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I misread your question. My comment alluded to the question whether you can have a basis such that $E_i * E_j = $ some single $E_k$, (as opposed to a linear combination of all of them) but what you as …

When talking about Zariski-topology it is important to specify which field you are working over! Over $\mathbb{C}$ the cirkel is not closed in the Zariski topology on the one-dimensional reductive com …

The book by Broecker and tom Dieck (Representations of Compact groups, Springer Graduate Texts in Math) has a very useful section about real, complex and quaternionic representations and how to pass b …

Here is an issue that thoroughly confuses me. I hope I can express it in a way that is clear cut enough for this site. Let $G$ be a real reductive Lie group and $\mathfrak{g}$ be the complexification …

The statement that this always CAN be done is called 'Ado's theorem'. One way to get an answer is to look for proofs of this theorem online. This blog post by Terrence Tao looks useful (but I didn't r …

It is quite some years ago that I used to know this, but as far as I remember the ansewr is YES. $J(Q_F, \sigma, \lambda)$ is the unique irreducible quotient of $Ind_{Q_F}^G(\sigma \otimes \lambda \ …

I think when it comes to second courses, the question OF WHAT you take your representations becomes more relevant. I will say something about representations of Lie groups because that is where I ende …

I think "Group Theory for Unified Model Building" by R. Slansky qualifies. As the title suggests it is written with an application in physics (beyond my understanding) in mind, but the tables are very …

[I started to type this as a comment but it took like ten comments to fit, so I paste it into an answer. It is an answer as it explains how you can use the references you already have to answer your q …