest weight $\...
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Questions About the Proof of Cauchy–Pompeiu Integral Formula.

I am studying function theory in several complex variables and the book I am using is "Tasty Bits of Several Complex Variables" by Jiří Lebl: https://www.jirka.org/scv/scv.pdf. At the moment I am ...
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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(\...
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Does every subspace of the exterior algebra of dimension $>1$ contain a decomposable element?

Let $V$ be a real $n$-dimensional vector space, and let $W \le \bigwedge^k V$ be a subspace . Suppose that $\dim W \ge 2$. Does $W$ contain a non-zero decomposable element? If $\dim W=1$, then ...
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The non-zero exterior product

Let $V$ denote a vector space (maybe not finite-dimensional) over a field $\mathbb k$ with basis $\{e_1, e_2, \ldots\}$. I have to prove that the set $\{e_{i_1} \wedge e_{i_2} \ldots \wedge e_{i_n}$ $|...
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Covariant Exterior Derivative action on the $\mathrm{End}(E)$-valued p-forms

Suppose I define an operator $d_A$ by its action on sections $s\in \Gamma(E)=\Omega_M^0(E)$ of some vector bundle $\Pi:E\rightarrow M$ in a trivializing neighbourhood $U\subset M$ as $$ d_As|_U=(ds+A\...
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$\omega$ a closed 2-form and $\bigwedge_{i=1}^n \omega \ne 0$ on a compact orientable smooth $2n$-manifold w/o boundary, $M$, then $H^2(M) \ne 0$.

Suppose $M$ is a compact orientable smooth $2n$-manifold without boundary, and let $\omega$ be a closed $2$-form such that $\bigwedge_{i=1}^n \omega_p \ne 0$ at every point $p$. Show that $H^2_{dR}(M) ...
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General commutators of derivations of the exterior algebra

Let $M$ be a smooth manifold and let $\Omega(M)$ be the exterior algebra of smooth differential forms over $M$. The $\mathbb R$-linear map $D:\Omega(M)\rightarrow\Omega(M)$ is a derivation of the ...
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How many independent components are there in the exterior product of two one forms?

Say we have two one-forms $\alpha = \alpha_1 dx^1+...+\alpha_n dx^n$ and $\beta = \beta_1 dx^1+...+\beta_n dx^n$ and $\gamma = \alpha \wedge \beta$. How many independent components will $\gamma$ have, ...
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Using 6th dimensional vector to rotate a tesseract

I'm trying to rotate a tesseract in 4D space for a project. This shows I can use bivectors to "to generate rotations in four dimensions." Following exterior algebra and finding the wedge product, I'...
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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 ...