Tagged Questions

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How do I get the generators of a group formed by combining two groups with known generators?

Consider two groups, $G$ and $H$, with generating sets $S$ and $T$, respectively. (That is, $G=\langle S \rangle$ and $H=\langle T \rangle$.) Let us say that we can represent elements of both $G$ ...
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Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course. Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
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Construction of the simply connected Lie group of a given Lie algebra

Given a finite dimensional real Lie algebra $\mathfrak{g}$, I am trying to obtain a concrete realization of its simply connected Lie group $G$, with $\mathrm{Lie}(G) \cong \mathfrak{g}$. Let us ...
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Cohomology group $H^n(G, \mathbb{R}/\mathbb{Z})$ as a continuous group or a Lie group?

Is there any example of Borel cohomology group $H^n(G, \mathbb{R}/\mathbb{Z})$ for any $G$ such that $H^n(G, \mathbb{R}/\mathbb{Z})$ is a continuous group? Such as a Lie group? Most of the examples ...
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Need help with this exercise about real division algebra

I am trying to solve the following exrcise: Prove that complex multiplication does not extend to a multiplication on $\mathbb R^3$ so as to make $\mathbb R^3$ into a real division algebra. I ...
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Linear algebraic group [closed]

Let A be a finite dimensional algebra over C . This means that there is a multiplication map $f : A \times A \to A$ that is bilinear ( it is not assumed to be associative). Define the automorphism ...
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Group Axioms Motivation

Group theory is all about symmetries. Can this be seen from the axioms defining a group? Or equivalently can the group axioms be motivated from this point of view? Of course one can look at several ...
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The decomposition of the exterior of the symmetric square over Lie algebra sl(3)

I am studying the representation theory of finite dimensional modules over the simple Lie algebra $\operatorname{sl}(3)$. I know some basics facts about the decomposition of some construction of ...
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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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Connected subgroups of SU(2) and SU(3)

I am reading 'Lie groups, Lie Algebras, and Representations : An Introduction' by Brian Hall and am unable to do the problem 17 in chapter 3. It says Show that every connected Lie subgroup of ...
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Find a homomorphism from $\mathbb{Z}$ to $SO(2,\mathbb{R})$

If we pick some element $x$ in the $SO(2,\mathbb{R})$, how do we write out the homomorphism explicitly?
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What are the one-parameter subgroups of GL?

Are the multiplicative one-parameter subgroups of the general linear group (i.e., morphisms $\lambda:\Bbbk^\times\to\mathrm{GL}_n\Bbbk$ of algebraic groups) completely classified? The obvious ...
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The Quaternions and $SO(4)$

I am interested in the map $\phi:S^3 \times S^3 \to GL_4(\mathbb{R})$ given as follows: Let $(p,q) \in S^3 \times S^3$. We identify $p$ and $q$ as real quaternions with unit norms and define ...
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Lie algebra of normal subgroup is an ideal

I want to prove that if $G$ is a connected Lie group, $H$ is a normal Lie subgroup of $G$, $\mathfrak{g}$ and $\mathfrak{h}$ their respective lie algebras, then $\mathfrak{h}$ is an ideal of ...
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Computing Sylow $p$-subgroups of classical groups

Let $p>4$ be prime, and let $G=GL_2(\mathbb{F}_p)$, $H=O_3(\mathbb{F}_p)$, and $K=Sp_4(\mathbb{F}_p)$. We know that $|G|=p(p-1)^2(p+1)$, so that a Sylow $p$-subgroup of $G$ is isomorphic to ...
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Dihedral and quaternion groups as subgroups of SO(n), SU(n), Spin(n), SO(n)$\times$SO(n), SU(n)$\times$SU(n)

This is a very simple question on whether these three discrete groups $D_4$,$Q_8$,$(\mathbb{Z}_2)^3$ are subgroups of certain Lie groups. More precisely, given discrete groups below (a), (b), (c): ...
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Discrete subgroups of SU(n) and SO(n).

Thank you very much for your concern. I am in physics background, any simpler but complete explanation would be helpful. I would like to know whether there is a complete understanding of discrete ...
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Relationship between representations of $\mathfrak{sl}_{2n}\mathbb{C}$ and $\mathfrak{sp}_{2n}\mathbb{C}$

If $V=\mathbb{C}^{2n}$ denotes the standard representation of $\mathfrak{sl}_{2n}\mathbb{C}$, what can we say about $\wedge^kV$ in terms of the standard representation $W$ of ...