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The title says it all: are there examples of Lie groups that cannot be represented as matrix groups?

Thanks in advance.

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

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The title says it all: are there examples of Lie groups that cannot be represented as matrix groups?

Thanks in advance.

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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.

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martinbn

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Your notation for the unitary groups is unconventional.fresh_42 said:

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.

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martinbn

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

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Thank you. Can you tell me how it is defined, or under what name I can look up information about that group?martinbn said:##\widetilde{SL}_2(\mathbb R)##

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martinbn

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Metaplectic

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

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Tank you to both of you! This is what I was looking for!martinbn said:Metaplectic

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WWGD

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

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I guess we want to have a faithful representation to call a group a matrix group.WWGD said:Don't you always have a trivial representation sending everything to the identity?

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WWGD

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How about the Cayley representation then, as a group of permutations?fresh_42 said:I guess we want to have a faithful representation to call a group a matrix group.

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Are we still talking about Lie groups?WWGD said:How about the Cayley representation then, as a group of permutations?

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WWGD

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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.fresh_42 said:Are we still talking about Lie groups?

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martinbn

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

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Infrared

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martinbn said:##\widetilde{SL}_2(\mathbb R)##

martinbn said: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}##.

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martinbn

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Yes, but that was clear from the link.Infrared said: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}##.

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Infrared said: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.

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Infrared

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The universal cover of a space is simply connected. The metaplectic group is not simply connected. So they are different.dextercioby said:The tilde on top of the group "name" exactly universal cover of that group means. So the two groups are not different.

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

A Lie group is a type of mathematical group that is continuous, meaning it can be described by a smooth curve, and is also a differentiable manifold, meaning it can be described by a set of equations. Lie groups are often used in physics and geometry to describe symmetries and transformations.

A matrix representation of a Lie group is a way of describing the group using matrices, which are arrays of numbers. This is often done by finding a set of matrices that satisfy the group's defining equations and can be used to perform the same operations as the group.

Not all Lie groups can be represented in matrix form because some groups have infinite dimensions, meaning they have an infinite number of elements. Matrices, on the other hand, have a finite number of elements, so they cannot fully represent these infinite-dimensional groups.

One example of a Lie group that cannot be represented in matrix form is the group of diffeomorphisms, which are transformations of a differentiable manifold that preserve its structure. This group has infinite dimensions and therefore cannot be represented using matrices.

Lie groups that cannot be represented in matrix form are studied using other mathematical tools, such as differential equations and functional analysis. These methods allow for a more abstract and general understanding of these groups, without relying on specific matrix representations.

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