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Questions tagged [lie-groups]

A Lie group is a group (in the sense of abstract algebra) that is also a differentiable manifold, such that the group operations (addition and inversion) are smooth, and so we can study them with differential calculus. They are a special type of topological group.

Consider using with the (group-theory) tag.

Lie groups are groups that are also differentiable manifolds that represent the best developed theory of continuous symmetry of mathematical objects.

Examples of lie groups are:

1) The Euclidean space $\mathbb{R}^n$ under addition is a lie group.

2) The special orthogonal group of real orthogonal matrices with determinant $1$ (note that $n=3$ is the rotation group in $\mathbb{R}^3$).

3) The spin group, which is the double cover of the special orthogonal group such that $\exists$ a sequence of lie groups:

\begin{equation*} 1\to Z_2\to~\text{Spin}(n)\to SO(n)\to 1. \end{equation*}

Note that it has dimension $\frac{n(n-1)}{2}.$

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Is the exponential function isometrical?

I'm reading Stillwell's Naive Lie Theory. At end of Ch4 he said, For readers acquainted with differential geometry, it should be mentioned that the exponential function can be generalized even beyond matrix groups, to Riemannian manifolds. In this…
athos
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Prove $\exists$ neighborhood of $I \in Gl(n,\mathbb{C})$ containing no nontrivial subgroup.

Prove that there exists a neighborhood of the identity $I \in Gl(n,\mathbb{C})$ that contains no subgroup other than $\left\{ I \right\}$. Thanks!
Doug
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Show that the following groups are Lie subgroups of $\mathrm{GL}(2)$.

Let $G = \left\{ \mathrm{exp}\, t \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \right\}$. I want to show that $G$ is a Lie subroup of $\mathrm{GL}(\Bbbk)$, where $\Bbbk = \mathbb{R}$ or $\mathbb{C}$. To do so I have to show that the $G$ can be…
Butters Stotch
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Behavior of Matrices in SO(3)

Show that each matrix in SO$(3)$ equals $e^X$ for some skew-symmetric $X$. Here, SO(3) refers to the special orthogonal group that is the rotation group of $\mathbb{R}^3$. I am also supposed to use the following facts that I have derived in…
user824237
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How to find all homomorphisms from SU(2) to U(1)

How to find all such non-trivial homomorphisms? At the Lie algebra level, should it just be a projection to the one-dimensional sub-space arbitrarily singled out? Could there be other Lie algebra homomorphisms? I searched on-line but failed to find…
X-Naut PhD
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understanding compact symplectic groups

I am following Brian Hall's "Lie groups, Lie algebra and representations". Compact Symplectic groups (Sp(2n;R)) are intersection of skew-symmetric bilinear forms preserving Symplectic groups and inner product preserving unitary groups. I am having…
Hemanta Mandal
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Is the dimension of $G/H$ even in the following context?

Let $G$ be a connected compact Lie group and let $H$ be a connected Lie subgroup of $G$ such that $G$ and $H$ have the same rank. I've come across a formula (given under the above assumptions ) Wich contain the expression $(-2\pi)^{-\operatorname…
Mira
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Are the difinition of the coset in the finite group theory and the definition of the coset space in the Lie group theory equivalent?

In the finite group theory, if we ignore the group structure, we can tell that the group can be written as the direct product of the subgroup and the representative elements of the cosets. However, even if we ignore the group structure, the lie…
金广羊
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Rank 4 Lie Group inclusion

The compact simple Lie groups $ SO_8(\mathbb{R}) $ and $ SO_9(\mathbb{R}) $ both have rank 4. The group $$ G=SU_3 \times SU_2 \times U_1 $$ also has rank 4. Does there exist a subgroup of $ SO_8(\mathbb{R}) $ or $ SO_9(\mathbb{R}) $ isomorphic to $…
Ian Gershon Teixeira
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What is the connection between the generator of a Lie group, the element of the group, and the exponential?

(I'm not a mathematician, unfortunately, and I don't really master my own question. I have looked at all questions close to my question, but I could not understand anything because the level is much too high in mathematics.). *Let's consider some…
Mathieu Krisztian
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How will I prove that determinant $det: GL (R) → R$ is a homomorphism? where R is a ring and $G=GL(R)_{n}$ a group

My Algebra teacher made that statement and asked why it was true. I am not able to formulate a correct demonstration. I thought of something using Det (AB) = Det (A). Det (B) but I think I'm wrong.
user899489
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Subgroup of $GL(n,\mathbb{C})$ that is not a Matrix Lie group

Given the definition of a Matrix Lie Group as a subgroup of $GL(n,\mathbb{C})$ such that for every sequence $A_n$ its limit $A$ is either in the subgroup or it is not invertible, what are the known examples for which the limit $A$ is invertible but…
Daniele Bernardini
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Lie Algebra over Quaternions

We define the skew-symmetric hermitian inner product $$\phi(x,y)=\bar{x}^tjy$$ over $\mathbb{H}^2$ and are asked to calulate the Lie algebra of the group $G\le GL(2,\mathbb{H})$ of automorphisms of $\phi$. My solution is the following: We need to…
hmmmm
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Lie map Question

I am trying to prove the below but I am having problems starting and I think that I am misunderstanding something? If we have that G is a connected linear group and $p:G\rightarrow GL(V)$ and ,V a finite dimensional vector space, is a homomrphism…
hmmmm
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Exponential chart G-equivariant

Consider G lie group, $(M,g)$ Riemannian Manifold and $\phi: G \times M \longrightarrow M$ isometric action, what does it mean to say that we have an exponential chart G-equivariant?
Toy
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