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Why is it true that with $x\in \mathfrak{m}$ also $p(x)\in \mathfrak{m}$ for all polynomials $p(x)$ without constant term ? The proof I know just uses the Jordan decomposition $x=x_s+x_n$, and shows f …

Using the parametrization $$ A= \begin{pmatrix} a^2+b^2-c^2-d^2&2bc-2ad &2bd+2ac \cr 2bc+2ad &a^2-b^2+c^2-d^2&2cd-2ab \cr 2bd-2ac &2cd+2ab &a^2-b^2-c^2+d^2\\ \end{pmatrix}, $$ which comes from …

This is Weyl's dimension formula. The dimensions of the fundamental modules $L(\omega_1), \ldots ,L(\omega_{n-1})$ for $A_{n-1}$ are $\binom{n}{1}, \binom{n}{2},\ldots ,\binom{n}{n-1}$. The dimensions …

Taking the finitary simple Lie algebras $L$, like $\mathfrak{sl}(\infty)$, $\mathfrak{o}(\infty)$, then these locally simple Lie algebras are known to have no non-trivial finite-dimensional module at …

There are many fundamental lemmas, not only Langland's fundamental lemma (with a proof by Ngo) in the theory of automorphic forms. Of course, this one is a particularly deep result. I do not really se …

Over the complex numbers, connected linear algebraic groups correspond to Lie algebras in the usual way. This Lie correspondence breaks down over number fields, and breaks down even more over fields o …

I have found a counterexample while studying Lie algebra degenerations. Consider the following filiform nilpotent Lie algebra $\mathfrak{g}$ of dimension $13$, given by the brackets with respect to a …

Kac and Peterson have classified all inequivalent Heisenberg subalgebras of a loop algebra. This is explained in the book "Lie Algebras, Part 2: Finite and Infinite Dimensional Lie Algebras and Applic …

Here are some references (which are not mentioned in Resolutions of Lie algebras). First I can recommend the book of Charles A. Weibel, An Introduction to Homological Algebra. It answers your questio …

Such generalizations exist. It is well known that the classical Clifford algebras can be used to construct Lie superalgebras. The main tool of the construction is the notion of the $\mathbb{Z}_2$-grad …

The Lie algebra is filiform nilpotent and is generated by $e_1$ and $e_2$. It is known that any faithful Lie algebra representation $\rho:\mathfrak{f}_n\rightarrow \mathbb{gl}(V)$ of a $n$-dimensional …

The following result is proved in Bourbaki's book on Lie algebras: Theorem (A converse to the First Whitehead Lemma). Any finite-dimensional Lie algebra over the field of characteristic zero such tha …

Specht polynomials and Specht modules have been applied not only to the representation theory of symmetric groups, but also for other reflection groups, e.g., for octahedral groups, see for example ht …

Heisenberg groups over a finite field $\mathbb{F_q}$ with $q=2^m$ are abelian and its representations are all one-dimentional, i.e., characters. The classification of irreducible representations is gi …

In the semisimple case it is really easy to calculate $ext_A^i(M,N)$, $i\ge 1$, with the above assumptions. It is zero. For Koszul rings this is almost true, i.e., $ext^i(M,N)$ is concentrated in degr …