# 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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### 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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### 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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### 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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### 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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### 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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### Supercommutative algebras except commutative algebras and exterior algebras

When I first saw a definition of a supercommutative algebra, first example that came to my mind was an exterior algebra on some vector space. Of course, purely even supercommutative algebra is just a ...
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### Why are “innermorphisms” not useful?

I commonly studied type of linear function in geometric algebra is the outermorphism. For reference, here's Wikipedia's definition: Let $f$ be an $\Bbb R$-linear map from $V$ to $W$. The ...
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### The exterior algebra is a superalgebra?

Can someone explain how the exterior algebra of a vector space or a module over a commutative ring is a superalgebra? The exterior algebra has an obvious $\mathbb{Z}$-grading, but I don't see where ...
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### Symmetric and wedge product in algebra and differential geometry

Which is the correct identity? $dx \, dy = dx \otimes dy + dy \otimes dx$ $~~~$or$~~~$ $dx \, dy = \dfrac{dx \otimes dy + dy \otimes dx}{2}~$? $dx \wedge dy=dx \otimes dy - dy \otimes dx$ $~~~$or$~~~$...
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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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