Tagged Questions

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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Irreducible Representation of $sl(3,C)$

We know that the the roots of $\mathbb{g} = sl(3,C)$ under the adjoint action are given by $L_i - L_j$ where $L_i (diag(a_1, a_2, a_3))=a_i$ for $i = 1,2,3$. If $V$ is any irreducible representation ...
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Infinite Lie algebra

It is well known fact that the finite dimensional Lie algebra will always be closed under commutation relation. But I have doubt about infinite Lie algebra. In some of the case it is closed under ...
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The cohomology of $\mathrm{GL}_n$ over an algebraically closed field

How does one go about computing the cohomology groups $H^*(\mathrm{GL}_{\kern{0.1em}{m}}(\overline{\mathbb{F}}_p),M)$? I am particularly interested in the case when $M$ is an algebraic representation. ...
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Non-Lie character of Leibniz algebra

Let $J$ be the largest ideal of Leibniz algebra $L$ which denotes the non-Lie character of $L$. Is it possible to write $L=L_{Lie}\cap J$? We know that $L_{Lie}= L/J$. I am going to give the following ...
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Algebraic groups and restricted Lie algebras

If $G$ is an algebraic group with coordinate algebra $A=\mathcal O(G)$, say over a field $k$ of characteristic $p$, then its Lie algebra $\mathfrak g$ can be endowed with the structure of a restricted ...
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Problems with Matrix of irreducible representation of SO(3)

I'm working out the irreducible representation of $SO(3)$. Let's call $R_{\theta}$ as a general rotation and $Y_{m}^{l}\left(\eta,\varphi\right)$ the spherical harmonics. I now like to have the matrix ...
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Importance of universal enveloping algebras and Poincare-Birkhoff-Witt theorem.

Let $L$ denote a Lie algebra and $U(L)$ denote its universal enveloping algebra. I am trying to see why universal enveloping algebra and PBW theorem are important. Precisely: Given any $L$-...
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Every Lie algebra contains a maximal proper Lie subalgebra

I am working though the proof of Proposition 6.2 in Erdmann's "Introduction to Lie Algebras". I can't verify that every Lie subalgebra of $L \subseteq \mathfrak{gl}(V)$ contains a maximal (proper) ...
Let $F$ be a field and let $L = b(n,F)$ be the Lie algebra of $n×n$ upper triangular matrices and $V = {F^n}$ .
$\space$ Let $F$ be a field and let $L = b(n,F)$ be the Lie algebra of $n×n$ upper triangular matrices and $V = {F^n}$ . Let $e_{1} , ... , e_{n}$ be the standard basis of $F^n$. For $1 \le r \le n$ , ...
In several QFT textbooks (namely, those of Peskin and Shroeder and of Schwartz) there is presented an identity for representations of Lie algebras, $$d(R) C_2(R) = T(R) d(G),$$ where $d(R)$ is the ...