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In general, if $V = \bigoplus_{\mu \in \Lambda} V_\mu$ and $W = \bigoplus_{\nu\in \Lambda} W_\nu$ are $\Lambda$-graded vector spaces (for some abelian group $\Lambda$), then $V \otimes W = (\bigoplus …

Let $a \in J(A)$, and suppose for a contradiction that the left ideal $A[x](1 - xa)$ is proper. We can then choose a maximal left ideal $J$ of $A[x]$ containing $A[x](1-xa)$, so that $M := A[x] / J$ …

How about the Harish-Chandra isomorphism, which computes the centre of the universal enveloping algebra of $\mathfrak{g}$?

Let $G$ be a simple algebraic group over a field $k$, and let $U$ be the unipotent radical of a Borel subgroup $B$. Because $B$ normalises $U$, the group $H = B/U$ acts on the coordinate ring $\mathca …

When $\mathfrak{g}$ is non-zero and finite dimensional over $k$, its restricted enveloping algebra $u(\mathfrak{g})$ is never a domain. To see this, note that $u(\mathfrak{g})$ is itself finite dimens …

The completed group ring of a compact $p$-adic Lie group with coefficients in a field of characteristic $p$ is topologically Noetherian. In fact, it is even abstractly Noetherian. This is a theorem of …

Just for the record, here is an example of a (necessarily infinite dimensional) $k$-algebra $A$ where the Jacobson radical is not equal to the intersection of all maximal two-sided ideals. Let $k$ be …

No. Let $R = \mathbb{Z}_p[C_p]$ and consider the $R$-modules $M = R$ and $N = \mathfrak{m}$, the maximal ideal of the local ring $R$. Let $A,B$ the matrices in $GL_p(\mathbb{Z}_p)$ giving the action o …

Chapter II.5 in Jantzen's Representations of Algebraic Groups offers an algebraic treatment of this theorem.

I don't have a full answer, but the following calculation may be useful. Consider the example $G = SL_3$ so that $N$ is the Heisenberg group on three generators and in characteristic zero, $\bar{U}(N …

Let $\mathcal{A}$ be the two-dimensional Lie algebra over a field $k$ with basis $\{x,y\}$ and relation $[x,y] = y$. Let $\mathcal{I}$ be the ideal of $\mathcal{A}$ spanned by $y$. Let $D' : \mathcal{ …