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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4
votes
1answer
47 views

What is the anticommutator of the interior product and codifferential (adjoint of exterior derivative)?

What is $\eta=i_X \delta + \delta i_X$ acting on differential forms? Here $\delta$ is the usual Hodge adjoint of the exterior derivative and $i_X$ is contraction of a form with the vector field $X$. ...
3
votes
1answer
170 views

decomposable elements of $\Lambda^k(V)$

I have conjecture I have problem to prove or disprove. Let's $ w \in \Lambda^k(V)$ is k-vector. $W_w=\{v\in V: v\wedge w = 0 \}$ is k-dimensional vector space if and only if $w$ is decomposable. ...
2
votes
1answer
47 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
61 views

“Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions

Is there some precise sense in which the "alternation" functor $A$ that maps a multilinear function $f\colon M^d\to N$ to the alternating multilinear function $A(f)\colon M^d\to N$ defined by $$ ...
2
votes
1answer
37 views

A basis in the $k$-th exterior power of a vector field

Definition: Let $\mathbb R^n$ be the $n$-dimensional real vector space. An exterior $k$-form call any skew-symmetric tensor on $\mathbb R^n$ of rank $k$. Denote the set of exterior $k$-forms by $E^k$. ...
1
vote
1answer
51 views

Pull back of a vector representing a 2-form in $\mathbb R^3$

Let $\omega$ be a $2$-form defined in $\mathbb R^3$. I know that it can be represented by a vector field $\xi$ in such a way: $$ \omega_x (v,w) = \xi(x)\cdot (v\times w) $$ ($x,v,w \in \mathbb R^3$ ...
1
vote
1answer
55 views

Putting the wedge product in standard/normal form

I have to compute the wedge product of $$(e_1^* + ze_2^*) \wedge (e_2^* + ze_3^*) \wedge \cdots \wedge (e_{n-1}^* + ze_n^*) \wedge (e_n^* + ze_1^*),$$ and then put it in normal/standard form. So I ...
0
votes
1answer
81 views

From Orthogonal vectors to Useful Bivector

If we have set of orthogonal vectors (X) can we form a set of orthogonal bivectors from that set? I am trying to find if there is a way to get 'more information' from an orthogonal matrix by some ...
0
votes
1answer
79 views

Operations in the exterior algebra. Multiplication in the direct sum of rings.

Let the exterior algebra $\Lambda(V)$ of a vector space $V$ over a field $K$ be the direct sum of the exterior powers $\Lambda^k(V),\quad k\in\overline{0,n}$. Then an element $x\in\Lambda(V)$ has the ...
0
votes
1answer
96 views

Bivector as a sum of exterior products of basis vectors.

Prove that for any 2-vector $\alpha=\Lambda^2(V)$ there is a basis $\{e_1,\ldots, e_k\}$ of $V$ such that $\alpha= e_1\wedge e_2 + e_3\wedge e_4 + \ldots +e_{k-1}\wedge e_k$, where $\wedge$ denotes ...
7
votes
0answers
284 views

Hodge-Star-Operator on arbitrary oriented basis

Assume that $V$ is oriented finite dimensional vectorspace with dimension $n$, $g \in T^0_2(V)$ a given symmetric and nondegenerate tensor. Let $\mu$ be the corresponding volume element of $V$. ...
6
votes
0answers
34 views

Relationship between exterior power of representation and variance?

I was reading the question: Symmetric and exterior power of representation regarding how to determine the character of an exterior power of a representation from the original representation. One of ...
5
votes
0answers
160 views

Free Graded Commutative Algebra on a Graded Vector Space

Let $V$ be a graded vector space, thought of as a collection $\{ V^n \}_{n \ge 0}$ of vector spaces. Let $V_{odd} = \bigoplus_{n \text{ odd}} V^n$ and $V_{even} = \bigoplus_{n \text{ even}} V^n$. I am ...
4
votes
0answers
70 views

What is the difference between tensor calculus and exterior derivative type concepts?

I am trying to clarify terms in order to help me figure out what I'd like to study. I understand that $p$-forms and $p$-vectors are used with things like wedge products, exterior algebras, and a ...
4
votes
0answers
87 views

How to compute Ext over an exterior algebra

I found this question in several places (even on mathoverflow and mathstackexchange), but I never found a satisfying answer. Let $k$ be a field and $V$ a finite dimensional $k$-vectorspace, I would ...
3
votes
0answers
49 views

Lie algebra: symmetric and exterior power of representation

If $\mathfrak{g}$ is a Lie algebra, $V$ and $W$ are representation of $\mathfrak{g}$ we define the action of $\mathfrak{g}$ on $V \otimes W$ in the following way: $X \cdot (v \otimes w)=(X \cdot v) ...
3
votes
0answers
87 views

Prove that $\phi_1 \wedge \cdots \wedge \phi_k (v_1, \cdots, v_k) = \frac{1}{k!}\det[\phi_i(v_j)].$

I have proved these two exercises: (1) Suppose that $T \in \Lambda^p(V^*)$ and $v_1, \ldots, v_p \in V$ are linearly dependent. Prove that $T(v_1, \ldots, v_p) = 0$ for all $T \in \Lambda^p(V^*)$. ...
3
votes
0answers
139 views

Multiplication in exterior algebra

Take $V = K^{n}$. Let $\omega$ be a non-zero element of $\bigoplus_{k=1}^n \bigwedge^k V$, where we have excluded the summand $\bigwedge^0 V = K$. (1) Prove that there exists an $m > 1$ for which ...
3
votes
0answers
218 views

Grassmann Algebras

The Grassmann algebra $G$ is the algebra over a field $\mathbb{F}$ generated by the variables $e_i$ such that $e_i^2=0$ and $e_i e_j = - e_j e_i$. I'm looking for some references on algebras $G ...
2
votes
0answers
41 views

How to phrase this identity in differential form language?

If the vector field $\mathbf B$ on $\mathbb{R}^3$ is constant, then the vector field $$ \mathbf A = \frac 1 2 \mathbf B \times \mathbf r $$ satisfies $$ \nabla \times \mathbf A = \mathbf B. $$ This ...
2
votes
0answers
34 views

Bivector into orthogonal components

Suppose I have a metric $g$ and a bivector $ F $ on a four-dimensional vector space. It seems I can always decompose $ F $ into four mutually orthogonal vectors $a,b,c,d$ $$ F = a\wedge b + c\wedge d ...
2
votes
0answers
36 views

Inner product exterior algebra

I have to prove that if V is a real vectorial space (dimV=n) with inner product (.,.) then if we define $$ (v_{1}\wedge v_{2}\wedge...\wedge v_{k},w_{1}\wedge w_{2}\wedge...\wedge w_{k}) ...
2
votes
0answers
52 views

Vector Laplace Beltrami operator on surface tangent and surface normal vector field

Consider a closed, compact, embedded surface $f:M \rightarrow \mathbb{R}^3$ and a vectorfield $X$ on the surface that can be decomposed in the surface frame basis $\{e_1,e_2,e_3\}$, where ...
2
votes
0answers
57 views

Exterior power of a space of maps $(\mathbb{K}^T)$

We are given a set $T \neq \emptyset, \ \ p \ge 1, \ \ p_i : T \rightarrow \mathbb{K}$ Could you help me prove that if $ \phi: (\mathbb{K}^T)^p \ni (f_1, ..., f_p) \rightarrow \rho \in ...
2
votes
0answers
87 views

Computing wedge product of two 1-forms.

Let $L$ be a lattice in $\mathbb{C}$ and let $\pi :\mathbb{C}\to X=\mathbb{C}/L$ be a quotient map. Show that the local formula $dz$ in every chart of $\mathbb{C}/L$ is a well-defined holomorphic ...
2
votes
0answers
153 views

Calulation of pullback of form

If $M$ is $2n+1$ dimensional manifold, and $M'= M\times \mathbb R$ Let $x_1,y_1,... x_n, y_n,t', t$ be coordiante of $M'$. With $t$ for coordinate for $\mathbb R$. Let $$ \omega= \sum_{i=1}^n ...
2
votes
0answers
63 views

Determining explicitly the action on the exterior products of a vector space

Let $V$ be a 2-dimensional complex vector space with basis $e_1,e_2$. Consider the endomorphism $f:V\to V$ given by $f(e_1) = e_2$ and $f(e_2) = -e_1$ with matrix $$ \left( \begin{matrix} 0 & -1 ...
2
votes
0answers
137 views

Exterior algebras and radicals

So without giving excessive background, I was working on baby Spivak & got to the stuff about exterior algebras, & now I'm going through the section on tensor algebras in my copy of ...
1
vote
0answers
138 views

base change of exterior powers

Let $n\geq 0$ be an integer, $R\to R'$ a ring homomorphism, and $M$ an $R$-module. Then the following holds: $$\bigl(\bigwedge^n_R M\bigr)\otimes_r R' \cong \bigwedge_{R'}^n\, (M\otimes_r R').$$ I ...
1
vote
0answers
54 views

Poincaré Lemma problems and computing contractions in an economical way

Let $x=(A,B,C,D)$ be coordinates on $\mathbb{R}^4$. $\displaystyle \beta = \frac{(AdB-BdA)\wedge(dC \wedge dD)+(dA \wedge dB)\wedge(CdD-DdC)}{(A^2+B^2+C^2+D^2)^2}$ I would like to compute ...
1
vote
0answers
43 views

Wedge product of differential forms

I'm trying to grasp the notation and concept of wedge products(, and tensors as well). In my lecture notes, the following expansion/notation for a $(n,r)$-tensor is used: In a basis $\left\{ ...
1
vote
0answers
36 views

Reference Request for differential ideals of Pfaffian forms on jet bundles

My setting is the following: Given two families of differential forms $\omega^i$ (with $i=1,...,m$) and $\mathrm d F^j$ (with $j=1...n-m$), define $$\eta^i := \sum_{j=1}^m a^i_j\omega^j + ...
1
vote
0answers
49 views

The canonical perspective on the Hodge star operator

I am looking for the canonical perspective on the Hodge star operator. I want to see it done properly, not using basis for its definition, saying clearly what we assume in its definition. ...
1
vote
0answers
77 views

Is $\Lambda(T^{*}E)=\bigoplus_{k=0}^n\Lambda^k(T^{*}E)$ a complex line bundle over $T^{*}E$?

Is $\Lambda(T^{*}E)=\bigoplus_{k=0}^n\Lambda^k(T^{*}E)$ a complex line bundle over $T^{*}E$? I know that ...
1
vote
0answers
105 views

Norm inequality with wedge product

Anyone could help me to prove this following inequality? $\displaystyle\frac{||(u+v)\wedge w||}{||u+v||}\le \frac{||u\wedge w||}{||u||} +\frac{||v\wedge w||}{||v||} $ where $u\wedge v$ is the wedge ...
1
vote
0answers
40 views

Grassmannian as a submanifold of the exterior product

I'm looking for a proof of the fact that if $V$ is a finitely dimensional vector space, then $G_p(V) \setminus \{0\}$ is a submanifold of $\Lambda_pV$. Here $G_p(V) = \{ v_1 \wedge ... \wedge v_p \ ...
1
vote
0answers
30 views

Exterior product of vectors in $\mathbb{R}^4$ with integer coefficients.

Let $a, b, c, d$ be vectors with integer coordinates in $\mathbb{R}^4$ such that $k a \wedge b = c \wedge d$ for some integer $k$ and $a \wedge b \neq l v$ for any $v \in \bigwedge^2 (\mathbb{R}^4)$ ...
1
vote
0answers
45 views

Vanishing criterion of pure wedges

Let $R$ be a commutative ring, $M$ some $R$-module, and $m,n \in M$. Is there some criterion when $m \wedge n = 0$ in $\Lambda^2(M)$? There are some sufficient criterions, for example that $m \in ...
1
vote
0answers
133 views

Explicit computations of tensor and wedge product

Let $f\colon K^3\to K^3$ be a map in Jordan canonical form having matrix $$ f=\begin{bmatrix} -1 & 1 & 0 \\ 0 & -1 & 0 \\ 0 & 0& -1 \end{bmatrix} $$ What is $f\otimes f$? What ...
1
vote
0answers
53 views

Alternating forms tangential to a subspace.

Let $V$ be a finite-dimensional vector space with euclidean product, and let $U$ be a subspace. Now let $P$ be the projection of $V$ onto $U$, and let $\omega$ be any alternating multilinear $k$-form. ...
1
vote
0answers
102 views

Given a surjective linear mapping of free modules how do you show the corresponding matrix has an invertible minor?

The following post can is related to part c) of this problem http://www.artofproblemsolving.com/Forum/viewtopic.php?f=349&t=124137 and boils down to some issues I am having with use of wedge ...
0
votes
0answers
7 views

Condition for local Lipschitzness of pullback map for exterior forms

Given $w\in\Lambda^k(\mathbb{R}^n)$, determine the condition under which the map $T\rightarrow T^*(w)$ is locally Lipschitz, where $T\in GL_n(\mathbb{R})$ and $T^*(w)$ denotes the Pullback of $w$ by ...
0
votes
0answers
27 views

Wedge product and derivatives

If I was given $$B=f(\phi,\psi,\theta)\,d\theta\wedge d\phi$$ for some function $f$, along with $$H_{\mu\nu\lambda}\equiv\nabla_\mu B_{\nu\lambda}+\nabla_\nu B_{\lambda\mu}+\nabla_\lambda ...
0
votes
0answers
45 views

Let $K$ a field, $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor. Prove that $f \wedge f =0$

Let $K$ a field, with $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor ($f \in {\mathcal T}_n(V):=\Lambda^{n}(V)$), i.e., $f$ is an multilinear ...
0
votes
0answers
28 views

The exterior product (given by an alternative definition) is associative

Definition: Let $w^k$ and $w^l$ be a $k$-exterior form and an $l$-form on $\mathbb R^n$. For any $k+l$ vectors $x_1,\dots,x_{k+l}$ define $$w^k\wedge w^l(x_1,\dots,x_{k+l})=\sum_{\begin{matrix} ...
0
votes
0answers
31 views

Why is the presheaf-“p-exterior power” of a sheaf separated?

In the first volume of the EGA ( http://www.numdam.org/numdam-bin/feuilleter?id=PMIHES_1960__4_ ), p38, Grothendieck says that the presheaf-p-exterior power of a module in ringed space a separated ...
0
votes
0answers
17 views

Exterior algebra evaluation

How do you evaluate the following expression $\hat{i}\wedge\hat{j}\wedge\hat{i}$? And why does $(\hat{i}\wedge\hat{j})\wedge(\hat{i}\wedge\hat{j}\wedge\hat{k})=-\hat{k}$ philosophically?
0
votes
0answers
89 views

Kernel of the Lie bracket $[,]\colon\wedge^2\mathfrak g\to\mathfrak g$

I believe the following is probably well-known, but so far I couldn't find the answer by myself: Let $\mathfrak g$ be a real (finite-dimensional) Lie algebra, and $\wedge^2\mathfrak g$ its second ...
0
votes
0answers
25 views

Did I give enough justification when I extend to $p$-dimensional?

I have proved these two exercises: (1) Suppose that $T \in \Lambda^p(V^*)$ and $v_1, \ldots, v_p \in V$ are linearly dependent. Prove that $T(v_1, \ldots, v_p) = 0$ for all $T \in \Lambda^p(V^*)$. ...
0
votes
0answers
53 views

Each entry in $\phi_1, \ldots, \phi_k \in V^*$

I have always been scared by exterior algebra, which means I don't really have any background. So here's a very basic question I would like to clear out: Consider $\phi_1, \ldots, \phi_k \in V^*$, ...