Podcast #128: We chat with Kent C Dodds about why he loves React and discuss what life was like in the dark days before Git. Listen now.

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.

Filter by
Sorted by
Tagged with
1
vote
0answers
21 views

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 ...
2
votes
1answer
64 views

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 ...
0
votes
0answers
22 views

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, ...
3
votes
0answers
48 views

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 ...
3
votes
1answer
37 views

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 ...
1
vote
1answer
24 views

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 $G$ ...
0
votes
1answer
71 views

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: ...
5
votes
1answer
369 views

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 ...
0
votes
1answer
240 views

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 ...
1
vote
0answers
37 views

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, ...
1
vote
2answers
301 views

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 $$\...
0
votes
1answer
85 views

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}\...