Insights Representations and Why Precision is Important

[fresh_42]

First of all: What is a representation? It is the description of a mathematical object like a Lie group or a Lie algebra by its actions on another space 1). We further want this action to preserve the given structure because its structure is exactly what we're interested in. And this other space here should be a vector space since we want to deal with operators and transformations.

Our main examples shall be the special unitary group and its Lie algebra. The special unitary group $SU(n)$ is the group of isometries of an n-dimensional complex Hilbert space that preserve the volume form on this space. How that? I thought it were $SU(n)=\{\text{ unitary matrices }\}$? To be a bit more precise $$SU(n)=\{A \in \mathbb{M}_n(\mathbb{C})\,\vert \, A\cdot A^\dagger = 1 \wedge \det(A)=1\}$$
Well, both is true. And the definition via matrices is already our first example of a representation. It is in a way nearby via the association

$$\it{isometry} \rightarrow \it{transformation} \rightarrow \it{ matrix}$$

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[Greg Bernhardt]

Great job on your first Insight @fresh_42!

 

[bhobba]

Nice.

Very important for understanding modern physics - and of course just for the math.

Thanks
Bill