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2 votes

Application of Stone's theorem to regular representation

Note that the Fourier transform of $\lambda_x f$ is, $$\begin{split} \mathcal{F}(\lambda_x f)(\xi) &= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(y-x)e^{-i\xi y} \, dy\\ &= \frac{1}{\sqrt{2\...
David Gao's user avatar
  • 5,205
1 vote

How to show that $\left[D_\mu, F^{\mu\nu}\right]=\left(\partial_\mu \delta_{ae}-gf^{bae}A_\mu^b\right)F^{\mu\nu a}t^e$?

Whenever you work with a commutator it is a good idea to add a generic test-function $g$ that the operator acts on. When doing this you also have to remember that a derivative operator acts on ...
Winther's user avatar
  • 24.5k
1 vote

Theorem of Highest Weight for Reductive vs Semisimple complex algebraic group

For a split reductive group there is exactly the same kind of story. I will describe the setup as precisely as possible so that you can see the connection. This may not be an answer to the question ...
Joppy's user avatar
  • 12.9k
1 vote

Irreducible but not absolutely irreducible representations

Abbreviate $K=\Bbb{F}_{q}$, $L=\Bbb{F}_{q^2}$, $k=\overline{K}$ to save a few keystrokes. Let $U$ be the space $k^2$ with $G$ acting on it via $\rho$ (composed by the inclusion $GL_2(K)\hookrightarrow ...
Jyrki Lahtonen's user avatar
1 vote

Decomposition of non-finitely generated modules over a finite dimensional algebra, $A$, where $A$ is of finite representation type.

If $\kappa$ and $\lambda$ are different infinite cardinals, then the direct sum of $\kappa$ copies of $M_i$ is different from the direct sum of $\lambda$ copies. So you should let $k_i$ be any ...
Jeremy Rickard's user avatar
1 vote

Semisimple Lie algebras: general features of ladder operators and towers of vectors

I think the best thing here would be to actually have a look at the representation theory of semisimple Lie algebras such as in Fulton and Harris's Representation Theory or maybe Humphreys' ...
Callum's user avatar
  • 4,456

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