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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60 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 ...
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27 views

Universal Linearizer of Alternating Multi-$F[x]$-Linear Maps is Same as that of Multi-$F$-Linear Maps.

Let $V$ be a an $n$-dimensional vector space over a field $F$. Let $M=F[x]\otimes_F V$. We can consider $M$ as an $F[x]$-module by extending scalars using the inclusion $F\to F[x]$. Fact 1. There is ...
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50 views

Cayley-Hamilton Theorem - Trace of Exterior Power Form

Let $V$ be an $n$-dimensional vector space over a field $F$ (the characteristic of which, for the purpose of this post, may be taken as $0$). Let $T$ be a linear operator on $V$ and $\lambda\in F$. ...
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76 views

Determinant from Paul Garret's Definition of the Characteristic Polynomial.

$\DeclareMathOperator{\id}{id} \DeclareMathOperator{\End}{End}$ On pg. 390 of Paul Garret's notes on Algebra, a definition for the characteristic polynomial is given, which I discuss here. Let $V$ be ...
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1answer
47 views

Differential of a form

I'm learning about Chern-Simons theory and my differential geometry is a bit rusty. The Chern-Simons 3-form is given by $\omega_3=tr(A\bigwedge\nolimits dA+\frac{2}{3}A\bigwedge\nolimits A ...
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1answer
48 views

Lack of associativity of cross product vs associativity of the exterior product

Can someone remind me in a nutshell why the associativity of the exterior product fails to transfer to the cross product? (It's been over a decade since I had to deal with the former back in school.) ...
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0answers
31 views

Is $\Lambda$ essentially the unique solution to $F(M)\cong\frac{F(M\oplus R)}{F(M)}$?

Let $R$ be a commutative ring and let $F$ be a functor $\mathbf{Mod}_R\rightarrow \mathbf{Mod}_R$. Then for a module $M$ the split mono $M\rightarrow M\oplus R$ gives a split mono $F(M)\rightarrow ...
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45 views

Decomposition into simple bivectors

According to Wikipedia, any element of $\wedge^2\Bbb R^n$ should be decomposable into $n/2$ simple bivectors for $n$ even or $(n-1)/2$ for $n$ odd. How do I count that? How do I check that ...
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3answers
55 views

In Wikipedia's motivating example of wedge product, what happened to $e_1 \wedge e_1$

Wikipedia (https://en.wikipedia.org/wiki/Exterior_algebra) has a motivating example for wedge product. In particular, it was written that: $ \begin{align} {\mathbf v}\wedge {\mathbf w} & = ...
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1answer
78 views

When considering vector spaces $A$ and $B$, what is the difference between $A \times B,$ $A \otimes B$ and $A \wedge B?$

I have looked at this resource http://hitoshi.berkeley.edu/221a/tensorproduct.pdf to instinctively differentiate between the tensor product and the direct sum of two vector spaces. I am currently ...
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48 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 ...
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52 views

Building a two-form with wedge product of one-forms

Suppose $\alpha,\beta \in \Lambda ^1(X)$ where $X$ is a smooth manifold; and let $v,w \in TX$. Is the following an identity? $(\alpha \wedge \beta)_x (v,w) = (\alpha_x(v)) \wedge (\beta_x(\omega))$
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36 views

Isomorphism between those two Algebras

Why is there isomorphism between Clifford Algebra and Exterior Algebra? Maybe, better said, what does it mean to have isomorphism between those two Algebras?
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1answer
105 views

Difference Between Tensoring and Wedging.

Let $V$ be a vector space and $\omega\in \otimes^k V$. There are $2$ ways (at least) of thinking about $\omega\otimes \omega$. 1) We may think of $\otimes^k V$ as a vector space $W$, and ...
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2answers
37 views

Dual space of exterior power and exterior power of dual space

Let $V$ be a finite-dimensional vector space. Is there an isomorphism between $\Lambda^k(V^\ast)$ and $\left(\Lambda^k(V)\right)^\ast$? I was able to prove this with the additional requirement of ...
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1answer
72 views

Slick proof of cross product identities

The cross product between vectors in $\mathbb{R}^3$ obeys two pleasant identities (sometimes named after Lagrange), namely $a\times(b\times c)=b(a\cdot c)-c(a\cdot b)$ $(a\times b)\cdot(c\times ...
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1answer
37 views

Exterior power of irreducible representation

I am new to representation theory. Suppose that $G$ is a finite group with an irreducible representation over a (real or complex) vector space $V$. In my application, $G$ is a symmetric group and the ...
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1answer
55 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 ...
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0answers
21 views

Frobenius Condition for Singular Integrable Distributions

A smooth "singular" distribution $D\subseteq TM$ on an $n$-dimensional manifold $M$ is integrable if it is tangent to immersed submanifolds $N_\alpha$ that are disjoint and cover $M$. If dim$D=k$ ...
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1answer
87 views

A coordinate free book on linear and multilinear algebra defining determinants using exterior algebra

I would like to find an advanced introduction to linear and multilinear algebra that is 1)Coordinate free 2)Use tensor products and exterior algebras to define determinants 3)DOES NOT assume a ...
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1answer
61 views

On volume forms and norms on exterior powers

Let $V$ be a $2$-dimensional vector space. Given an inner product on $V$ one may define an inner product on the simple $k$-vectors of $\Lambda^k(V)$ by $$\langle v_1 \wedge \cdots \wedge v_k, w_1 ...
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38 views

Is it true that $V^{\otimes k}=S^k (V)\oplus \Lambda^k V$ for $\dim V\ge 2$?

I was trying to show that $V^{\otimes 2}=S^2(V)\oplus \Lambda^2V$ and this is what I came up with. Obviously $S^2(V)\oplus \Lambda^2V\subset V^{\otimes 2}$. Now take $a\in V^{\otimes 2}$.Let $\dim ...
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10 views

exterior powers and isomorphisms [duplicate]

Apparently, it possible to show that if there is a lin. mapping $f$ ($\not=0$) on $\Lambda^nV$ ($\mbox{dim}(V)=n$), then, by using the same $f$, the following is true $$\Lambda^kV\cong ...
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1answer
30 views

Where does the wedge product arise in the definition of an integral?

For a given function $f(x,y)$, its double integral is defined as: $$\iint_R f(x,y)\;\mathrm{d}x\;\mathrm{d}y=\lim_{n\to\infty}\; \sum_{i=1}^{n }f(x_{i},y_{i})\Delta A_{i}$$ Where $\Delta ...
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1answer
52 views

wedge product is linearly dependent

Suppose $a,b,c$ are in $\Lambda^1 V$. Then I must show that they are linearly dependent iff $a \wedge b \wedge c=0$. I am new to tensors and wedges and I don't know how to proceed. I could only ...
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2answers
50 views

if $\omega$ is a $2$-form, and $\Bbb d \omega = 0$ what can we conclude?

Let $f_1, f_2, f_3$ be smooth functions on an open subset $\Omega \subset \Bbb R ^3$, which contains the standard cube $I^3$. We define the differential $2$-form $$ \omega = f_1 \Bbb d x^2 ...
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1answer
34 views

Understanding a step in a computation involving dual basis and permutations

Since $\varphi_{i_j} \in \mathcal{T}(\mathbb{R}^n)$, for every $j = 1, \dots, k$, we have \begin{align*} \varphi_{i_1}\wedge\dots\wedge\varphi_{i_k}(e_{i_1}, \dots, e_{i_k}) &= ...
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1answer
30 views

Wedge product: prove $\omega \wedge \eta= (-1)^{kl} \eta \wedge w$

prove $\omega \wedge \eta = (-1)^{kl} \eta \wedge w$ where $\omega \in \Lambda^{k}(V)$ and $\eta \in \Lambda^{l}(V)$. This is from page 79 on spivak calculus on manifolds. my progress: $\omega ...
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71 views

Show that $\omega^{2}\wedge\cdots\wedge\omega^{2}$ n times is equal to

Consider $\mathbb{R}^{2n}$ with coordenates $x^{1},\cdots,x^{2n}$ and the following differential form of grade two $$\omega^{2}=dx^{1}\wedge dx^{n+1}+dx^{2}\wedge dx^{n+2}+\cdots+dx^{n}\wedge ...
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2answers
33 views

The exterior derivate

I am trying to show that $d(a \wedge da)=0$ if $k$, the degree of k-form $a$ is even. I have said: $=da \wedge da + (-1)^k a \wedge d^2a$ I believe the first term is zero due to repeated indices and ...
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35 views

Exterior power of cotangent bundle of a cartesian product of manifolds

Let $M \times F$ be a product manifold. The identity $T^*(M \times F) = T^*(M) \times T^*(F)$ holds generally (I hope). What can be said about the exterior powers of it? $$\bigwedge^r T^*(M \times ...
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votes
1answer
54 views

Signature of the inner product.

I need some help with this problem. Consider in $\mathbb{R}^{4}$ the lorentz's metric $h$ which has a signature $(3,1)$, this mean that diagonal matrix $M(h)$ is the form $$M(h)=\begin{pmatrix} 1 ...
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votes
1answer
52 views

Problem proving Cartan's identity

There is a famous identity stating that, if $X$ is a field and $\omega$ a form, then: $$\mathcal{L}_X\omega=\iota_Xd\omega+d\iota_X\omega.$$ I'm trying to prove it. Thanks to Anthony Carapetis, I ...
2
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0answers
30 views

Associativity of the exterior multiplication of forms.

Let $\omega^{k}\in\bigwedge^{k}(V^{*})$ and $\eta^{l}\in\bigwedge^{l}(V^{*})$, be two exterior forms of degrees $k$ and $l$. The exterior product $\omega^{k}\wedge\eta^{l}$ is defined as $(k+l)$ form, ...
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1answer
64 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, ...
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2answers
55 views

Confusion with notation on Indices, differential forms

I think I have a little confusion with index notation (concerning p-form). For example, do the coordinates on $\mathbb{R}^n$ $\{x^1,x^2, \dots , x^k\}$ and $\{x^{i_1},x^{i_2}, \dots , x^{i_k}\}$ mean ...
2
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2answers
62 views

Identify $\bigwedge^{N-1}V$ with $V$

Suppose we have a $N$-dimensional vector space $V$, equipped with a scalar product $\langle\cdot{ , }\cdot\rangle$ and let $\{e_1,\dots,e_N\}$ be a orthonormal basis. I've seen that there are ...
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1answer
14 views

Exercise of directors cosine of a vector in the dual space

For $\vec{x}\in\mathbb{R}^3-\{\vec{0}\}$, let for every $i=1,2,3$ $\alpha_i =\cos^{-1}\langle\hat{\vec{x}}|\underline{e_i}\rangle$, the angle que between the $\vec{x}$ with the $ith$ vector in the ...
3
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1answer
35 views

The $-1$-eigenspace of the exchange map is $V \wedge V$

Let $V$ be a vector space, and $T(\mathbf{v}_0 \otimes \mathbf{v}_1) = \mathbf{v}_1 \otimes \mathbf{v}_0: V \otimes V \to V \otimes V$ be a linear map. Theorem: The eigenspace of $T$ with ...
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46 views

Difficulty comprehending this sentence on Wikipedia

Via Wikipedia https://en.wikipedia.org/wiki/Exterior_algebra#Differential_geometry, there is this given definition "In particular, the exterior derivative gives the exterior algebra of differential ...
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votes
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79 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 ...
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1answer
42 views

Why are “innermorphisms” not useful?

I commonly studied type of linear function in geometric algebra is the outermorphism. For reference, here's Wikipedia's definition: Let $f$ be an $\Bbb R$-linear map from $V$ to $W$. The ...
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37 views

Geometric meaning of Berezin integration

Berezin integration in a Grassmann algebra is defined such that its algebraic properties are analogous to definite integration of ordinary functions: linearity (taking anticommutativity into account), ...
3
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2answers
46 views

Reference needed: Exterior covariant calculus

I would like to understand Cartan's formalism of exterior covariant calculus. I think it could be useful for some calculations in physics (But If I am wrong here and it's only good for abstract ...
3
votes
1answer
55 views

Different definitions for exterior derivative? [duplicate]

Let $\omega \in \Omega^k(M)$ and $X_i \in \Gamma(TM)$. In Spivak Volume I page 213 the exterior derivative is given invariantly by the formula \begin{align*}d \omega(X_0, \ldots, X_k) = & \sum_i ...
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1answer
58 views

Interior product of a differential form with respect to a commutator of two vector fields

I've been working my way through Nakahara's book "Geometry, topology & physics" and I've reached chapter 5 section 4 in which he discusses the interior product and lie derivative of differential ...
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12 views

ℂ version of alternating property

The alternating property says that if arguments $x_i$ and $x_j$'s places are switched, then the $\mathtt{output} \mapsto - \mathtt{output}$. Is there an extension where the index set of the ...
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1answer
42 views

Is matrix algebra a special case of Grassmann (exterior) algebra, and if so what is more general case?

Just a little question. I only recently heard about Grassmann algebra while reading a book on the history of vector algebra and quaternions. I still don't understand what does exterior product mean ...
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1answer
134 views

Vectors, Forms, Multivectors, and Tensors

In researching some of the ways that vectors (and vector fields) generalize I find that there are apparently many different objects that generalize them -- matrices, differential forms/ covectors, ...
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1answer
56 views

Finding a form to solve a wedge product equation

We start out with $(\mathbb{R}^{2n},\omega_0)$, the "standard" symplectic manifold. We define $\Omega=\omega_0\wedge\dotso\wedge\omega_0$, i.e. $\Omega$ is the product of $\omega_0$ with itself $n$ ...