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I was especially glad to find out that the platform that I had used for all my studies .Q:
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 $I$?
A:
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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Quickstudy And Sparkcharts Over 200 Study Guideszip10/21/2016 – From 0 to 60 mph in 6.5 seconds, or 390 kph, the new and highly anticipated Volkswagen Golf R is the most powerful Golf ever. Thanks to an upgraded turbocharged engine, the Golf R sprints with more than 60 kilograms of thrust and offers better top-gear acceleration with 470 horsepower and 479 lb-ft of torque – all for a price of less than $38,500. The 2019 Volkswagen Golf R delivers a driving experience like no other, with an emphasis on sport and handling even at full power. The platform has been stretched by 2.6 inches (62 millimeters) and weighs 20 kilos less than the previous Golf R.
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