# Questions tagged [lie-algebras]

For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

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### On the Killing splitting of some algebras

It is well-known that the algebra $so(4,1)$ admits the Killing splitting as vector spaces as follows: $so(4,1)=so(3,1) \oplus \mathbb{R}^{3,1}$. In the same context, is there a Killing splitting ...
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### Applying theorems from Fulton and Harris to Lie groups/algebras with non-$\mathbb{C}$ coefficients

I recently asked this question about $SL_4(\mathbb{Q})$ representations. Commenters warned me that Fulton and Harris is about representations of complex Lie groups/algebras so I should be careful ...
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### Book on $SL(2,C)$

Is there a book, which treats $SL(2,C)$ in detail as a group, Lie group, its Lie algebra, geometry of its subgroups etc.? It is often seen as an example in Lie Algebra/Group books but it always ...
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### Moving between $Sp_{2n}(\mathbb{C})$ reps and $SL_{n}(\mathbb{C})$ reps

Say I have some irreducible $Sp_{2n}(\mathbb{C})$ representation, such as $\Gamma_{0,1,0,1}$. Consider the subgroup of $Sp_{2n}(\mathbb{C})$ isomorphic to $SL_n(\mathbb{C})$, consisting of matrices of ...
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### Jacobi identity as a cocycle condition

The Jacobi identity in a Lie algebra $\mathfrak{g}$ looks like this (for $x,y,z\in \mathfrak{g}$): $$[[x,y],z]-[x,[y,z]]+[y,[x,z]]=0.$$ This just says that the map $y\mapsto [x,y]$ is a derivation ...
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### Elementary consequences of the root system axioms

On Root_system: Elementary_consequences_of_the_root_system_axioms (wikipedia), from the relation $\langle \alpha, \beta \rangle = (2\cos(\theta))^2 \in \mathbb Z$, the value $\cos(\theta)$ can only be ...
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### Representations of a simply-connected non-compact Lie group induced from Lie algebra representations

It is a known fact that for simply connected Lie groups, each representation of the Lie algebra comes from a representation of the Lie group [See this]. Consequently, we study representations of the ...
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