# 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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### Exterior Derivative vs. Covariant Derivative vs. Lie Derivative

In differential geometry, there are several notions of differentiation, namely: Exterior Derivative, $d$ Covariant Derivative/Connection, $\nabla$ Lie Derivative, $\mathcal{L}$. I have listed them ...
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### Wedge product and cross product - any difference?

I'm taking a course in differential geometry, and have here been introduced to the wedge product of two vectors defined (in Differential Geometry of Curves and Surfaces by Manfredo Perdigão do Carmo) ...
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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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### What's the connection between derivatives and boundaries?

The (second) fundamental theorem of calculus says that $$\int_a^b f'(x) dx = f(b) - f(a)$$ which can also be stated, if one knows enough about what's coming next, as: The integral of the ...
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### Exterior power “commutes” with direct sum

I know that for vector spaces $V, W$ over a field $K$, we have the following identity : $$\bigoplus_{k=0}^n \left[ \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \right] \simeq \Lambda^n(V \oplus W)$$ ...
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### Decomposable elements of $\Lambda^k(V)$

I have a conjecture. I have a problem proving or disproving it. Let $w \in \Lambda^k(V)$ be a $k$-vector. Then $W_w=\{v\in V: v\wedge w = 0 \}$ is a $k$-dimensional vector space if and only if $w$ ...
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### Signs in the natural map $\Lambda^k V \otimes \Lambda^k V^* \to \Bbbk$

Let $V$ be a finite-dimensional vector space over a field $\Bbbk$. Let $V^*$ denote its dual. I strongly suspect that there is a natural map $$\Lambda^k V \otimes \Lambda^k V^* \to \Bbbk$$ that ...
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### Wedge Product, A Novel Interpretation or Just Plain Wrong?

I have read (I think) all of the previous threads on this website (and many others) on this topic & unfortunately have not found an answer to my question. Due to the fact that I am only beginning ...
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### Geometric meaning of Berezin integration

Berezin integration in a Grassmann algebra is defined such that its algebraic properties are analogous to definite integration of ordinary functions: linearity (taking anticommutativity into account), ...
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### Is There a Basis Free Definition of the Pfaffian

$\DeclareMathOperator{\pf}{pf}$ I recently came across a delightful fact that: The determinant of a $2n\times 2n$ skew-symmetric matrix is a the square of a certain polynomial called the pfaffian. I ...
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### Geometric Intuition about the relation between Clifford Algebra and Exterior Algebra

It is common to see a relation being established between the Clifford Algebra and the Exterior Algebra of a vector space. Recently reading some texts written by Physicists I've seem applications of ...
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### Proving the Poincare Lemma for $1$ forms on $\mathbb{R}^2$

I am trying to prove the Poincare Lemma for $1$ forms on $\mathbb{R^2}$. So I said that if I doing this, I start of with $$\omega = f_1(x_1,x_2) dx_1 + f_2(x_1,x_2)dx_2.$$ First thing I want to ...
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### What are “Super Numbers”?

I'm reading Hyperspace by Michio Kaku and in the chapter on SuperGravity "Super Numbers" are mentioned and are described as a number system where for any super number $a$, $a*a=-a*a$. I was wondering ...
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### Symmetric and exterior powers of a projective (flat) module are projective (flat)

Assume that $R$ is a commutative ring with unity and $P$ a projective (flat) $R$-module. Why $\mathrm{Sym}^n(P)$ and $\Lambda^n(P)$ are projective (flat) for every $n$?
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Let $V$ be a finite-dimensional vector space over $F$ and let $\tau:V \to V$ be a linear operator. Here's my definition of the determinant: If $t:U \to U$ is a linear operator and $\dim(U)=n$ then $... 3answers 513 views ### Without choosing bases, how to show that the determinant is multiplicative in this sense? I was recently considering this statement: Let$V$be a finite-dimensional$k$-vector space, and let$\phi:V\to V$be an endomorphism. Suppose that$W\subseteq V$is a subspace that is stable under ... 1answer 295 views ### A scalar product in the space of oriented volumes? Let$L\colon \mathbb{R}^n \to \mathbb{R}^N$be an injective linear map. By the Cauchy-Binet formula,$\det(L^TL)$equals the sum of the squares of all minors of$L$of order$n$: this looks just like ... 1answer 548 views ### Determinant bundle of a tensor product Let$X$be a ringed space (for example, a scheme or a manifold). If$V$is a locally free$\mathcal{O}_X$-module of rank$n$, then$\mathrm{det}(V) := \Lambda^n V$is a locally free$\mathcal{O}_X$-... 1answer 156 views ### What is the image of the map$\hom(V,V) \to \hom(\wedge^k V,\wedge^k V)$? The title says it all. For the uninitiated: Any map$f:V \to W$induces a map$\wedge^k V \to \wedge^k W$by$v_1 \wedge \cdots \wedge v_k \mapsto f(v_1)\wedge \cdots \wedge f(v_k)$, so$\wedge^k(-)$... 0answers 50 views ### How many matrix minors determine all the minors? Let$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 ... 1answer 102 views ### 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 ... 1answer 185 views ### Which metrics on exterior power are induced from metrics on the base?$\newcommand{\id}{\text{id}}\newcommand{\Hom}{\text{Hom}}$Let$V$be a$d$-dimensional real vector space, and let$2 \le k \le d-1$. Every inner product on$V$induces an inner product on$\...
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Let $V$ be a graded vector space, thought of as a collection $\{ V^n \}_{n \ge 0}$ of vector spaces. Let $V_{odd} = \bigoplus_{n \text{ odd}} V^n$ and $V_{even} = \bigoplus_{n \text{ even}} V^n$. I am ...
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### Is there a formula for the determinant of the wedge product of two matrices?

I was going over the Wikipedia page for exterior products of vector spaces and we can define the determinant as the coefficient of the exterior product of vectors with respect to the standard basis ...
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### How to perform wedge product

I have heard all kinds of great things about Clifford/Geometric algebra, but I can't find any good resources. I have been looking EVERYWHERE for just one actual example of a wedge product being ...
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### Decomposition of product of exterior products

Suppose $V$ is a $n$-dimensional vector space. What is the kernel of $$\bigwedge^p V \otimes \bigwedge^q V\longrightarrow \bigwedge^{p+q} V$$ here $p+q \le n$.
I was under the impression that all 2-forms are the wedge $(\wedge)$ of two 1-forms. Is it possible to have a 2-form that you can't write as $A\wedge B$ with $A,B$ 1-forms?