# 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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### How to explain this contradiction about Weyl group of $SL_n(K)$?

I have some difficulties in understanding why the Weyl group of algebraic group $SL_n(K)$ is isomorphic to symmetric group $S_n$. Let $G=SL_n(K)$ be the simply-connected algebraic group over the ...
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### Table of e8 representations

I want to understand the representation theory for the (complex-valued) $e8$ exceptional Lie algebra. An ideal answer to this question would contain a link to a text file (or any other format) ...
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### Baker-Campbell-Hausdorff/Zassenhaus formula to first order in one matrix

Is there a closed-form expression for the term of $e^{t(c \hat{X} + d \hat{Y})}$ that is first-order in $d$, where $t$, $c$, and $d$ are scalars and $\hat{X}$ and $\hat{Y}$ are finite-dimensional ...
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### Tensor product of representations of a Lie algebra (or Lie Superalgebra)

Let $V$ and $W$ be finite dimensional irreducible representations of a Lie Algebra or a Lie Superalgebra. If $V$ is one dimensional, is $V\otimes W$ necessarily irreducible? I know this to be true ...
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### Trace identities for $\text{SO}(n)$

The Green-Schwarz mechanism in Type I string theory involves certain identities relating traces in the vector and adjoint representations of $\text{SO}(n)$ of dimension $n$ and $n(n - 1)/2$ ...
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### Multiple of roots in symmetric spaces

Fix a Cartan subalgebra $\mathfrak{h}$ on a (compact simple) Lie algebra $\mathfrak{g}$ and consider the associated root system. If $\alpha$ is a root, it is well-known that $k\alpha$ is also a root ...
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### Explicit matrix representation of $\mathfrak{sl}_3$

Given a semisimple Lie algebra $\mathfrak{g}$ and a dominant integral weight $\lambda$ (and all the other necessary data), I want to be able to write down a matrix representation for $V(\lambda)$, the ...
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### Is the assignment of a root system to a semisimple Lie algebra functorial?

As described here, we have a category of root systems, where a morphism from a root system $\Phi$ in a Euclidean space $E$ to a root system $\Phi'$ in $E'$ is given by a linear map $f: E \to E'$ such ...
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### Non Inner Automorphism of Lie Algebras

I have seen some examples of inner automorphisms of Lie algebras. Can anyone please give me an example of an automorphism of Lie algebras that is not inner (with proof). Note - An automorphism is said ...
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### Finite Dimensional Representation of Lie Algebra.

Let $V, W, U$ be finite dimensional representations of a lie algebra $\mathfrak{g}$. Show that $\hom(V \otimes W, U) \cong \hom (V, U \otimes W^*)$. I think I have to use the enveloping algebra of ...
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### Is the universal enveloping algebra functor exact?

The universal enveloping algebra is a functor from Lie algebras to unital associative algebras, and is left adjoint to the functor which sends a unital associative algebra to a Lie algebra with ...
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### infinitesimal generators of the conformal inversions

More broadly speaking, this question involves the subgroups of the conformal Lie groups on Euclidean space. With some insight, one will know that these consist of the infinitesimal rotations, ...
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### How to find a Cartan subalgebra of $so(3)$.

Let $so(3)$ be the Lie algebra given by $$so(3) = \{X \in \text{Mat}_{3 \times 3}: X^T = - X \}.$$ Here $\text{Mat}_{3 \times 3}$ is the set of all $3 \times 3$ matrices and $X^T$ is the transpose ...
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### Regular Representation of Lie Algebras

I have a basic understanding of Lie Algebra and it may be naive but is there a regular representation of lie algebras as in case of Finite Groups ? Do the generators form a representation ?
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