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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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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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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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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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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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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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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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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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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^...
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38 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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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}...
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Clarifying a step in an answer about exterior products of coherent sheaves

The question is about the accepted answer here. I decided not to ask in a comment there since the original asker is no longer active. In "Step 2" of the answer given there, Roland claims, "This is ...
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Finding the Wedge Product of Two Multivariable Vectors and Making Sense of It

I was attempting to solve: $$ -2dx_{1} \land dx_{4}\left( \begin{bmatrix} 2 \\ 3 \\ -5 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 4 \\ -5 \end{bmatrix}\right) $$ I solved this by pulling out the ...
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understanding orientable manifolds

I'm reading Warner. "Foundations of Differentiable Manifolds and Lie Groups." p. 138. I don't get the statement in the definition of orientable manifolds. 4.1 Definitions $\;$ (the preface omitted) ...
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Skew-symmetric implies alternating for $2$ a zero divisor in $R$?

In Keith Conrad's notes: https://kconrad.math.uconn.edu/blurbs/linmultialg/extmod.pdf Theorem 2.10 reads: Let $k\geq 2$. If $2\in R^\times$, then a multilinear function $f:M^k \to N$ which is ...
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A construction of the Hodge Dual operator

This question about showing that an alternative construction of the Hodge dual operator satisfies to the universal property through which the Hodge dual is usually defined. Let me give the ...
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34 views

wedge product (exterior algebra)

I got confused on the operator of the wedge product on other 2 vectors. Please help. Let $V=\mathbb R^3,e_1= (1,0,0),e_2= (0,1,0)$, and $e_3= (0,0,1)$. Find: $3e_1∧4e_3((1,α,0),(0,β,1))$, where α,β ...
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32 views

Prove Poincaré Lemma for $1$-form

Let $U\subseteq\mathbb{R}^n$ be an open set that contains $0$, and for all $t\in[0,1]$ and $ x\in U$, $tx\in\mathbb{R}^n$. Show that every closed differentiable 1-form $w$, (i.e. $dw=0$) is an exact ...
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Which subspaces of exterior power have decomposable bases?

Let $V$ be a real $n$-dimensional vector space, and let $1<k<n,r>1$. I wonder: Is there a way to characterise which $r$-dimensional subspaces of the exterior power $\bigwedge^k V$ have ...
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Arithmetics of '$\wedge$' and '$d$' operators

I don't find arithemtic rules of the operators $\wedge$ and $d$. For example, why does this equality hold? $$ \\ (u^2\cos^2v+u^2\sin^2v)[\cos vdu-u\sin vdv]\wedge [\sin vdu+u\cos vdv] \ \\ +u\cos v[\...
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$G$ is Lie group and $V$ is a representations of $G$,prove representations $V \otimes V \cong S^2(V) \oplus \Lambda^2(V)$

Let $G$ a Lie group and let $V$ a representations of $G$. Then we have the following representations are isomorphic: \begin{align} V \otimes V \cong S^2(V) \oplus \Lambda^2(V) \end{align} I have no ...
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33 views

Connection between ranks of an endomorphism and its linear image on the exterior power

Let $V$ be an $n$-dimensional real vector space, and let $1<k<n$. Let $\psi:\text{End}(V) \to \text{End}(\bigwedge^kV)$ be the exterior power map, $\psi(A)=\bigwedge^k A$. For $B \in \text{End}...
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Why is the volume element unique in the way Spivak develops it?

In Spivak's Calculus on Manifolds, he develops the volume element in the following way: The fact that $\dim \Lambda^n(\mathbb{R}^n) = 1$ is probably not new to you, since det is often defined as ...
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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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If $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$ is a complex power, is it a real power up to a sign?

Let $1<k<d$ be an integer. Let $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$, and suppose that $A=\bigwedge^k B$ for some complex $B \in \text{End}(\mathbb{C}^d)$. Does there exist $M \in \...
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Is every basis for $\bigwedge^kV$ satisfying a “complementary” property a rescaling of a “standard” basis?

This question was inspired by this beautiful answer: Let $V$ be a $4$-dimensional real vector space. Let $\omega_{i_1,i_2}$ ($1 \le i_1 < \ldots < i_2 \le 4$) be a basis for $\bigwedge^2V$, ...
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Characterising minors of diagonal matrices

Let $k,d$ be positive integers, $1<k<d$. Let $\lambda_I=\lambda_{i_1,\ldots,i_k}$ be real numbers, indexed by multi-indices $I=(i_1,\ldots,i_k)$, where $1\le i_1<\ldots<i_k \le d$. Are ...
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Is every decomposable basis for $\bigwedge^kV$ “standard”?

This is a curiosity: Let $V$ be a $d$-dimensional real vector space, and let $1<k<d$. Set Let $\omega^{i_1,\ldots,i_k}$ be a basis for $\bigwedge^kV$, whose elements are all decomposable. Is $\...
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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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1answer
148 views

Functorial proof of Cayley-Hamilton using exterior powers

Let $V$ be a rank $n$ free module over a commutative ring $R$. Let $\dagger$ denote the adjoint with respect to the natural perfect pairing given by the wedge product $$\textstyle \bigwedge^k\otimes \...
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Decomposing $k$th exterior powers $\Lambda^kV(\omega_1)$

Let $\Phi$ be a $G_2$ root system, $\omega_1$ the fundamental weight corresponding to the shorter root and consider the unique irreducible highest weight module $V = V(\omega_1)$ of highest weight $\...
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Questions About the Proof of Cauchy–Pompeiu Integral Formula.

I am studying function theory in several complex variables and the book I am using is "Tasty Bits of Several Complex Variables" by Jiří Lebl: https://www.jirka.org/scv/scv.pdf. At the moment I am ...
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Why is the exterior power $\bigwedge^kV$ an irreducible representation of $GL(V)$?

$\newcommand{\GL}{\operatorname{GL}}$ Let $V$ be a real $n$-dimensional vector space. For $1<k<n$ we have a natural representation of $\GL(V)$ via the $k$ exterior power: $\rho:\GL(V) \to \GL(\...
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Does every subspace of the exterior algebra of dimension $>1$ contain a decomposable element?

Let $V$ be a real $n$-dimensional vector space, and let $W \le \bigwedge^k V$ be a subspace . Suppose that $\dim W \ge 2$. Does $W$ contain a non-zero decomposable element? If $\dim W=1$, then ...
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1answer
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The non-zero exterior product

Let $V$ denote a vector space (maybe not finite-dimensional) over a field $\mathbb k$ with basis $\{e_1, e_2, \ldots\}$. I have to prove that the set $\{e_{i_1} \wedge e_{i_2} \ldots \wedge e_{i_n}$ $|...
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Covariant Exterior Derivative action on the $\mathrm{End}(E)$-valued p-forms

Suppose I define an operator $d_A$ by its action on sections $s\in \Gamma(E)=\Omega_M^0(E)$ of some vector bundle $\Pi:E\rightarrow M$ in a trivializing neighbourhood $U\subset M$ as $$ d_As|_U=(ds+A\...
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$\omega$ a closed 2-form and $\bigwedge_{i=1}^n \omega \ne 0$ on a compact orientable smooth $2n$-manifold w/o boundary, $M$, then $H^2(M) \ne 0$.

Suppose $M$ is a compact orientable smooth $2n$-manifold without boundary, and let $\omega$ be a closed $2$-form such that $\bigwedge_{i=1}^n \omega_p \ne 0$ at every point $p$. Show that $H^2_{dR}(M) ...