Skip to main content

Questions tagged [exterior-algebra]

For questions on the exterior algebra, and related concepts such as the wedge product, the tensor algebra and differential forms.

Filter by
Sorted by
Tagged with
2 votes
0 answers
46 views

Hodge star decomposition in non-diagonal manifold product

I'm studying differential forms and I came across the following problem. From what I learnt in another question, when a manifold can be decomposed as $X \times Y$, then the formula found there works ...
Fredrigo6's user avatar
2 votes
1 answer
72 views

Can SAGE help me simplify wedge products of wedge products?

In particular, I have a vector space $V$ with basis $\{e_1, ..., e_n \}$ and I want to "foil" elements of $\bigwedge^k \bigwedge^m V$. For example, given a vector $((e_1 + e_2) \wedge e_3) \...
Chase's user avatar
  • 326
1 vote
0 answers
36 views

Question about moving Hodge $\star$ to the argument of a 1-form

Let $\alpha$ be a 1-form on an $n$-dimensional vector space $V$ and $v_1,...,v_{n-1}$ (1-)vectors in $V$. Is it true that $$ \star \alpha(v_1,...,v_{n-1}) = \alpha\big(\star(v_1 \wedge ... \wedge v_{n-...
Niels Slotboom's user avatar
2 votes
3 answers
474 views

Two definitions of antisymmetrization of a tensor?

I am currently learning about tensors and the exterior product, and I have found some contradictory information. I have seen some sources define the antisymmetrization of a tensor as the following: ...
Christian S.'s user avatar
-1 votes
2 answers
106 views

Confusion about differential forms and integration

I'm self-studying general relativity using Sean M. Carroll's textbook. I recently made it to sections 2.9 and 2.10, which talk about differential forms and integration of functions on manifolds. I ...
Aidan Beecher's user avatar
0 votes
1 answer
32 views

Why is interior multiplication by $v$ an antiderivation implying $v\lrcorner\ \omega ^n = n(v\lrcorner\ \omega)\wedge\omega^{n-1}$?

In Proposition 22.8 of Lee's Introduction to Smooth Manifolds it is written that [For $V$ a $2n$-dimensional vector space, $v$ a vector in $V$ and $\omega$ a degenerate $2$-covector of $V$] interior ...
Sam's user avatar
  • 5,156
0 votes
0 answers
53 views

Confusion over tensor definition of exterior power of a vector space and exterior algebra

I am new to and currently learning about Tensor Algebra and Exterior Algebra. I am confused about the definition of the exterior power of a vector space $V$, $\textstyle \bigwedge^k (V)$, and the ...
Christian S.'s user avatar
0 votes
0 answers
64 views

Wedge Product and Differential Forms, example

Let $x=id_{\mathbb{R}^4}$, $\alpha=dx^1+x_2dx^2\in \Omega^1\mathbb{R}^4$, $\beta=\sin(x_2)dx^1\wedge dx^3+\cos(x_3)dx^2\wedge dx^4\in \Omega^2\mathbb{R}^4$, $h(x_1, x_2, x_3, x_4)=(x_1, x_2, x_3x_4, ...
Lu1998's user avatar
  • 27
0 votes
0 answers
49 views

A Question Regarding Cartan’s Absorption Method

I want to ask a question from the book named “ Cartan for Beginners : Differential Geometry via Moving Frames and Exterior Differential Systems” as to how one can absorb an apparent torsion. Suppose ...
iliTheFallen's user avatar
1 vote
1 answer
79 views

Nice proof that $\text{Alt}$ is natural.

There is a pair of functors $T:\text{kVect}\rightarrow \text{kAlg}$ and $\Lambda:\text{kVect}\rightarrow \text{kAlg}^-$ which are left adjoints to the forgetful functors $U$ (forget the multiplication ...
Wyatt Kuehster's user avatar
0 votes
0 answers
62 views

How to show that a differential form is not exact?

I want to show that the differential form $w = x^2\sin(y) dx \wedge dy + 2x \sqrt{1+y^4} dx \wedge dz \in \Omega(\mathbb{R}^3)$ is not exact. Would it be enough to show that $w$ is not closed? Or does ...
seitanist.snail's user avatar
0 votes
0 answers
24 views

Can the wedge product acting on continuous functions be written differently?

I have $M=\mathbb{R}^2$, $N=\mathbb{R}^3$ $$F(\theta,\phi)=\big((\cos\phi+2)\cos\theta,(\cos\phi+2)\sin\theta,\sin\phi\big)$$ $$\omega=y\text{d}z\wedge\text{d}x$$ Calculating: $$F^*\omega=F^*(y\text{d}...
Superunknown's user avatar
  • 2,973
1 vote
0 answers
49 views

Compute d$\omega$ in Cartestian coordinates for a given $\omega$

Define a $2$-form $\omega$ on $\mathbb{R}^3$ by $$\omega=x\text{d}y\wedge\text{d}z+y\text{d}z\wedge\text{d}x+z\text{d}x\wedge\text{d}y$$ Compute d$\omega$. Using the formula for $$\text{d}\omega=\text{...
Superunknown's user avatar
  • 2,973
2 votes
2 answers
78 views

Do functions distribute over the wedge product?

I am wrapping my head around the arithmetic properties of the wedge product. I understand that constants do distribute over the wedge product, i.e. for $c_1,c_2\in\mathbb{R}$, it holds \begin{equation}...
seitanist.snail's user avatar
4 votes
0 answers
53 views

Can we construct the exterior algebra just from simple multivectors?

$ \newcommand\K{\mathbb K} \newcommand\Ext{\mathop{\textstyle\bigwedge}} \newcommand\Lip{\mathrm{Lip}} \newcommand\ev{\mathrm{ev}} \newcommand\Gr{\mathrm{Gr}} $Let $V$ be a finite-dimensional $\K$-...
Nicholas Todoroff's user avatar
2 votes
1 answer
43 views

Show that the dual Lefschetz operator applied to a two-form $\alpha$ is explicitly given by $\Lambda \alpha =\sum_i \alpha(x_i, y_i)$.

Choose an orthonormal basis $x_1, y_1 = J(x_1), \dots , x_n, y_n = J(x_n)$ of an euclidian vector space $V$ endowed with a compatible almost complex structure $J$. Show that the dual Lefschetz ...
Rene's user avatar
  • 363
2 votes
1 answer
65 views

Wedge product and isomorphism between $\bigwedge^{k}T_{p}^{*}M$ and $\left(\bigwedge^{k}T_{p}M\right)^{*}$

Wikipedia states that there is the following isomorphism, $$ \bigwedge^{k}T_{p}^{*}M \cong \left(\bigwedge^{k}T_{p}M\right)^{*} $$ More concretely, we think of the k-form, $\omega$ as either a linear ...
Jeff's user avatar
  • 895
-3 votes
1 answer
45 views

Is the 'square matrix' in a 2d linear transformation just a bivector? [closed]

$\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}=\begin{pmatrix} b_1 \\ b_2 \end{pmatrix}$ If so, how do we call the operation of ...
user66901's user avatar
0 votes
0 answers
20 views

Exterior powers of Weyl algebra

I'm working with first Weyl algebra $\frac{k[x,y]}{<xy-yx-1>}$and I want to compute its cohomology using Koszul complex, so I need to find its exterior powers as a module over ground field k, ...
VadimStacheff's user avatar
4 votes
3 answers
177 views

What operation on matrices corresponds to the curl of a vector field?

Given the total derivative $Df$ of a (sufficiently) smooth function $f:\mathbb{R}^n \to \mathbb{R}^n$, the trace of the total derivative matrix corresponds to the divergence of $f$ (considered as a ...
hasManyStupidQuestions's user avatar
1 vote
0 answers
15 views

Question on homotopy for Koszul Complex

I am working on illustrating the relationship between the originally defined Koszul complex and its dual version by examining the definitions of the differentials we define for the Koszul complex. For ...
Beginner's user avatar
3 votes
1 answer
176 views

A simple question about the Hodge star

The usual definition of the Hodge star says that it maps $\Lambda^k(V)$ to $\Lambda^{n-k}(V)$ in such a way that for each pair $\omega, \eta \in \Lambda^k(V)$ holds $\omega \wedge *\eta = \langle \...
tsnao's user avatar
  • 320
0 votes
0 answers
23 views

Linear relations between minors of a matrix

Let $X = (x_{ij})_{1 \leq i, j \leq n}$ by an $n \times n$-matrix over the field $Q = \mathbf{F}(x_{ij})_{ij}$ for some field $\mathbf{F}$. Let $[n] = \{1,\dotsc,n\}$. For subsets $I, J \subseteq [n]$,...
Bubaya's user avatar
  • 2,244
0 votes
1 answer
66 views

Dot product of two exterior products and associativity of geometric product?

This is a quick and basic question. I looked online (Wikipedia articles, Wolfram, etc..., and poked inside of Hestenes and Snygg's books, but couldn't easily pull out an answer). I'm going to define ...
Nate's user avatar
  • 894
2 votes
1 answer
118 views

Prove that $d_3d_3^*+d_3^*d_3=-\nabla^2$

Consider the geometry in $\mathbb R^3$, define $$d_3=dx\frac{\partial}{\partial x}+dy\frac{\partial}{\partial y}+dz\frac{\partial}{\partial z}.$$ We then define the Hodge star operator $*_3:\Omega^p(\...
Ho-Oh's user avatar
  • 919
2 votes
1 answer
43 views

What's the definition of dual number at perspect of exterior algebra?

In Dual Number it said that "It may also be defined as the exterior algebra of a one-dimensional vector space with $\varepsilon$ as its basis element." But I can't find the detailed rigorous ...
Richard Mahler's user avatar
0 votes
1 answer
53 views

Prove that the isotopy generated by a time-dependent symplectic vector field is a symplectomorphism

Let $M$ a compact and connected smooth manifold. Suppose $X_t$ is a time-dependent symplectic vector field and let $\phi_t$ be the isotopy generated by $X_t$. Prove that $\phi_t ∈ Symp(M, \omega)$ for ...
some_math_guy's user avatar
0 votes
0 answers
24 views

How to determine that the quotient of the top power of two $2$-forms is $\leq 1$

For full context, I work on a complex manifold $M$ of dimension $n$ where I have a Hermitian $(1,1)$-form $\Omega$ and $(1,1)$-form $\tau$ which is negative semidefinite at a point. I am trying to see ...
rosecabbage's user avatar
  • 1,697
5 votes
1 answer
81 views

Kernel of the action of GL(V) on exterior square of V

I wonder whether anyone knows a reference for the following result? I can give a shortish proof, but would prefer to cite the literature if possible. Theorem Let $V=F^n$ be an $n$-dimensional $F$-...
Glasby's user avatar
  • 544
0 votes
1 answer
75 views

Simplifying $d(\frac{2y_1}{r^2+1})\wedge d(\frac{2y_2}{r^2+1})\wedge\cdots\wedge d(\frac{2y_n}{r^2+1})$, where $r^2=\sum y_i^2$

I've been trying to find a nice, simple form for the following expression: $$d\left(\frac{2y_1}{r^2+1} \right)\wedge d\left(\frac{2y_2}{r^2+1} \right)\wedge ... \wedge d\left(\frac{2y_n}{r^2+1} \right)...
user13121312's user avatar
0 votes
0 answers
55 views

Contraction of the determinant

In $\mathbb{R}^{n+1}$, we consider an orthonormal basis $e_0,\dots,e_n$ and $\alpha_0,\dots,\alpha_n$ its dual basis. Then, the determinant is $\alpha_0 \wedge \cdots \wedge \alpha_n$. I am seeking a ...
JulianDoyle's user avatar
0 votes
1 answer
74 views

Natural isomorphism between tensor product and exterior product

I am requesting help with the following problem. Below all rings are commutative with unit, and for a ring $R$, we define an $R$-algebra to be a ring $R'$ with ring homomorphism $f: R \to R'$. This ...
Abced Decba's user avatar
1 vote
1 answer
101 views

$\mathbb{C}-$linear extension of 2-forms to (1,1)-forms

I am trying to analyze a bit how we can extend a differential form to the complexification $V\otimes\mathbb{C}=V_{\mathbb{C}}$ of the vector space. Of course, you can do this for a general $k$-form, ...
領域展開's user avatar
  • 2,397
1 vote
0 answers
22 views

Two view points of exterior algebra - k-vectors and tangent spaces

I'm trying to see how two ideas are related: In (Discrete) Exterior Calculus, we defined k-vectors as volumes, and that k-forms are like "measurement tools", that allow us to measure k-...
blz's user avatar
  • 615
3 votes
1 answer
69 views

L'Hopital's rule with dual numbers

Background: For the dual numbers, we extend the reals with an additional unit vector $\epsilon$ subject to the constraint that $\epsilon^2 = 0$. We can write dual numbers as $x_0 + x_1 \epsilon$ for $...
kc9jud's user avatar
  • 248
3 votes
1 answer
149 views

Hodge star operator evaluated at an orthonormal basis (Proposition 1.2.20 Huybrechts)

We want to show: $$*(e_{i_1} \wedge\ldots \wedge e_{i_k})=\epsilon e_{j_1} \wedge\ldots \wedge e_{j_{n-k}}$$ where $\epsilon$ is the sign of the permutation $i_1 \cdots i_k \bar i_1 \cdots \bar i_{n-k}...
領域展開's user avatar
  • 2,397
0 votes
1 answer
52 views

Fundamental form $\omega=\sum_{i\leq m}v^*_i\wedge (Jv_i)^* $ with a complex structure $J$

Let $V$ be a $\mathbb{C}-$ vector space, $J$ an almost complex structure on $V$ and take a real orthonormal basis $\langle v_1,Jv_1,\ldots,v_n,Jv_n\rangle $ with a scalar product $\langle,\rangle = \...
領域展開's user avatar
  • 2,397
0 votes
1 answer
61 views

Interior, cross and outer products between two multivectors?

For two arbitrary multivectors $\mathbf u$ and $\mathbf v$, what are the definitions of the interior (or scalar) product $\mathbf u\cdot \mathbf v$, the cross product $\mathbf u\times \mathbf v$ (if ...
HelloGoodbye's user avatar
0 votes
0 answers
54 views

Exterior algebra decomposition natural projections $ Π^{p,q}:\bigwedge {V}^*_{\mathbb{C}} \to \bigwedge^{p,q}V$ (Huybrechts book)

Let $\bigwedge^{p,q}V:=\bigwedge^{p}V^{1,0}\wedge \bigwedge^{p}V^{0,1}$ and $\bigwedge {V}^*_{\mathbb{C}}=\bigoplus_k \bigwedge^{k} V_{\mathbb{C}} $. Then one defines the natural projection $$ Π^{p,q}...
領域展開's user avatar
  • 2,397
0 votes
0 answers
39 views

A question on alternating product and $SL_n(R)$

I am reading Fulton and Harris representation theory. In the section 8.2 - Examples of Lie Algebras, while calculating the Lie algebra of $SL_n(\mathbb R),$ the author tell that by definition $$ A_t(...
Eloon_Mask_P's user avatar
2 votes
1 answer
35 views

Minors of change of basis matrix are Kronecker deltas

All the vector spaces I will consider in the following could be thought to have a base field of characteristic $ 0 $ (just to be safe). I know that if $ A\colon V\to W $ is a linear mapping between to ...
GeometriaDifferenziale's user avatar
5 votes
1 answer
205 views

Alternative of exterior power as a tensor algebra

This question is related to my previous question. The $n$-th exterior power of a vector space (over some field of characteristic zero) $U$ is a pair $(\wedge^{n}, \bigwedge^{n}U)$ where $\bigwedge^{n}...
Idontgetit's user avatar
  • 1,901
0 votes
1 answer
91 views

Does this alternating map satisfies universal property?

In what follows, $U$ is a complex vector space. I have the following abstract definition. Definition: Let $\wedge^{n}: \overbrace{U \times \cdots \times U}^{\text{$n$ times}} \to V$ be an alternating $...
Idontgetit's user avatar
  • 1,901
0 votes
0 answers
83 views

Wedge product of two functions

This is what I understand in the case of finite dimensions: Consider $1-$vectors in $R^2$. Given two $1-$vectors, $v_1$ and $v_2$, we can form a wedge product $v_1 \wedge v_2$, and if we are given $...
user2167741's user avatar
0 votes
0 answers
32 views

Does the Hessian correspond to the exterior derivative of the gradient 1-form? Or does its skew-symmetrization?

Question: Given a twice totally differentiable (not necessarily $C^2$) function $f: \mathbb{R}^m \to \mathbb{R}^n$, do its $n$ Hessian matrices correspond to the exterior derivatives of its $n$ ...
hasManyStupidQuestions's user avatar
2 votes
1 answer
76 views

Computing wedge product and exterior differential

I am studying for first time Smooth manifolds and I have some issues understunding the wedge product and the exterior differential since my teacher does not provide examples. For example, one of my ...
Daniel García's user avatar
1 vote
0 answers
92 views

Abstract and concrete wedge product

Some notation Let $ V $ be a finite dimensional vector space (say, over the real numbers). Let's suppose that all we know about the exterior power $ \bigwedge^k V^* $ of the dual space $ V^* $ of $ V $...
GeometriaDifferenziale's user avatar
0 votes
1 answer
67 views

How to define the determinant in a basis-independent form using alternate maps, without using properties of the „usual” determinant?

Let $V$ be a vector space of dimension $n$ over some field. Let $Alt_k(V)$ be the set of $k$-multilinear alternate maps. My question boils down to this, I think (although I will provide more context ...
rosecabbage's user avatar
  • 1,697
1 vote
1 answer
115 views

Riemannian geometry, manifolds and volume elements

I have two quesitons about a book by Nakahara: Geometry,topology and physics In the snippet below how do I compute that $$|\det(\frac{\partial x^\mu}{\partial y^\kappa}\frac{\partial x^\nu}{\partial y^...
user122424's user avatar
  • 3,978
1 vote
2 answers
99 views

What is the Grassmann algebra of $2\times 2$ complex matrices?

The traceless hermitian $2\times 2$ complex matrices form a real Euclidean space $\mathfrak E_3$ with dot product $a\cdot b:=\frac{1}{2}(ab+ba)/I$ where $I$ is the $2\times 2$ identity matrix. A ...
mma's user avatar
  • 2,065

1
2 3 4 5
26