The ordinal number $\omega^2$ can be visualized as $\omega$-many copies of $\omega$. Likewise, the ordinal number $\omega^3$ can be visualized as $\omega^2$-many copies of $\omega$, arranged as appropriately (= lexicographically) ordered rows in the cube $\omega^3$ (rows being sequences of cells ...

I'm not familiar with algebra theory, but I'm interested to know if there is an ensemble in mathematics for which the power of ensemble could be a real number, we take for example $\mathbb{R^{\frac{1}{2}}}$, Does this meant the space dimension is $\frac{1}{2} $? Note:I want $\mathbb{R^{\frac{1}{...

I am looking for a clear and simplified definition of the homogeneous dimension of a vector space or a Lie group. If anyone has the patience to explain me clearly his definition and if possible an example, I would be very grateful. Thank you in advance

Can the notion of vector space or algebra over a field be meaningfully extended to fractional dimensions, so that for example $\mathbb{R}^{-2/3}$ makes sense? Has this been explored somewhere? I know that super vector spaces can be thought of as one way of generalizing vector spaces to negative ...

I need some help to understand the following proof. Let $k$ a field and $V$ an algebraic set. I note $\mathfrak{m}_P$ the ideal generated by $X_1-a_1,\dots ,X_n-a_n$ in $k[V]=k[X_1,\dots ,X_n]/I(V)$ with $P=(a_1,\dots ,a_n)$. Theorem. $\left\lbrace P\in V\mid \dim_k ~^{\mathfrak{m}_P} \big...

How to add and multiply on fractional vector space Please, answer in layman terms. I don’t understand the notation of Supersimetry and Super vector spaces. If a fractional “vector space” (or his fractional equivalent) has dimension between 1 and 2: $1 \,\leq\, (d=1+\frac{1}{n}) \,\leq\, 2 \;\;...

Why is there an Zariski open set of $P\in GL(n,\mathbb{R})$ such that $P\,\text{diag}(1,\dots,1,-1)P^{-1}$ can be conjugated by a diagonal matrix $D$ to get an orthogonal matrix? Note that $M=P\,\text{diag}(1,\dots,1,-1)P^{-1}$ is involutory ($MM=I$) and $\det(M)=-1$. Possible starting point: ...

Let $G= (\mathbb{C}^3, A)$ be the $\mathbb C[x]$-module given by $$ A=\left( \begin{matrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{matrix}\right). $$ For a vector $v ∈ \mathbb C^3$ let $L(v) := \{f(x)v \mid f ∈ \mathbb C[x]\}$ be the submodule of $...