Questions tagged [central-extensions]
Use this tag for questions about short exact sequences of groups 1 → A → E → G → 1 such that A is in the center of E.
32 questions
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Does every central extension come from a 2-cocycle?
Let $G$ be a Lie group and $[C]\in H^2 (G;\mathbb{R})$. We can define a group $G_C\equiv G\times \mathbb{R}$ with the operation $$(f,\alpha )\cdot (g,\beta)\equiv (fg,\alpha +\beta +C(f,g)).$$ One can ...
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Central extensions versus 2-cycles
Let $G$ be a Lie group and $[C]\in H^2 (G;\mathbb{R})$. Then $\hat{G}_C :=G\times \mathbb{R}$ with multiplication $(g,r)\cdot (f,s):=(gf,r+s+C(g,f))$ (i.e. $\hat{G}_C=G\ltimes \mathbb{R}$) is a ...
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Classifying central extensions of perfect groups
In Weibel's book (Exercise 6.9.1), it is claimed that central extensions of the form $0\to A\to X\to G\to 1$ with G perfect are classified by $Hom(H_2(G,\mathbb{Z}), A)$.
My thoughts so far lead me ...
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Comparing conjugacy classes in central extensions
Let $G$ be a group containing a finite central subgroup $Z$. Considering the conjugation action of $G$ on $G/Z$, define the stabilizer subgroups $$\operatorname{Stab}_G(gZ) = \{h \in G\colon hgh^{-1} =...
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What exactly is a universal central extension?
Let $G$ be a group. In An introduction to homological algebra, Chapter 6.9 Weibel defines a universal central extension as a central extension
$$0 \to A \to X \to G \to 1,$$
which is initial with ...
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Reference request: Universal central extension of $\operatorname{PSl}_2(\mathbb Z)$ is the braid group $\mathcal B_3$.
$\DeclareMathOperator{\PSl}{PSl}$
According to Wikipedia, the universal central extension of $\PSl_2(\mathbb Z)$ is given by the braid group $\mathcal B_3$ on three strands,
\begin{equation} \label{...
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If central extensions are isomorphic as principal bundles, are they isomorphic as extensions?
Let
$$1 \to G \overset{\iota_i}{\to} H_i \overset{\pi_i}{\to} K \to 1 $$
be two central extensions of $K$ by $G$. ($i=1,2$). Assume for simplicity that this is an extension of Lie groups, and $G$ is ...
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Internal symmetries in abelian or non abelian groups
so I've been studying the centrally extended Galilei group, by restricting to 1D translation, the group becomes abelian and that means that all the irreps of the groups are of dimension 1. A ...
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Why does the definition of cocyle in a central extension of a Lie algebra work?
I am currently reading Edward Frenkel's "Langlands Correspondence for Loop Groups", freely available here: https://math.berkeley.edu/~frenkel/loop.pdf.
In the appendix $A.4$ he describes ...
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Group Cohomology, Module Extensions, and Group Extensions, and $Ext^2_{\mathbb{Z}G}(\mathbb{Z},A)$
I've read that for some $G$-module $A$, group cohomology can be defined as $$H^{n}(G,A)=Ext^{n}_{\mathbb{Z} G}(\mathbb{Z},A).$$
I've also read that for two $R$-modules $C,D,$ $Ext^{n}_{R}(C,D)$ can be ...
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Proof of $H^2(\mathfrak g_1,\mathfrak g_1)=0$
I want to prove $H^2(\mathfrak g_1,\mathfrak g_1)=0$ where $\mathfrak g_1=\mathbb R$, the $1$-dimensional (abelian) Lie algebra.
I think I need to show two things:
All central extensions $\mathbb R \...
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Does $\mathbb R \overset{\iota}{\hookrightarrow} \mathfrak{aff}(1,\mathbb R) \overset{\pi}{\twoheadrightarrow} \mathbb R$ Lie algebra extension exist?
I think I have an example for a $\mathbb R \overset{\iota}{\hookrightarrow} \mathfrak{aff}(1,\mathbb R) \overset{\pi}{\twoheadrightarrow} \mathbb R$ Lie algebra extension.
$$\iota:\mathbb R\to \...
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Group Cohomology and Pontryagin Duality
My question is related to this question, which I tried to post an answer. I think it is better to ask a question directly.
My question is from the book "Foundations of Quantum Theory: from ...
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Connection between central extensions and universal covering groups for projective representations
Disclaimer, I'm a physics student, so some of my arguments might be a little bit sloppy. I'm trying to find a somewhat straightforward general argument to explain why we need central extensions when ...
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Quaternion group as a central extension of a $2$-group [closed]
I'll appreciate it if you help me to tackle this situation.
I'm going to characterize 2-group $G$ whose two main properties such as $cd(G)=\{1,2\}$ and there is normal abelian subgroup $P$ such that $...
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Nilpotent groups can be constructed by means of abelian groups.
I am studying A course in theory of groups by Robinson. When defining the central extension of groups, the author says that
Every nilpotent group can be constructed from abelian groups by means of a ...
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Why are there only two central extensions corresponding to $H^2(C_p; \Bbb Z/p)$ which has $p$ elements?
I'm trying to understand why $H^2(C_p; \Bbb Z/p)$ despite having $p$ elements there are only two distinct central extensions of $C_p$ by $\Bbb Z/p$. My lecture notes says:
In particular, $H^2(C_p;\...
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Some clarifications required about the two extremes of general extensions (semi-direct products and central extensions)
This is a sequel to Why can the homomorphism $\phi$ in semi-direct products only be varied by inner automorphisms upon changing the complement group?
My professor made another remark that:
Let's go ...
user568976
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Is every generator of $Z({\rm Spin}_n^{\epsilon}(q))$ a square element in finite spin group ${\rm Spin}_n^{\epsilon}(q)$?
A. I wonder if every generator of $Z({\rm Spin}_n^{\epsilon}(q))$ is a square element in ${\rm Spin}_n^{\epsilon}(q)$?
B. When $Z(\Omega_{2m}^{\epsilon}(q))\cong C_2$, is the generator of $Z(\Omega_{...
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Nilpotent Lie algebras are closed under extensions
Problem: Let $L$ be a Lie algebra and $K$ an ideal such that $L/K$ is nilpotent and such that $ad(x)|_K$ is nilpotent for all $x \in L$. Prove that $L$ is nilpotent.
By Engel's Theorem, I know that $...
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What controls the central extensions of Lie algebras when the Lie Group is not compact?
If the Lie algebra $\mathfrak{g}$ can be realized as the tangent space of a compact Lie group $G$, then all the possible central extensions of $\mathfrak{g}$ are in one to one correspondence which the ...
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What is the explicit central extension given by the prequantization procedure (from functions on phase space to vector fields)?
The Question:
The Lie algebra of functions on phase space (under the Poisson bracket) is a central extension of the Lie algebra of Hamiltonian vector fields on phase space (under the vector field ...
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Haar measure construction for extended Lie Algebra
Consider a Lie algebra $\mathcal{G}$ of the form $$[T_i, T_j] = f_{ij}^k T_k$$ which has an Abelian (corresponding to the Abelian Lie algebra $A$) central extension $\mathcal{H}$ of the form $$[T'_i, ...
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Structure of affine Lie algebras
It is well-known that to every simple Lie algebra $\mathfrak{g}$ one can associate an affine Kac-Moody algebra by a double extension (once by a 2-cocycle and once by a derivation). One can then show ...
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Virasoro algebra question: Is there a two-surface in Diff($S^1$) with a non-zero integral over the cocycle in $H^2(\mathfrak{g}, \mathbb{C})$?
I am a physicist so forgive me if this question doesn't make sense.
You can start off by defining the Witt algebra, which I'll call $\mathfrak{g}$, as the complexified Lie algebra of vector fields on ...
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Basic assertion about central extensions
I'm having trouble verifying what ought to be a relatively simple detail in a proof of Milnor's book on algebraic K-theory, in the section on universal central extensions.
Here is the set up. Let $G$ ...
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Algorithms to determine the explicit forms of possible group extensions
I learned from link 1, link 2 link3 that sometimes it is possible to write down all possible group extension. I also know that the group extension is classified by group cohomology.
My questions are: ...
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Central extensions versus semidirect products
Consider an extension $E$ of a group $G$ by an abelian group $A$.
$$1 \to A \overset{\iota}{\to} E \overset{\pi}{\to} G \to 1$$
Two special kinds of extensions are:
Central Extensions: $A$ is ...
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Classification of projective representations in terms of linear representations of central extensions
Let $k$ be a field, and let $k^\times$ denote its multiplication group. Further let $\mathrm{PGL}(V,k)$ denote the projective general linear group of some vector space $V$ over the field $k$. Denote ...
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Central Extensions and Homomorphisms
We have the short exact sequence:
$$1\rightarrow C\rightarrow \widetilde{G}\rightarrow G\rightarrow 1$$
Equipped witha map $i$ from $C$ to $\widetilde{G}$, and a map $p$ from $\widetilde{G}$ to G, ...
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Central extension of a Lie algebra, why is the bilinear form a 2-cocycle?
My professor talked about a central extension of a Lie algebra $\mathfrak g$, which he defined as $\tilde{\mathfrak{g}}=\mathfrak g\oplus\mathbb C$. The Lie bracket on $\tilde{\mathfrak{g}}$ is
$$\...
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Why do linear splitting maps of Lie Algebra central extensions induce cocycles?
If we consider a central extension $\mathfrak h$ of a Lie algebra $\mathfrak{g}$ by the abelian $\mathfrak a$:
$$0 \longrightarrow \mathfrak a \longrightarrow \mathfrak h \stackrel{\pi}\...
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