# Tagged Questions

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### Baker–Campbell–Hausdorff formula for [exp(x),exp(y)]

Can someone provide a explicit (the first priority with leading orders, then the secondary consider as complete as possible, or) expansion like Baker–Campbell–Hausdorff formula for the commutator: ...
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### Simple Lie algebra is a matrix algebra?

Wedderburn's Theorem. Let $A$ be a simple finite $k$-algebra. Then $A$ is a matrix algebra over a finite $k$-algebra $K$ which is a skew field. (Here matrix algebra means $A=M_n(K)$ for some $n$.) ...
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### Explicit Weyl group invariant polynomials

Quoting this post, "Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset\mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to g. Let ...
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### Kernel of homomorphism on unit circle S1

Let $f : S^1 \to S^1$ be defined such that $f(z) = z^2$, where $z$ is a complex number. It's easy to check that this is a homomorphism on $S^1$. However, how would you find the kernel and the coset ...
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### Models for Lie algebra E8 and octonions

I've heard that one can construct the exceptional Lie algebra $E_8$ as the Lie algebra of the group of isometries of projective plane over octonions, or something of this form. Unfortunately, I do not ...
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### Corestriction map in lie algebra cohomology

Given a lie algebra $\mathfrak{g}$ over a field $k$, we can define the cohomology groups of $\mathfrak{g}$ as follows: $$H^n(\mathfrak{g},k):=\mathrm{Ext}_{U(\mathfrak{g})}^n(k,k)$$ where ...
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### With the branching rules of subalgebra, how can I write down explicit matrix elements for a representation?

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not ...
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### Linear structure on the category of formal groups

Let $R$ be a commutative ring. If $R$ is a $\mathbb{Q}$-algebra, then the category of formal groups over $R$ (or the category of formal group laws) carries the structure of an $R$-linear category; ...
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### Status of a question from Freeman Dyson's 1972 article

In a famous article, Freeman Dyson mentions an interesting relationship between the $\tau$ functions of number theory and the dimensions of finite-dimensional simple Lie algebras (section 2). He ...
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### Computing the fundamental groups of simple algebraic groups of type $A$

I'm interested in seeing the computation for the fundamental groups of the simple algebraic groups of type $A$. Below is the definition of the fundamental group for a simple algebraic group $G$. Let ...
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### How should I show that the Lie algebra so(6) of SO(6) is isomorphic to the Lie algebra su(4) of SU(4)?

As far as I can see, an isomorphism of Lie algebras is a bijective map which preserves the Lie bracket. I need to show that $\mathfrak{so}(6)$ (the Lie algebra of SO(6)) is isomorphic to the ...
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### Question regarding isomorphisms in low rank Lie algebras

I am reading Brian Hall's book 'Lie Groups, Lie Algebras, & Representations' and on p.271 I find that in low rank Lie algberas there are some isomorphisms. For example, ...
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### Irreducibility of Lie algebra representations

Let $\mathfrak{g}$ be a finite dimensional real Lie algebra and $\pi: \mathfrak{g} \to \mathfrak{gl}(V)$ be a homomorphism of real Lie algebras where $V$ is a finite dimensional real vector space. But ...
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### Nil and nilpotent restricted lie algebras

Let $k$ be a field of characteristic $p$, and let $L$ be a restricted Lie algebra over $k$. Thus $L$ is a lie algebra together with a map $(-)^{[p]}:L\to L$ satisfying the three axioms found here. ...
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### Image of a nilpotent Lie algebra

Suppose $L_1$ and $L_2$ are two Lie algebras, and that $f: L_1\to L_2$ is a Lie algebra homomorphism. If $L_1$ is nilpotent, does it follow that $f(L_1)$ is nilpotent? Remark 1. The corresponding ...
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### Does every Lie algebra come from commutator of some associative product operation?

Suppose $\mathfrak{g}$ is an Lie algebra. Is it possible to define an associative product operation $\star$ on $\mathfrak{g}$ such that $[A,B]=A\star B - B \star A$ ? If it is not possible to do so ...
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### Show that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is isomorphic to V + iV

Let V be a real n-dimensional vector space. Show that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is isomorphic to V + iV. Note that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is a real vector space and is ...
### Existence of an irreducible $L$-submodule
Suppose $L$ is a finite dimensional Lie algebra. Let $V$ be an $L$-module (i.e. $V$ is a vector space which $L$ acts upon). We are assuming that $V$ has a finite dimension. My question is the ...