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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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Relationship between exterior power of representation and variance?

I was reading the question: Symmetric and exterior power of representation regarding how to determine the character of an exterior power of a representation from the original representation. One of ...
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606 views

Hodge-Star-Operator on arbitrary oriented basis

Assume that $V$ is oriented finite dimensional vectorspace with dimension $n$, $g \in T^0_2(V)$ a given symmetric and nondegenerate tensor. Let $\mu$ be the corresponding volume element of $V$. ...
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Free Graded Commutative Algebra on a Graded Vector Space

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 linear injection $ \Lambda^k V^* \otimes \Lambda^k V^* \to \Lambda^k (V^* \otimes V^*)$ which preserves decomposability?

Let $V$ be an $n$-dimensional real vector space, and let $2 \le k \le n-2$. Definitions We say an element $\omega \in \Lambda^k V$ is decomposable if $\omega=\alpha_1 \wedge \dots \wedge \alpha_k$, ...
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255 views

What is the relation between the determinant of the adjoint representation, and volume or eigenvalues?

I'm relatively new to Lie Groups, and finding in a text the following statement, as it is supposed to be obvious. I would love some reference or context to either of these two (probably related) ...
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201 views

Why is the symbol for the exterior product a meet rather than a join?

I've moved this over to HSM. It seems odd that something that looks so much like a join [see below] would get given "the wrong symbol". It's even worse when you dualise it and get something called (...
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What is the difference between tensor calculus and exterior derivative type concepts?

I am trying to clarify terms in order to help me figure out what I'd like to study. I understand that $p$-forms and $p$-vectors are used with things like wedge products, exterior algebras, and a ...
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728 views

Lie algebra: symmetric and exterior power of representation

If $\mathfrak{g}$ is a Lie algebra, $V$ and $W$ are representation of $\mathfrak{g}$ we define the action of $\mathfrak{g}$ on $V \otimes W$ in the following way: $X \cdot (v \otimes w)=(X \cdot v) \...
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Clifford algebra is isomorphic to exterior algebra, proof in Lawson's

This Proposition 1.2, pg 10 of Lawson's Spin Geometry. Context: We have a homomorphism of graded algebra $$ \wedge^*(V) \rightarrow G^*$$ induced on each component from the map $$\wedge^r(V) \...
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Understanding the derivation of this form of the coderivative

Let $M$ be a smooth Riemannian manifold; Let $E$ be a vector bundle over $M$, equipped with a metric and a compatible connection $\nabla$. Denote by $d:\Omega^k(M,E) \to \Omega^{k+1}(M,E)$ the ...
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Generalization of Liouville's formula to other coefficients of the characteristic polynomial

If $X(t)$ is an $n \times n$ matrix solving linear homogeneous ODE $$ \frac{d}{dt} X(t) = A(t)X(t), $$ then for $\det X(t)$ we have Liouville's formula: $$ \frac{d}{dt} \det X(t) = \text{tr} A(t) \det ...
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Is this a valid way to think of decomposable $k$-forms?

Let $V$ be a finite dimensional vector space and $\eta \in \Lambda^k(V^\ast)$ be decomposable and non-zero. Then there exists covectors $\omega^1,\dots,\omega^k$ such that $$\eta(v_1, \dots, v_k) = \...
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212 views

Frobenius Condition for Singular Integrable Distributions

A smooth "singular" distribution $D\subseteq TM$ on an $n$-dimensional manifold $M$ is integrable if it is tangent to immersed submanifolds $N_\alpha$ that are disjoint and cover $M$. If dim$D=k$ ...
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Is $\Lambda$ essentially the unique solution to $F(M)\cong\frac{F(M\oplus R)}{F(M)}$?

Let $R$ be a commutative ring and let $F$ be a functor $\mathbf{Mod}_R\rightarrow \mathbf{Mod}_R$. Then for a module $M$ the split mono $M\rightarrow M\oplus R$ gives a split mono $F(M)\rightarrow F(M\...
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When does the duality functor commute with the wedge power functor?

When working with modules over a fixed commutative ring, I know that $(M \otimes N)^* \cong M^* \otimes N^*$ provided either $M$ or $N$ is finitely generated projective. Does it follow that $(\...
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Prove that $\phi_1 \wedge \cdots \wedge \phi_k (v_1, \cdots, v_k) = \frac{1}{k!}\det[\phi_i(v_j)].$

I have proved these two exercises: (1) Suppose that $T \in \Lambda^p(V^*)$ and $v_1, \ldots, v_p \in V$ are linearly dependent. Prove that $T(v_1, \ldots, v_p) = 0$ for all $T \in \Lambda^p(V^*)$. ...
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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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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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Does this geometric rigidity condition forces the map to be the identity?

$\newcommand{\Cof}{\operatorname{cof}}$ $\newcommand{\id}{\operatorname{Id}}$ $\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$ Let $V$ be a real $d$-dimensional vector ...
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Classifying the orbits of the natural $\text{GL}(V)$-action on the exterior power $\bigwedge^k V$

Let $V$ be a real $d$-dimensional vector space, and let $1 < k < d$. Consider the following action of $\text{GL}(V)$ on $\bigwedge^k V$: $(T,\omega) \to (\bigwedge^k T) \omega$. Can we ...
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Complexified cotangent bundle

I have never worked with differential geometry over $\mathbb{C}$, and I feel a little bit confused. If $(M,J)$ is an almost complex manifold of dimension $2n$, we can extend $J_{p}$ to $T_{p}(M)\...
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Regarding vector-valued (differential) forms

I denote the space of all $V$-valued differential $k$-forms on $M$ with $\mathcal A^k(M,V)$. Let $\omega\in \mathcal A^k(M,V)$ and $\eta\in \mathcal A^l(M,W)$, where $V,W$ are finite real vector ...
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47 views

Translating between two different definitions of exterior derivative

If $\Omega^k(M)$ is the space of differential $1$-forms on a manifold $M$, one may define the operator $\mathop{}\!\mathrm{d} : \Omega^k(M) \to \Omega^{k+1}(M)$ in a coordinate-independent way as ...
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29 views

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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How do mathematicians think about multivectors/exterior algebra?

I recently came across the notion of multivectors and exterior algebras recently and while I find these notions to be very enlightening, I also feel a strong temptation to interpret these objects too ...
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159 views

The determinant of $X_I$ = wedge product of elementary tensors

Let $(x_1,\dots,x_k)$ be vectors in $\mathbb{R}^n$; let $X$ be the matrix $X = [x_1 \cdots x_k]$. If $I = (i_1, \dots, i_k)$ is an arbitrary $k$-tuple from the set $\{1,\dots, n\}$, show that $\phi_{...
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544 views

Koszul sign convention and symmetric group action on the graded n-th tensor product

Let $V_\bullet = (V_k)_{k \in \mathbb{Z}}$ and $W_\bullet = (W_k)_{k \in \mathbb{Z}}$ be two graded vector spaces on 0 caracteristic field. We define the tensor product of $V_\bullet$ by $W_\bullet$ ...
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207 views

Tensor algebra becomes an R-algebra. Theorem, Proof, Dummit and Foote

I have the definition of tensor algebra as follows: $T(M) = \bigoplus_{i =1}^{\infty} T^i(M)$, where $M$ is an $R$ module, where $R$ is commutative and contains the element $1$. Finally $T^k(M) = \...
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841 views

Inner product exterior algebra

I have to prove that if $V$ is a real vector space ($\dim V=n$) with inner product $(.,.)$ then if we define $$ (v_{1}\wedge v_{2}\wedge\cdots\wedge v_{k},w_{1}\wedge w_{2}\wedge\cdots\wedge w_{k}) =...
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407 views

Vector Laplace Beltrami operator on surface tangent and surface normal vector field

Consider a closed, compact, embedded surface $f:M \rightarrow \mathbb{R}^3$ and a vectorfield $X$ on the surface that can be decomposed in the surface frame basis $\{e_1,e_2,e_3\}$, where $\{e_1,e_2\}$...
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Multiplication in exterior algebra

Take $V = K^{n}$. Let $\omega$ be a non-zero element of $\bigoplus_{k=1}^n \bigwedge^k V$, where we have excluded the summand $\bigwedge^0 V = K$. (1) Prove that there exists an $m > 1$ for which $\...
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285 views

Grassmann Algebras

The Grassmann algebra $G$ is the algebra over a field $\mathbb{F}$ generated by the variables $e_i$ such that $e_i^2=0$ and $e_i e_j = - e_j e_i$. I'm looking for some references on algebras $G \...
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80 views

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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47 views

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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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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66 views

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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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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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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63 views

General commutators of derivations of the exterior algebra

Let $M$ be a smooth manifold and let $\Omega(M)$ be the exterior algebra of smooth differential forms over $M$. The $\mathbb R$-linear map $D:\Omega(M)\rightarrow\Omega(M)$ is a derivation of the ...
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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 ...
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22 views

Two related(?) definitions on the wedge product

I read from a textbook that one defines the wedge product of basis elements as, for example, $$e^{i_1}\wedge...\wedge e^{i_k}\equiv k!\text{Alt}(e^{i_1}\otimes...\otimes e^{i_k})$$ and of two forms $$...
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62 views

Can we stay invertible while approximating linear maps in Sobolev spaces?

Let $\Omega \subseteq \mathbb{R}^d$ be an open bounded domain. Fix an integer $1<k<d$. Let $f \in W^{1,k}(\Omega;\mathbb{R}^d)$ be a continuous map with $\det df > 0$ a.e. Consider the map ...
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80 views

Determining the isomorphism classes of these symmetry groups (exterior algebra)

Let $V$ be a real $d$-dimensional vector space. Let $\omega \in \bigwedge^kV$ be a fixed non-zero multivector for some $1 < k < d$. Define $ G_{\omega}=\{ T \in \text{Aut}(V) \, | \, (\bigwedge^...
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Is there a non-degenerate solution for this PDE on $\mathbb{R}^3$?

$\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\R}{\mathbb{R}}$ I am looking for a smooth map $f:\mathbb{R}^3 \to \mathbb{R}^3$, which satisfy $$\delta \big( df \wedge df \big) \neq 0, \, \...
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154 views

Uniqueness of connection $1$-forms and curvature $2$-forms

If $(M,\langle\cdot,\cdot\rangle)$ is a pseudo-Riemannian manifold and $\nabla$ denotes the Levi-Civita connection of the metric, we can define the connection $1$-forms and curvature $2$-forms ...
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160 views

Wedge product of vector and tensor

As near as I can figure it: (a⊗b)∧c = a⊗b⊗c - a⊗c⊗b + c⊗a⊗b But I'm not sure if thats right. a, b, and c are vectors. I know that a∧b = a⊗b - b⊗a which is a bivector. The equation I am most ...
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90 views

Lie derivative in geometric algebra/Clifford algebra

What is the form of the Lie derivative in Clifford algebra? Context: Consider the Clifford algebra $\mathcal{C}l (p,q) $ with basis $\{e_i \}$. The geometric derivative following Hestenes is defined ...
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111 views

Wedge product of regular $r$-forms

I need to solve this exercise: A continuous form $w$ of degree $r$ in the open $U\subset\mathbb{R}^m$ is regular $\iff$ for all open $V$ with compact closure $\overline{V} \subset U$, there is a ...