The symmetric algebra of an object exists in every cocomplete $\otimes$-category. For the category of sets $\mathrm{Sym}(X)$ is the set of multi-subsets of $X$. The usual definiti …

Hello! in Little Spivak, p. 79, we find this: We will therefore define a new product, the wedge product $\omega\wedge\eta\in \Lambda^{k+\ell}(V)$ by $$ \omega\wedge\eta …

Hello everyone. Let $V$ and $W$ be finite dimensional vector spaces over some field $K$. Consider $\rho:Sym(V)\to End(W)$ a homomorphism of algebras with unit (i.e., a representat …

Hello All, I apologize if the following question is too elementary. Any suggestion is greatly appreciated. Thank you. Let $n\geqslant 4$, $X$ be an $n$-dimensional inner product …

I have another exterior differential system for one forms $U^i$, where the $\theta^i$ are a cotangent basis on $SO(3)$, i.e. they satisfy $d \theta^i = \epsilon_{ijk} \theta^j \wed …

Hello, I'm looking for references on various inequalities involving the wedge product and exterior forms. The only references I could hunt down are references on Hadamard-Schwarz …

Hello, The following question appears as a step in my proof. It seems easy but somehow I have not been able to prove this. I could solve few special cases though. Any help in thi …

I have the following exterior differential system for one forms $\alpha, \beta, \gamma$, where the $\theta^i$ are a cotangent basis on $SO(3)$, i.e. they satisfy $d \theta^i = \eps …

There are different conventions for defininig the wedge product $\wedge$. In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$, in Spivak, we find $\alpha\we …

As addressed in this past question, there are many applications of linear algebra to combinatorics. What about examples of applications of exterior algebras? Part 4 here is one suc …

I'm looking for something like a Grassmannian, but which parameterizes the submodules of a module rather than the subspaces of a vector space. Most specifically, I'm looking for so …

I have written a paper which involves a "cross product" in $\mathbb{R}^n$ and I would like to have a reference to point to. Let ${\bf e_1}, \dots, {\bf e_n}$ be the standard basi …

I don't understand wedge product and Grassmann algebra. However, I heard that these concepts are obvious when you understand the geometrical intuition behind them. Can you give thi …

Under what conditions is it possible, using a suitable change of variables, to eliminate 1st order terms in an elliptic partial differential equation, so that the equation involves …

Let $\mathcal{C}$ be a cocomplete $R$-linear tensor category. Many notions of commutative algebra may be internalized to $\mathcal{C}$. For example an algebra is an object $A$ in …