Questions tagged [exterior-algebra]
For questions on the exterior algebra, and related concepts such as the wedge product, the tensor algebra and differential forms.
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Determinant Formula for Wedge Product via Universal Property of Exterior Powers
I'm currently learning about differential forms in my analysis class, and I thought I'd dig a bit more into the linear algebra of exterior powers. I've seen the universal property of $\bigwedge^k(V)$: ...
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Cross-product Exercise
I am doing some problems in Spivak's calculus on manifolds, and I was stuck on this one. It says that:
If $w_1, ... ,w_{n-1} \in \mathbb{R}^n$, show that $$| w_1 \times ... \times w_{n-1}| = \sqrt{\...
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Hatcher 3.2.17 cohomology ring of James reduced product of a sphere
the goal of this exercise is (after getting inspired by Hatcher's proof in the even dimensional case (prop 3.22)) to compute
$$H^*(J(S^n), \mathbb{Z}), n=2k+1$$
And we are told that this should be $\...
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Decomposition of the symmetric group $\mathfrak{S}_{p+q}$
Let $p,q \ge 1$ be two integers and $\mathfrak{S}_{p+q}$ the symmetric group of $\{1, \dots, p, p+1, \dots, p+q\}$. Denote:
by $\Gamma_{p,q}$ the subgroup of permutations $\alpha \in \mathfrak{S}_{p+...
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How is quotient of the tensor algebra calculated to form the exterior algebra
As stated on wikipedia and elsewhere the exterior algebra is the quotient of the tensor algebra by the ideal generated by $x \otimes y + y \otimes x$. But nowhere is this calculation given explicitly ...
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Using interior multiplication to define basis for dual
Let $\omega\in\bigwedge^2(V)$ where $V$ is a real vector space with $\text{dim }V=n=2m$. There exists a basis $\{e_1,f_1,e_2,f_2,\dots,e_m,f_m\}$ of $V$ for which $\omega(e_i,e_j)=0=\omega(f_i,f_j)$ ...
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Grading of exterior algebra given by generators
I'm having troubles understanding exterior algebra graded structure.
I need to write the polynomial (a sort of Poincare polynomial) given by
$P(t)=\sum_{p=0}^{N} (\dim \Lambda^p )\ t^p,$
where $\...
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On vector multiplication [closed]
In this video (check it out, it's worth it), F. Holmér nicely derives the dot and cross product (with some insights into quaternions, the wedge product and much more), just by using ordinary ...
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Understanding a figure showing the wedge product
On the wikipedia article explaining exterior algebra, this figure is shown, visualizing the relationship between the wedge product of 2 vectors (or differential forms for that matter), and the "...
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Mixed symmetric algebra and applications to linear algebra?
For dual finite-dimensional vector spaces $V,V^*$ the "mixed exterior algebra" $$\textstyle\bigwedge(V^*,V)=\bigwedge V^*\otimes\bigwedge V$$
is a powerful tool for studying linear ...
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Sufficient conditions for a vector bundle to be an exterior bundle of some vector bundle
Given a smooth rank $2^n$ real vector bundle $\pi:E\to M$ over a smooth manifold $M$. I want to determine sufficient conditions for $E$ to be isomorphic to an exterior bundle of some vector bundle. ...
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Intrinsic definition of tensor, tensor density, pseudotensors, even and odd tensors
Motivation
I am trying to figure out what the intrinsic definition of different variants of "tensors" are supposed to be. I managed to find the transformation rules for the coefficients of ...
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$R$-action of a group-ring $\mathbb{Z}[G]$ on the free module $M = \mathbb{Z}e_1 + \mathbb{Z}e_2$ of rank $2$.
Let $R = \mathbb{Z}[G]$ be the group ring of the group $G = \{1, \sigma\}$ of order $2$. Let $$M = \mathbb{Z}e_1 + \mathbb{Z}e_2$$ be
a free $\mathbb{Z}$-module of rank $2$ with basis $e_1$ and $e_2$. ...
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Levi-Civita symbol for cross product generalisation, geometric interpretation, orthogonal subspace
Cross Product in $\mathbb{R}^3$
In $\mathbb{R}^3$ one can compute the coordinates of the cross product between $b_2$ and $b_3$ using the Levi-Civita symbol as follows:
$$b_{1,i_1} = \sum_{i_2i_3} \...
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How to compute exterior derivative of generalized angular form in $\mathbb R^n$
On $\mathbb R^n$, with $m\in\mathbb Z$, I want to calculate the exterior derivative of the angular form
\begin{align}
w=\sum_{i=1}^n(-1)^{i-1}\frac{x_i}{\|x\|^m}dx_1\cdots dx_{i-1}\hspace{0.03cm}\...
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If $M$ is a free $R$-module of rank $n$, then the $i^{\operatorname{th}}$ exterior algebra is free of dimension $n$ choose $i$.
I want to show that
If $M$ is a free $R$-module of rank $n$, for a unital, commutative ring $R$, then the $i^{\operatorname{th}}$ exterior power, $$\bigwedge^{i}(M)$$ is free of dimension $\binom{n}{...
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Higher product rule for graded derivations
If $f^{i_1...i_p}$ and $g$ are smooth functions on some open set $U\subseteq\mathbb R^n$, with the former symmetrically indexed, then we have the higher product rule $$ \sum_{i_\bullet=1}^n\partial_{...
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A vector bundle whose fibers are isomorphic to an exterior algebra of a vector space is an exterior bundle.
Given a smooth vector bundle $\pi: E\to M$ it is a standard result that the exterior vector bundle $\pi':\wedge E\to M$ where $\wedge E= \sqcup_{p}\wedge E_p$ is also a smooth vector bundle. My ...
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When wedge power of a vector nulls
Let $l$ be even and $v \in \bigwedge^\ell \mathbb{C}^{\ell n}$. What conditions on vector $v$ imply $v^{\wedge n} \neq 0$?
Note, that for $\ell = 2$ the criteria is that $v$ must be non-degenerate (...
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Do unit bivectors square to -1 in vector spaces of any dimension?
We know that in a 2D vector space (VS) the unit bivector squares to -1. My question is:
Is the geometric product of a unit bivector with itself equal to -1 in any VS, independently of the VS dimension?...
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Orthogonal complements in exterior powers
Consider the standard induced inner product structure on $\wedge^k\mathbb{R}^d$ given by defining $$\langle u_1\wedge \cdots \wedge u_k, v_1\wedge \cdots \wedge v_k\rangle:=\det ([\langle u_i, v_j\...
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What is the operation between a bivector and a vector that outputs another vector?
I'll start with a physical example. Let's say we have the angular velocity (in 3D euclidean space with orthonormal basis spanning it), which has the bivector representation $$\Omega = \omega_x \mathbf ...
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Resolution for exterior power of a quotient
Let us assume that we have exact sequence of vector spaces:
$$0\to U\to V\to W\to 0.$$
We can think of $0\to U\to V$ as a resolution of $W$.
Can we construct some canonical resolution of $\Lambda^n W$...
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Decompositions of exterior products
Let $V$ be a real vector space equipped with an almost complex structure $J$ and denote the complexification by $V_\mathbb C$. The complexified vector space splits into the eigenspaces $V^{1,0}$ and $...
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Are there other "expressions" of $a \otimes b$ that are isomorphic to $V \wedge V := \textrm{span}\{a \otimes b - b \otimes a\}$? [duplicate]
We usually define $V \wedge V := \textrm{span}\{a \otimes b - b \otimes a; \ a, b \in V\}$. If $V$ is a vector space over $\mathbb{K}$, it's obvious that $k(V \wedge V) \cong V \wedge V$ for any ...
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Equivalence of p-vectors and skew-symmetric tensors
It is said (e.g. Lovelock and Rund) that p-vectors are equivalent to skew-symmetric tensors. However, a skew-symmetric form, such as $A_{ijk}$, is a specific form on $\mathbb{R}^n\times\mathbb{R}^n\...
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Are there Plucker-like relations for the tensor product of two decomposable differential forms?
Let $V$ be a $k$-dimensional vector space, and consider a decomposable tensor in $\bigwedge^\ell V \otimes \bigwedge^m V$ having the form
$$
\mathbf{v} \otimes \mathbf{w} := v_1 \wedge \cdots \wedge ...
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Deriving the general form of the exterior product from the quotient space definition.
In the linear algebra book that I'm reading, the exterior product is defined explicitly as $a \wedge b := a \otimes b - b \otimes a$. Obviously this satisfies the required properties of the exterior ...
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Exterior algebra question
I don't know how to approach this question from Flanders' Differential Forms. I see it was discussed here, but I don't believe that argument is correct and would apply to something like $2v_1 \wedge ...
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exterior derivative of wedge of two one-forms
Let $\alpha, \beta$ be $k$-forms, then, where $d$ is the exterior derivative, we have
$$ d (\alpha \wedge \beta) = d \alpha \wedge \beta + (-1)^k (\alpha \wedge d \beta)$$
However, in many cases, e.g. ...
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Linear dependence and nonzero p-vectors
If a, b, and c are linearly dependent, then it's easy to show that $ a\wedge b\wedge c = 0$. However, if a, b, and c are linearly independent, how do you show that $ a\wedge b\wedge c \ne 0$?
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trace of wedge product and cyclic property [closed]
Surprisingly, while their are similar but more advanced questions on this site, I don't see any answers to the basic version I am asking herein.
If I am taking the trace of a wedge product of matrices,...
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The dot product is the norm of what vector (product)?
Let $a$, $b$, $c$ be vectors in $\mathbb{R}^3$ that form a triangle.
(How this even makes sense formally, I don't know, since in a vector space all vectors are "glued" to the origin. In an ...
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Wedge product defined on not alternating tensors?
I am currently reading Calculus on Manifolds by Spivak. In there, it defined wedge product as follows
To determine the dimensions of $\Lambda^k(V)$, we would like a theorem analogous to Theorem 4-1. ...
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Direct sum of a vector space V and a field R
What does it mean, that the exterior algebra of a vector space V over R is a superset of direct sum of real numbers and real vector space V? I thought that given vector spaces must have trivial ...
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Integral involving Hadamard product with volume element [closed]
Consider the following integral:
$$\int_{\mathbb{R}^{n\times m}}f(X)(H^T(A\odot dX)V)^{\wedge},$$
where $f:\mathbb{R}^{n\times m}\to \mathbb{R},$ $H\in O(n)$, $V\in O(m),$ $A$ is an $n\times m$ matrix ...
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Exterior square of rational function of one variable
Let us consider vector space $$V = (\Lambda^2 (\mathbb C(t))^\times)\otimes_\mathbb Z\mathbb Q.$$
Here I consider the group $A=(\mathbb C(𝑡))^\times$
of non-zero rational functions under ...
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Orientation preserving diffeomorphism $\iff$ positive determinant
I am stuck on a step in a proof, so I will write out the statement and the proof (it is not too long). I would like someone to explain the last 2 steps of the proof, if possible.
Statement: Let $U,V \...
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Why is $d(xdx) = 0$?
If the idea behind the exterior derivative $d$ is that it tells us how quickly a $k$-form changes along every possible direction, why is $d(xdx)=0$ even though $xdx$ varies with $x$?
I understand the ...
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Does an algebra over a $\mathbb{F}_2$ of countable dimension have a grading where each component is finite?
I have a commutative unital $\mathbb{F}_2$-algebra $Q$ of countably infinite dimension, and I want to see if it has a $\mathbb{Z}$-grading where $Q_i=0$ for $i<0$ and $Q_0=\mathbb{F}_2$ and $Q_i$ ...
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Linear transformations who keeps wedge product
We have a $\mathbb K-$v.s. V.
We call a linear transformation $A \in \mathscr L(V)$ keeps the wedge product of two vectors of $V$ $a$ and $b$ ($a\wedge b$) iff $(Aa) \wedge (Ab) = a\wedge b$.
I want ...
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Confusion with covariant derivatives with vielbeins
I have some confusion regarding how the covariant derivative is defined for one forms on a manifold in the context of frames/vielbeins. I am a physics student and my reference is Sec 4.3 of the ...
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Explicit computation of exterior power of vector bundle
I am following the excellent book by Ellinsgrud and Ottem, Introduction to Schemes.
When proving that the $n$-th exterior power of the cotangent bundle of $\mathbb{P}^n$ is $\mathcal{O}_{\mathbb{P}^n}$...
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$ω = dx_1 ∧ dy_1 + dx_2 ∧ dy_2 + dx_3 ∧ dy_3$. Prove: $∧^1 ( \mathbb{R}^6 ) \ni η \to η ∧ ω ∧ ω \in ∧^5(\mathbb{R}^6)^{*}$ is a linear isomorphism.
In $\mathbb{R}^6$ we take variables $x_1, x_2, x_3, y_1, y_2, y_3$ and a bilinear form $\omega \in \wedge^2(\mathbb{R}^6)^{*}$ given by:
$$\omega = dx_1 \wedge dy_1 + dx_2 \wedge dy_2 + dx_3 \wedge ...
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Do all bivectors simplify to 2-blades in seven-dimensional space?
The wedge product of two vectors $\vec{v}, \vec{w}\in\mathbb{R}^{n}$ can be defined as an anti-symmetrized tensor product. In three dimensions, there is a correspondence between the wedge product of ...
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Exterior Power of a Tensor Product in case one is a line bundle
I was reading this thread Exterior power of a tensor product and I found that the result cited in first answer is very useful to me, but I couldn't prove it myself and what's said there is not enough ...
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How big can a wedge of 2-forms be?
The comass of a 2-form $\alpha$ is the maximal value of $\alpha(u,v)$
for a pair of unit vectors $u,v$. The symplectic form $\alpha$ on
$\mathbb R^{2n}$ has the property that $|\alpha^{\wedge n}| = n!...
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Coordinate-free proof of the identity for two times cross product operator
Consider an oriented three-dimensional Euclidean space $V$. Let $[a]$ be the operator of cross product by the vector $a$: $[a] b = a \times b$. It is easy to check that
$$[a]^2 = a \otimes \flat a - \...
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Caulculation of the Supertrace of an operator
I am trying to understand the following statement. We have operators $A_1 ,\ldots , A_k$ for $k\leq d$. Then suppose that
$$
\det (x_1A_1+\cdots +x_d A_d)=a_1x_1^d+\cdots +a_d x^d+\cdots a_{12\cdots d}...
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Explicit calculation of exterior products
Let $E$ be a vector space. Then, for any pair of vectors $e_1, e_2\in E$ the following holds:
$$ e_1\wedge e_2 = - e_2\wedge e_1$$
where $\wedge$ denotes the exterior product. Now, in principle, this ...
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