# Tagged Questions

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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### Does the Hodge dual (star) operator make the exterior algebra an involutive (*-) algebra?

Does the Hodge dual (star) operator make the exterior algebra an involutive (*-) algebra? https://en.m.wikipedia.org/wiki/Hodge_dual https://en.m.wikipedia.org/wiki/*-algebra This would seem to be a ...
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### A possible generalized determinant?

This will likely seem a bit contrived, and admittedly it is, but I wanted to see just how "close" we could get to generalizing the concept of a determinant. In what follows, we will lose quite a few ...
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### Simple properties of wedge product [closed]

How to prove a) $\omega \wedge \eta =(-1)^{kl}\eta\wedge\omega, \omega$ is $k$-tensor and $\eta$ is $l$-tensor. b)$f^*(\omega \wedge \eta)=f^*(\omega)\wedge f^*(\eta)$ where $f:V\rightarrow W$ ...
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### Relation between inner product and wedge product

$M$ is an Riemannian n-manifold, $p\in M$. $x, y\in T_pM$. Is it true that $\langle x,x \rangle \langle y, y \rangle-\langle x, y \rangle^2=|x \wedge y|^2$? I'm reading Do Carmo's Riemannian ...
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### Why this map implies the decomposition of the tensor product space?

Let $V$ be a vector space and consider the tensor product of $V$ with itself, that is $V\otimes V$. Define $\alpha' : V\times V\to V\otimes V$ by $$\alpha'(v,w)=w\otimes v.$$ In that case, $\alpha'$ ...
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### A question about the definition of the exterior product

Let $T\in\wedge^p(V^*)$ and $S\in \wedge^q(V^*)$. Then $T\wedge S$ is defined to be $\text{Alt}(T\otimes S)$ How is $\text{Alt}(T\otimes S)$ defined? I know that $\text{Alt}(T)$ is defined as an ...
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### Restriction of an $n$-form to a submanifold

I am currently studying exterior calculus on manifolds and am not sure if I understand things correctly. In my textbook (R.W.R. Darling, Differential Forms and Connections) there is an example of a $2$...
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### When do exterior and tensor algebras commute with dual spaces?

Suppose $V$ is a vector space, and $V^*$ is its dual space. Furthermore, let $\Lambda(V)$ be the exterior algebra of $V$, and let $T(V)$ be the tensor algebra. When do the following two statements ...
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### What is the motivation for wanting to remove redundant terms that arise from the wedge product of two multilinear functions?

I am currently working through Loring Tu's "An Introduction to Manifolds," specifically the section in which the exterior algebra of multicovectors is introduced, and I am having trouble grasping the ...
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### Is this an alternate characterization of $\lambda$-rings? Or, what is like a $\lambda$-ring but for symmetric rather than exterior powers?

This is a question about $\lambda$-rings. A $\lambda$-ring is a commutative ring together with operations $\lambda^n$ for each whole number $n$ which are analogous to the $n$th exterior power and ...
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### Exterior algebra subspace of all grade-n wedge products of a vector

Let $V$ be a finite-dimensional vector space, and let $\Lambda(V)$ be its exterior algebra. Then if $S_k = \text{span}(k_1,k_2,...,k_n)$ and $\hat k = k_1 \wedge k_2 \wedge ... \wedge k_n$, there is ...
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### Noncanonical isomorphism of spaces of differential forms

Let $\pi: V \to M$ be a smooth $n$-dimensional vector bundle over $M$. Are the spaces of differential forms $\Omega^i(V)$, $\Omega^i(V^*)$ noncanonically isomorphic? If so, how do I see this? Is there ...
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### Frank Warner's definition of the Hodge star

Frank Warner's book, chapter 2, excercise 13 states the following: If $V$ is an oriented inner product space ($n$ dimensional) there is a linear map $\ast \colon \Lambda (V) \to \Lambda (V)$, ...
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### Parameterizing the set of subquotients of a Hilbert space

If you want to parameterize the set of subspaces of a finite-dimensional Hilbert space $V$, and naturally induce a norm derived from $V$ on the resulting moduli space, the classical approach is to ...
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### Prove that $p-$form $\omega$ on $M\times N$ is $\delta \pi(\alpha)$ iff $i(X)\omega=0$ and $L_X\omega=0$

Let $M$ be connected and let $\pi:M\times N \rightarrow N$ be the natural projection. Prove that $p-$form $\omega$ on $M\times N$ is $\delta \pi(\alpha)$ for some $p-$form $\alpha$ on $N$ if and only ...
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### Alternating k-tensors and determinant (theorem 4-6, Spivak)

I am having troubles understanding this proof. I don't understand how, if $\eta \in \Lambda^n(\mathbb{R^n})$ implies that $\eta = \lambda \cdot \det$ - really I don't see how determinants come from ...
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### Unifying Transformations in Complex 3-Space

I am currently researching the vector space $\mathbb{C}^{3}$ and I was wondering if it is possible to generate a scheme of unifying the rigid transformations in $\mathbb{C}^{3}$. I know that in the ...
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### Anisotropic scaling in geometric/Clifford algebra

Take the geometric algebra over $\Bbb R^n$. Suppose we have a blade multivector in this algebra. Now we want to anisotropically scale this multivector. Is there a general closed-form expression for ...
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### Computing $\alpha^*w$ in general

Let $A$ be open in $R^k$, let $\alpha : A \rightarrow R^n$ be of class $C^\infty$. Let $x$ denote the general point of $R^k$, let $y$ denote the general point of $R^n$ If $I = (i_i, ...,i_l)$ is an ...
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### Bases and wedge product

Let $\{v_1,\dots, v_r\}$ and $\{w_1,\dots, w_r\}$ be linearly independent sets of a vector space $V$. If $v_1 \wedge \dots \wedge v_r =c w_1 \wedge \dots \wedge w_r$ for some nonzero number $c$, ...