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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Does existence of a parallel section on $E$ imply local existence of a parallel section on $\bigwedge^k E$?

Let $E$ be a smooth vector bundle equipped with an affine connection $\nabla$. Suppose that $(E,\nabla)$ admits a non-zero parallel section. I think that $(\bigwedge^k E,\bigwedge^k \nabla)$ does ...
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Is there a natural way to view $\bigwedge^k_{\mathbb{C}}\mathbb{C}^d$ as a subspace of $\bigwedge^k_{\mathbb{R}}\mathbb{C}^d$

Does there exists an $\mathbb R$-linear embedding $\bigwedge^k_{\mathbb{C}}\mathbb{C}^d \to \bigwedge^k_{\mathbb{R}}\mathbb{C}^d$ which maps decomposable tensors to decomposable tensors? (The ...
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Can we “mod out” a common subspace in the Grassmannian inside the exterior algebra?

While reading this paper, I have seen the following claim stated without a proof: Let $V$ be an $n$-dimensional vector space over a field, and let $\alpha,\beta \in \bigwedge^k V$ be decomposable ...
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Is “being decomposable” preserved under taking a subspace?

Let $V$ be a vector space over some field, and $W \le V$ a vector subspace. Let $1<k<\dim V$ be an integer. Suppose $\omega \in \bigwedge^k W$ is decomposable as an element in $\bigwedge^k V$....
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If $\omega \in \bigwedge^k\mathbb{R}^d$ is decomposable over $\mathbb C$, is it decomposable over $\mathbb R$?

Let $k,d$ be positive integers, and let $\omega \in \bigwedge^k\mathbb{R}^d$ be decomposable in $\bigwedge^k\mathbb{C}^d$. Is $\omega$ decomposable in $\bigwedge^k\mathbb{R}^d$? Edit: Let me be ...
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Is the Hodge dual the unique map which commutes with exterior powers of isometries?

Let $V$ be a real oriented $d$-dimensional inner product space, $d \ge 3$. For $1 \le k \le d-1$, the Hodge dual map $\star: \bigwedge^k V \to \bigwedge^{d-k} V$ commutes with orientation-preserving ...
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The Jacobian and Particular Solutions to an Underdetermined Equation

I was wondering if the factor $\sqrt{(x')^2 + (y')^2}$ in the line integral formula $$\int_a^b\ f(x,\ y)\ \sqrt{(x')^2 + (y')^2}\ dt$$ can also be thought of as a Jacobian determinant, due to the ...
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Do complexification and exterior power commute?

Let $V$ be a $d$-dimensional real vector space, and let $1<k<d$. Are $(\bigwedge^k V)^{\mathbb{C}}$ and $\bigwedge^k (V^{\mathbb{C}})$ naturally isomorphic? They both have the same complex ...
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In which degrees there exist non-decomposable elements in the exterior algebra?

I am trying to get a better understanding of the concept "decomposable" element in an exterior algebra. Let $V$ be a $d$-dimensional real vector space, and let $1<k<d$. For which tuples $(k,d)$,...
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A characterization of the subgroup of $\text{GL}(\bigwedge^k V)$ which preserves pure tensors?

Let $V$ be a $d$-dimensional real vector space, and let $1<k<d$. Set $$H=\{B\in\text{GL}(\bigwedge^k V) \, | \, B \,\text{ preserves pure tensors }\}$$ (i.e. $B \in H$ if it maps ...
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Leibniz rule for exterior derivative of wedge product

I'm trying to show $\text{d}(\alpha\wedge\gamma)=\text{d}\alpha\wedge\gamma+(-1)^p\alpha\wedge \text{d}\gamma$ for all $p$-forms $\alpha$ and $\gamma$, $\text{d}$ is exterior derivative. I want to ...
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Identifying a wedge-to-metric formula

In this question, the original poster wrote: On every Riemannian manifold $M$, we can consider the Hodge $*$-operator, which is characterised by the following formula: $$a\wedge *b = (a,b)\nu.$$...
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Equivalence of definition of symplectic form

Suppose that $V$ is a vector space of dimension $2n$, and let $\omega \in \Lambda^2(V)$. Prove that the following two statements are equivalent. (1) $\tilde{\omega} : V \rightarrow V^*$ defined ...
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Trouble in finding the Galois group of the covering spaces of $S^1 \vee S^1$.

I am studying covering spaces and deck transformations from the book Algebraic Topology written by Allen Hatcher. While reading deck transformations I came across the concept of Galois group of ...
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This is my first time posting so I apologise for the stupid question. I am trying to understand the idea of exterior algebra as a quotient algebra. Let $T(V)$ be the tensor algebra of a vector space $... 0answers 64 views Exterior Derivative over Quaternions I was wondering if it is possible to define the exterior derivative of a quaternionic valued function. I am doing the quaternionic analogue of a previously complex valued computation, namely something ... 1answer 82 views Are all bivectors in three dimensions simple? I want to show that all bivectors in three dimensions are simple. If I understand correctly, a bivector is simply an element from the two-fold exterior product$\bigwedge^2V$of a vector space$V$, ... 0answers 107 views How to calculate the wedge product of differential forms with arbitrary coefficients I need to calculate the wedge product between some differential forms of the type:$\omega=P_1(x_1, ..., x_n)dx_1+\cdots+P_n(x_1, ..., x_n) dx_n$and$d\omega$, i-e,$\omega\wedge d\omega$. where ... 1answer 33 views Exterior algebra: If$\alpha \land \beta =0$for all$\beta \in \Lambda ^{n-k} V$then$\alpha =0$. How do I prove the following: If$V$is$n$-dimensional, and$\alpha \in \Lambda ^{k} V$, if$\alpha \land \beta =0$for all$\beta \in \Lambda ^{n-k}$then$\alpha =0$. For$k=1$, then we can form ... 1answer 88 views One-dimensional null space for a 2-form Consider the following 2-form on$\mathbb R^{2n+1}$with coordinates$x_1...x_n;y_1...y_n;t$: $$\omega^2=\sum dx_i \land dy_i-\omega^1 \land dt$$ where$\omega^1$is any 1-form on$\mathbb R^{2n+1}$. ... 1answer 71 views Showing that a 2-form on an odd dimensional space is not degenerate On an odd-dimensional space$\mathbb R^{2n+1}$with coordinates$x_1...x_n;y_1...y_n;t$consider the following 2-form: $$\omega^2=\sum dx_i \land dy_i-\omega^1 \land dt$$ where$\omega^1$is any 1-... 1answer 164 views Lagrange identity for determinants Let A$\in M_{(n-1)},n(\mathbb{R})$and for each$1\leq j \leq n$,let$A_j$the matrix obtained from A by removing the j-th column. Show that:$det (AA^t)= \sum\limits_{j=1}^n det(A_j)^2$My first ... 0answers 70 views Product manifolds and exterior derivative with interior product While studying differential geometry, I read this part of a proof and I didn't understand it. Given a$2$-manifold$\Omega$and an interval$I=(-\epsilon, \epsilon)$, consider the cartesian product$M=...
I am a bit confused when it comes to wedge product and differential forms. I know the following property: $\omega\wedge\eta=(−1)^{kl}\eta\wedge\omega$ Also I know that when $k$ is odd and I am ...