All Questions
Tagged with cohomology lie-groups
40
questions
7
votes
0
answers
216
views
Relation beween Chern-Simons and WZW levels, and transgression
3d Chern-Simons gauge theories based on a Lie group $G$ are classified by an element $k_{CS}\in H^4(BG,\mathbb{Z})$, its level. Via the CS/WZW correspondence the theory is related with a 2d non-linear ...
- 761
1
vote
0
answers
60
views
How can we construct a non-trivial central extension of a Lie group
Let $G$ be a connected and simply connected Lie group with its Lie algebra $\mathfrak{g}$. Assume that $[c]\in H^2 (\mathfrak{g};\mathbb{R})$ is a non-trivial 2-cocycle. Then we can construct a non-...
- 277
6
votes
0
answers
147
views
Cohomology of $\mathrm{BPSO}(2d)$ with $Z_2$ coefficients
$\DeclareMathOperator\BPSO{BPSO}\DeclareMathOperator\PSO{PSO}\DeclareMathOperator\SO{SO}\DeclareMathOperator\BSO{BSO}$The projective group is $\PSO(2d)=\SO(2d)/Z_2$.
$\BPSO(2d)$ is the classifying ...
- 487
0
votes
0
answers
86
views
Cohomology group of a submanifold or Lie subgroup
In general: if one knows the cohomology group of some manifold ${\cal M}$, i.e. $H^n ({\cal M})$, are there known results for the same cohomology group $H^n (X)$ of a submanifold $X \subset {\cal M}$? ...
- 101
4
votes
2
answers
303
views
Low dimensional integral cohomology of $BPSO(4n)$
Toda has calculated the $\mathbb{Z}/2$‐cohomology ring of $BPSO(4n+2)$, and also gave the simple exceptional calculation of the $\mathbb{Z}/2$‐cohomology of $BPSO(4)$, in
Hiroshi Toda, Cohomology of ...
- 34.8k
8
votes
2
answers
1k
views
Ideals generated by regular sequences
In Vasconcelos' paper (Ideals generated by R-sequences), he proved
If $R$ is a local ring, $I$ an ideal of finite projective dimension, and $I/I^2$ is a free $R/I$ module, then $I$ can be ...
- 719
2
votes
0
answers
95
views
Compact $G$-ENR's and Euler characteristic computed with Alexander-Spanier cohomology with compact support
Let $(Z,A)$ a compact ENR pair, then
$$\chi(Z)=\chi_c(Z-A)+\chi(A)$$
where $\chi_c$ is the Euler characteristic taken in Alexander-Spanier cohomology with compact support (ENR means euclidean ...
- 795
8
votes
0
answers
265
views
Integral cohomology of compact Lie groups and their classifying spaces
Let $G$ be a compact Lie group and let $BG$ be its classifying space. Let $\gamma\colon \Sigma G \to BG$ be the adjoint map for a natural weak equivalence $G \xrightarrow{\sim} \Omega BG$. Taking ...
3
votes
0
answers
172
views
Lie algebra cohomology of loop algebra
Let $G$ be a simple algebraic group over the complex numbers. Is it true that the Lie algebra cohomology $H^*(L\mathfrak{g}, \mathbb{C})$ of the loop Lie algebra $L\mathfrak{g}=\mathfrak{g} \otimes \...
- 2,969
4
votes
0
answers
234
views
$L_\infty$-quasi inverse for the contravariant Cartan model on principal bundles
First of all I want to apologize for the much too long post.
A Lie group $G$ is acting on a smooth manifold $M$, then we define
\begin{align*}
T^k_G(M)=
(S^\bullet \mathfrak{g}\otimes T^k_\mathrm{...
7
votes
1
answer
712
views
Differential forms of a Lie group giving cohomology of the Lie group
Consider a manifold $M$. Then, we have the notion of differential forms on $M$ and complex associated to that, denoted by $$\cdots\rightarrow \Omega^{k-1}(M)\rightarrow \Omega^k(M)\rightarrow \Omega^{...
- 5,983
1
vote
1
answer
255
views
Why is the Chern Number Invariant under A Continuously Shrinking of the Structure Group?
In Witten's paper Three Dimensional Gravity Revisited and Quantization of Chern-Simons Theory with Complex Gauge Group, he used a fact that for a principal $G$-bundle, the quantization of the Chern ...
- 615
5
votes
0
answers
109
views
Restricting projective representations of Lie groups to lattices
Let $G$ be a simple Lie group with trivial center, and let $\Gamma$ be a lattice in $G$. Is it true that an infinite-dimensional projective representation of $G$ restricted to $\Gamma$ can be de-...
- 229
5
votes
1
answer
224
views
Which compact (finite dimensional) Lie groups have $H^1_{DR}(G)\neq 0$
In particular, I am wondering if $H^1_{DR}(G)\neq 0$ implies that the group can written as a semidirect product of $\mathbb{S^1}$ and something else, with the $\mathbb{S^1}$ factor being responsible ...
- 969
3
votes
0
answers
170
views
Finding generators of equivariant cohomology
Let $(M,\omega)$ be a symplectic manifold with symplectic form $\omega$, carrying a Hamiltonian action of a compact connected Lie group $G$ with moment map $\mu:M\to \mathfrak{g}^\ast$, where $\...
- 1,920