Quickstudy And Sparkcharts Over 200 Study Guideszip _VERIFIED_

October 18, 2022

Quickstudy And Sparkcharts Over 200 Study Guideszip _VERIFIED_

Quickstudy And Sparkcharts Over 200 Study Guideszip

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Problems with modular tensor algebra

Let $\mathfrak g$ be a finite dimensional, $U(\mathfrak g)$-module. By definition it has a $U(\mathfrak g)$-basis $(x_i, y_i)$ over $\mathbb C$ where $x_i\in \mathfrak g^*$ and $y_i\in \mathfrak g$. Define $\mathfrak g\rtimes \mathfrak g =U(\mathfrak g)/I$ where $I$ is the ideal generated by the set of equations
$$[x_i, y_j] – [y_i, x_j] + [y_i, y_j] = 0$$
for all $1\le i, j \le r$ where $r$ is the dimension of $\mathfrak g^*$.
My problem is the following: if I construct a basis over $\mathbb C$ by
$$\{x_i\otimes y_j – y_i\otimes x_j\mid 1\le i, j\le r\}$$
what is the corresponding ideal $I$?

Let us consider a concrete example. Let $\mathfrak{g}$ be a 2-dimensional vector space over $\Bbb C$ and $\mathfrak{g}^*$ be dual of $\mathfrak{g}$. An arbitrary basis of $\mathfrak{g}$ is $\{u,v\}$ and its dual basis of $\mathfrak{g}^*$ is $\{x,y\}$.
To define a non-zero ideal of $U(\mathfrak{g})\rtimes U(\mathfrak{g})$, it is enough to specify polynomials which are proportional to some scalar multiple of $[u,x]$, $[v,y]$, $[u,y]$ and $[v,x]$. Let us look at the easiest case $[u,x]$ and denote it by $a$. Then, we need to specify a polynomial $p=p(a,y)$ of two variables satisfying
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