Example of a Lie group that cannot be represented in matrix form?

[nrqed]

I am not sure if this is the right forum to post this question.
The title says it all: are there examples of Lie groups that cannot be represented as matrix groups?

Thanks in advance.

 

[fresh_42]

Here (at the beginning) is an example of a local Lie group
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/

However, we have the adjoint representation $\operatorname{Ad}\, : \,G\longrightarrow \operatorname{GL}(\mathfrak{g})$, and $\operatorname{Ad}(G)$ is a Lie subgroup of $\operatorname{GL}(\mathfrak{g})$. If it is a monomorphism, we automatically get $\operatorname{G}\cong \operatorname{Ad}(G)$ and have a matrix group.

 

[martinbn]

Here (at the beginning) is an example of a local Lie group
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/

However, we have the adjoint representation $\operatorname{Ad}\, : \,G\longrightarrow \operatorname{GL}(\mathfrak{g})$, and $\operatorname{Ad}(G)$ is a Lie subgroup of $\operatorname{GL}(\mathfrak{g})$. If it is a monomorphism, we automatically get $\operatorname{G}\cong \operatorname{Ad}(G)$ and have a matrix group.

Your notation for the unitary groups is unconventional.

 

[martinbn]

$\widetilde{SL}_2(\mathbb R)$

 

[nrqed]

$\widetilde{SL}_2(\mathbb R)$

Thank you. Can you tell me how it is defined, or under what name I can look up information about that group?

 

[martinbn]

Metaplectic

 

[fresh_42]

Here's a description: https://en.wikipedia.org/wiki/Metaplectic_group

 

[nrqed]

Metaplectic

Tank you to both of you! This is what I was looking for!

 

[WWGD]

Don't you always have a trivial representation sending everything to the identity?

 

[fresh_42]

Don't you always have a trivial representation sending everything to the identity?

I guess we want to have a faithful representation to call a group a matrix group.

 

[WWGD]

I guess we want to have a faithful representation to call a group a matrix group.

How about the Cayley representation then, as a group of permutations?

 

[fresh_42]

How about the Cayley representation then, as a group of permutations?

Are we still talking about Lie groups?

 

[WWGD]

Are we still talking about Lie groups?

Cant every group be described as a permutation group? Unless you want to preserve any other than algebraic properties, it seems it would work, though I don't see how to do it with Lie groups.

 

[martinbn]

Cant every group be described as a permutation group? Unless you want to preserve any other than algebraic properties, it seems it would work, though I don't see how to do it with Lie groups.

If the group is not finite, these permutations are not going to be matrices.

 

[Infrared]

$\widetilde{SL}_2(\mathbb R)$

Metaplectic

These groups are different. The metaplectic group $Mp(2)$ is not simply connected because it is a double cover of $Sp(2)=SL_2(\mathbb{R}),$ which has fundamental group of $\mathbb{Z}$.

 

[martinbn]

These groups are different. The metaplectic group $Mp(2)$ is not simply connected because it is a double cover of $Sp(2)=SL_2(\mathbb{R}),$ which has fundamental group of $\mathbb{Z}$.

Yes, but that was clear from the link.

 

[dextercioby]

These groups are different. The metaplectic group $Mp(2)$ is not simply connected because it is a double cover of $Sp(2)=SL_2(\mathbb{R}),$ which has fundamental group of $\mathbb{Z}$.

The tilde on top of the group "name" exactly universal cover of that group means. So the two groups are not different.

 

[Infrared]

The tilde on top of the group "name" exactly universal cover of that group means. So the two groups are not different.

The universal cover of a space is simply connected. The metaplectic group is not simply connected. So they are different.

 

[dextercioby]

Absolutely, it seems my memory betrays me. I stand corrected.