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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The different products on $k$-alternating forms and their relationship with the exterior product

Let $V$ be a vector space and $k\in \mathbb{N}$. Denote $\Lambda^k V$ the exterior $k $-power of $V$. Let $f:\Lambda^k V^*\to (\Lambda^k V)^*$ be the map such that a $k$-covector $\eta_1\wedge \...
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Degree part of exterior algebra

I have a graded vector space $V$ and exterior algebra $\bigwedge V$.Suppose further that $V^0=V^1=0$. I don't understand why $(\wedge V)^2=V^2,(\wedge V)^3=V^3$ and $(\wedge V)^4=P^2V^2$. notation: $...
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Degree of elements in exterior algebra

How one determines the degree of an element in the exterior algebra $\bigwedge V$, for a graded vector space V. e.g.Is it true that $\wedge^n V^2=0$, as elements of $V^2$ are of degree 2? In ...
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Exterior algebra of a direct sum

Can someone provide a formula for $\bigwedge( V_1\oplus V_2\oplus V_3\oplus..)$. The formula for two summands is $\bigwedge^1 (V_1\oplus V_2)=(\wedge^1V_1\otimes \wedge^0V_2) \oplus (\wedge^0V_1\...
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Exterior algebra over graded space

I can't understand how to compute exterior powers in the exterior algebra $\bigwedge V$ where $V$ is a graded vector space. I know for a single (not graded), $k$-vector space, that $\bigwedge V=\...
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Calculating exterior powers

I am confused with an example I found, which says that for a given graded vector space $V$ such that $V^0=0$ and $V^1=0$, then $(\bigwedge V)^2=\bigwedge^2V =V^2$. Two questions: 1.$V^0$ is not the ...
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Exterior derivative independence from coordinate systems

In the book Mathematical Methods of Classical Mechanics by V.I. Arnold, the author introduces (p.189) the concept of exterior derivative as "the principal linear part of the increment" of the function ...
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Is every rotationally-invariant $k$-covector zero?

Let $V$ be a real $n$-dimensional oriented inner product space, and let $1 \le k < n$. I am trying to find different simple proofs for the following claim: There is no non-zero $\omega \in \...
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Does the kernel of every alternating form contain a decomposable element?

Let $V$ be a real $n$-dimensional vector space, and let $1 < k < n$. Let $\alpha \in \bigwedge^k (V^*) \cong (\bigwedge^k V)^*$. Thinking of $\alpha$ as a linear functional $\bigwedge^k V \to \...
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Confusion with exterior algebra

I have a graded vector space $V$ such that $V=V^2\bigoplus V^3$, i.e $V^k=0\quad \forall k\neq2,3.$ Now I can't understand what is $\bigwedge V^{\leq 2}$. Is it $\bigwedge V^{\leq 2}=\bigwedge^0 V^...
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Sullivan minimal models

I am confused with a notation for the minimality of Sullivan model. In particular in some books it's written that $d(V)\subset \bigwedge^{\geq2}V$ and in others it's written that $Im(d)\subset \...
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Prove a change of coordinates is symplectic

I have to find the value $\alpha\in\mathbb{R}$ such that the following change of coordinates is symplectic: $ \varphi(p,q)\rightarrow (P,Q)$ where $Q = q^2 + \alpha\sqrt{q^2+p}$ $P = q + \sqrt{q^2+...
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Bilinear Map on Exterior Algebra That Gives Determinant

How do I see that for a $K$-vector space $V$ the map $\bigwedge^d(V^*) \times \bigwedge^d(V) \rightarrow K, (f_1 \wedge ... \wedge f_d, x_1 \wedge ... \wedge x_d) \mapsto det(f_i(x_i)_{i,j})$ is ...
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How many matrix minors determine all the minors?

Let $n$ be a positive integer, and let $1<k<n$. Suppose we have an "unknown" real $n \times n$ matrix $A$. (we do not know the entries of $A$). Can we recover all the $k$-minors of $A$ from ...
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( Proof Explanation ) Modified Euler scheme preserves the weighted area $ dx \wedge dy/xy$

I already showed that the Lotka Volterra Equations preserve the weighted area $ dx \wedge dy/xy$ see here. Now I need that a modification of the Forward Euler Method preserves the same weighted area. ...
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Equivalence of area forms on the sphere

This is Problem 23.5 from Tu's An Introduction to Manifolds. Prove that the area form $\omega$ on $S^2$ in Example 23.11 is equal to the orientation form $$x dy \wedge dz - y dx \wedge dz + z dx \...
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Elementary Question on Sections of $\Lambda^2(X \times Y)$

Let $X$ and $Y$ be complex manifolds with local coordinates $(x_1, ..., x_n)$ and $(y_1, ..., y_m)$, respectively. The exterior algebra $\Lambda^2(X \times Y)$ decomposes as $$\Lambda^2(X \times Y) = \...
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Determinant and wedge product confusion

Beginner with differential forms – please go easy. I'm trying to understand how the wedge product can be used to define the determinant. In Lee's Introduction to Smooth Manifolds (bottom of page 210) ...
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1answer
94 views

problem with Exterior Derivative

I have a 3-manifold with Riemannian metric $g=\omega_1^2+\omega_2^2+\omega_3^2$, where $\omega_i$ are 1-forms (coframe fields). I am in this situation: $- (u^{-1}p+\frac{3}{2}u^{-1})du \wedge \...
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Are vector bundles with isomorphic determinant bundles isomorphic?

Let $A$ and $B$ be $2n$-dimensional complex vector bundles and $\det A=\Lambda^{2n}(A)$ and $\det B=\Lambda^{2n}(B)$. Can you prove $A\cong B $ if and only if $\det A\cong \det B $? Is it a correct ...
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Simple computation with the $n$-form $dz_1\wedge…\wedge dz_n$ in $\mathbb{C}^n$

Let $z_j=x_i+iy_j$ be the coordinates for $\mathbb{C}^n$ and consider the $n$-form $\eta:=dz_1\wedge...\wedge dz_n$. I've just read the following (the contex is probably not important): Let $N\...
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Results from wedge product [closed]

Please let me know the following results $(a\wedge b).c$ , where $\wedge$ is the $\wedge$-product Also the following one $(a\wedge b).(c\wedge d)$
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When does $v_0\wedge\dots\wedge v_{k-1}=0$ when working over a ring that's not a field?

Let $M$ be a module over a commutative ring $R$, and let $v_0,\dots,v_{k-1}$ be elements of $M$. If $R$ is a field then $v_0\wedge\dots\wedge v_{k-1}$ is equal to $0$ if and only if $v_0,\dots,v_{k-1}$...
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1answer
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Isomorphism between exterior algebras

Let V is a m-dimensional vector space and $V^{*}$ is dual vector space. How can define isomorphism between exterior algebra $Λ(V)$ and exterior algebra $Λ(V^{*})$ with use a volume element $f\in Λ^{m}...
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What is the covariant derivative of a wedge product?

If we have a covariant derivative for vector fields given by an affine connection on a manifold, can we extend that to a covariant derivative for k-vectors by assuming that the product rule hold for ...
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1answer
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Show that the wedge product $ dX \wedge dX = 0 $ and $dY \wedge dY = 0$

So first I want to give you some background information: begin of the background information I'm currently reading an abstact about the Lotka Volterra differential equations: $$ x^{'} = x -xy $$ $$ ...
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1answer
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Clifford algebra and exterior algebra

Let $E$ be a finite dimensional real vector space with $E^*$ its dual, and let $\langle \; , \; \rangle$ be an inner product on $E$. For any $e \in E$, denote by $e^* = \langle e, \; \rangle \in E^*$ ...
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$\ell_1$ norm of multivector in exterior algebra

Suppose you have a set of $n$ linearly independent vectors $v_1, v_2, ..., v_n$. Then we can call their wedge product $W = v_1 \wedge v_2 \wedge ... \wedge v_n$. The $\ell_2$ norm $\|W\|_2$ is equal ...
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What “cross products” do I need to find the volume of a cuboid?

In two dimensions, one can find the area for a quadrilateral by calculating two "cross products". If the vertices of the quadrilateral are $a, b, c, d$ clock-wise, consider the vectors $A = \vec{ab}$,...
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1answer
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Inner product on diffential forms independence ortonormal basis

Suppose $\{e_1,...,e_n\}$ is a positive orthonormal basis for the tangent space at a point $p$ in an oriented n-manifold $M$, then define the inner product on $\Omega^k(M)$, for each $k$, by: $$\...
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Minimal embedding of the Grassmannian into Euclidean (or projective) space

Let $Grass(r,k)$ be the set of all $r$-dimensional subspaces of $\Bbb R^k$. It is well known that $Grass(r,k)$ embeds isometrically as a projective variety into the projectivization of the r'th power ...
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Inner product of $k$-forms

I'm working on the following problem from Lee's Introduction to Riemannian Manifolds: Let $(M,g)$ be a Riemannian $n$-manifold. show that for each $k=1,\ldots, n$, there is a unique fiber ...
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1answer
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Clarifying the definition of Wedge Product — a detail about factorial prefactor

Munkres book on Manifolds constructs a wedge product by defining the following sum on $f$ (an alternating $k$-tensor on $V$) and $g$ (an alternating $l$-tensor on $V$): $$(f \wedge g)(v_1,...,v_{k+l}) ...
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exterior algebra of representations of $S_n$

$$ \begin{aligned} \Lambda^{r} \mathbb{C}^{n} &=\bigoplus_{i=0}^{r}\left(\Lambda^{(r-i)} V \otimes \Lambda^{i} U\right) \\ &=\left(\Lambda^{r} V \otimes U\right) \oplus\left(\Lambda^{r-1} V \...
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1answer
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Decomposition of permutations and wedge products.

Let $V$ be an $\mathbb{R}$-vector space. Denote the space of all alternating $k$-linear forms from $V^k$ to $\mathbb{R}$ by ${\cal A}_k(V, \mathbb{R})$ Suppose $f\in{\cal A}_p(V, \mathbb{R})$ and $g\...
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Grassman numbers in physics, what are they?

Ok, so I asked a question on physics stack exchange about Grassman numbers used in quantum field theory. In physics books, they are introduced as "numbers" satisfying alternativity: $\chi ^2=0$, and ...
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1answer
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Questions on Cartan's magic formula $\mathcal{L}_X=i_X \circ d + d\circ i_X$

Algebra $A$ is called graded algebra if it has a direct sum decomposition $A=\bigoplus_{k\in\Bbb Z} A^k$ s.t. product satisfies $(A^k)(A^l)\subseteq(A^{k+l}) \text{ for each } k, l.$ A ...
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Basis for exterior algebra (infinite dimensional)

Let $\{e_i\}_{i\in I}$ be a basis for V where $I$ is some totally ordered indexing set. Fix $k\in \mathbb{Z}^{\geq 0}$. Do we get an induced basis on $\bigwedge^k V$, where $$\{e_1\wedge...\wedge e_k|...
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1answer
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Write a bivector as the exterior product of two vectors

The Wikipedia article https://en.wikipedia.org/wiki/Bivector#Simple_bivectors states that "A bivector that can be written as the exterior product of two vectors is simple. In two and three ...
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1answer
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Use abstract index notation to prove Leibniz rule for exterior derivative

I want to use abstract index notation to prove Leibniz rule for exterior derivative of wedge product: For $\omega\in \Omega^k(U),\eta\in\Omega^l(U)$, d$(\omega\wedge\eta)=\text{d}\omega\wedge\eta +(...
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1answer
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How to express $\omega$ as linear combination of wedge products?

I know that $$\omega \begin{bmatrix}a\\b\\c\end{bmatrix},\begin{bmatrix}d\\e\\f\end{bmatrix}=3ae-2af-3bd+4bf+2cd-4ce$$ How would I write this as a linear combination of: $dx\wedge dy$, $dx\wedge dz$, ...
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Solution of a wedge equation

Let $\Omega\subset \mathbb{C}$ be an open subset, $f\in C^\infty (\Omega)$ and $\alpha(z)= df= \partial_x f\, dx + \partial_y f\, dy$ be such that $\alpha(z)\neq 0,\forall\,\, z\in \Omega$. Let $\...
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1answer
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Disproving Submersion

Q.2 in the text. By the hint I have shown that it has 2 tangent directions. Now how does it follows that there is no submersion?
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1answer
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The kernel of $\bigwedge^r(T): \bigwedge^r(W)\to \bigwedge^r(V)$?

The full problem is: Given $T: W\to V$, a linear transformation of $F$-vector spaces, such that $\text{ker} T = 0$, show that the kernel of $\bigwedge^r(T): \bigwedge^r(W)\to \bigwedge^r(V)$ is ...
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Why is it that $\displaystyle\bigoplus_{k=0}^\infty A_k(V)=\bigoplus_{k=0}^{\dim V}A_k(V)$?

In W. Tu's An Introduction to Manifolds, the following definition is given: For a finite-dimensional vector space $V$, say of dimension $n$, define $$A_*(V)=\bigoplus_{k=0}^\infty A_k(V)=\...
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1answer
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Question on the coefficient in the definition of wedge product

In my textbook, W. Tu's An Introduction to Manifolds (page 26), the wedge product is defined to be After the definition, the following explanation regarding the coefficient in the definition, namely $...
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A way to define the wedge product

There are various equivalent ways of introducing the wedge product. One possibility is to first define $\,\wedge\,$ for basis forms, $$ e^{i_1}\,\wedge .\,.\,. \wedge\,e^{i_r}\;\equiv\;r!\,\left[\,e^...
3
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1answer
44 views

Does every eigenspace of the exterior power $\bigwedge^k A$ corresponds to an invariant subspace?

Let $V$ be an $n$-dimensional real vector space, and let $1<k<n$ be fixed. Given an automorphism $A \in \text{GL}(V)$, consider its $k$-th exterior power $\bigwedge^k A \in \text{GL}(V)$. ...
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Abtract explanation for $\det(A+B) = ∑_{k=0}^n \langle \Lambda^k A, \Lambda^{n-k} B \rangle$

Let $A$ and $B$ be $n×n$ matrices. If we expand the determinant of $A+B$ as a sum over all permutations of $[\![1,n]\!]$ and all choices of whether the coefficient comes from $A$ or from $B$, this ...
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2answers
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Are standard bases for the exterior power essentially unique?

Let $V$ be a real $d$-dimensional vector space, and let $1<k<d$ be fixed. Let $v_i$ be a basis for $V$. Consider the induced basis for the $k$-th exterior power $\bigwedge^k V$, given by $v_{i_1}...