0
votes
1answer
37 views

Linear algebraic group

Let A be a finite dimensional algebra over C . This means that there is a multiplication map $f : A \times A \to A$ that is bilinear ( it is not assumed to be associative). Define the automorphism ...
2
votes
1answer
54 views

Group Axioms Motivation

Group theory is all about symmetries. Can this be seen from the axioms defining a group? Or equivalently can the group axioms be motivated from this point of view? Of course one can look at several ...
1
vote
0answers
36 views

Exact sequences of $1 \to A \to SO(N) \to B \to 1$, special orthogonal group

Inspired by the nice post and this, apart from SU(N), now I am particularly looking into the exact sequences of SO(N), but I will like to loosen the conditions of the previous post, Q1. $$1 \to A ...
1
vote
0answers
38 views

Exact sequences of $1 \to A \to SU(N) \to B \to 1$, special unitary group

Inspired by the nice post, I am particularly looking into the exact sequences of SU(N), but I will like to loosen the conditions of the previous post, Q1. $$1 \to A \to SU(N) \to B \to 1$$ where ...
4
votes
0answers
44 views

Exact sequences of SU(N) and SO(N)

We know that the spin group $Spin(N)$ has a short exact sequence of Lie groups. $$1 \to Z_2 \to Spin(N) \to SO(N) \to 1$$ I wonder whether there are some examples for SU(N) and SO(N) group, such ...
0
votes
0answers
21 views

Are there some examples of Kac-Moody groups which are not reductive?

Are there some examples of Kac-Moody groups which are not reductive? Thank you very much.
1
vote
0answers
21 views

Subgroups of SO$(2)$

I'm doing an independent study with John Stillwell's Naive Lie Theory and I wanted to know if I'm on the right track. I'm just looking for some confirmation that these are acceptable answers. Find ...
2
votes
1answer
34 views

Is $sp(4)$ a subalgebra of $su(5)$?

Is $sp(4)$ a subalgebra of $su(5)$? And how can I prove/disprove this? I know already that it cannot be a regular maximal subgroup of $su(5)$ since the Dynkin diagram (which has two roots of unequal ...
2
votes
1answer
37 views

Isomorphism $U(p,q)/U(1)=SU(p,q)/Z_{n}$

I have a little experience in Lie groups, so I have met the strange isomorphism: $$U(p,q)/U(1)=SU(p,q)/Z_{n}$$ Here $U(p,q)$ is a set of complex $n\times n$ matrices ($p+q=n$), which satifies the ...
0
votes
2answers
44 views

Lie bracket of vector fields on $R^2$

Compute the Lie bracket$$\Big[-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y},\frac{\partial}{\partial x}\Big]$$ on $R^2$ Can you help me please?
1
vote
1answer
31 views

Obtaining representations of $G$ from $\mathrm{Lie}(G)$.

Suppose $\mathfrak{g}$ is a semisimple Lie algebra over $\mathbb{C}$, and $\tilde{G}$ is the unique connected, simply connected Lie group whose Lie algebra is $\mathfrak{g}$. Let $C$ be any discrete ...
2
votes
0answers
25 views

Rigidity for Lie Groups

This may be a very dumb question but I was wondering if the following train of logic is correct: We know a connected Lie group $G$ is isomorphic to the quotient $G\cong \tilde{G}/\Gamma$ where ...
0
votes
0answers
28 views

Automorphism Groups of Lie Groups

Take $X$ to be a Lie group and $Aut(X)$ to be its automorphism group (group isomorphisms which are also homeomorhisms). In general, are there some Lie groups in which this can be computed? For ...
0
votes
0answers
24 views

Universal Cover of $SL_2(\mathbb{R})$

I know that there is a way to define a multiplication map on a universal cover of a Lie group given in this post. However, I was wondering if there is a way to write this multiplication explicitly ...
0
votes
0answers
34 views

Ideals in the unitary group

What would be examples of one-dimensional ideals in the lie algebra of the unitary group? Moreover, how would one show that it is in the tangent space of the center of the unitary group and that the ...
3
votes
1answer
155 views

Baker–Campbell–Hausdorff formula for [exp(x),exp(y)]

Can someone provide a explicit (the first priority with leading orders, then the secondary consider as complete as possible, or) expansion like Baker–Campbell–Hausdorff formula for the commutator: ...
2
votes
1answer
196 views

How do I represent such a transformation?

Let's say I have a 2d rectangle defined by $ [0,x_0] \times [0,y_0]$. Now lets say I cut out the middle rectangle $[\frac{1}{3} x_0, \frac{2}{3} x_0] \times [\frac{1}{3} y_0, \frac{2}{3} y_0]$. Now ...
1
vote
0answers
33 views

Maximal tori in lie groups?

How would one prove that a maximal torus in a lie group is a maximal abelian subgroup? For example, in the specific cases of SO(n) or SU(n). I know that the maximal torus of SO(2n) and SO(2n+1) is Tn ...
1
vote
2answers
70 views

Quaternion representation of rotations

How would one show that 1/3 turns correspond to the eight antipodal pairs among the 16 quaternions : $$ \pm \frac{1}{2} \pm \frac{i}{2} \pm \frac{j}{2} \pm \frac{k}{2} $$ knowing that the rotation ...
3
votes
1answer
55 views

— Cartan matrix for a semisimple Lie algebra with an extension

The question is a modified one inspired by this post: What is the Cartan matrix for this Lie algebra below? (for this semisimple Lie algebra $g(X) \oplus h(Y)$,) $$ [X_i, X_j] = f_{ij}{}^k X_k ...
0
votes
1answer
63 views

The decomposition of the exterior of the symmetric square over Lie algebra sl(3)

I am studying the representation theory of finite dimensional modules over the simple Lie algebra $\operatorname{sl}(3)$. I know some basics facts about the decomposition of some construction of ...
1
vote
1answer
52 views

With the branching rules of subalgebra, how can I write down explicit matrix elements for a representation?

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not ...
2
votes
1answer
95 views

How to determine the dual space of se(2)?

In an article there are the following sentences: The euclidean group $SE(2)=\left\{\left[\begin{array}{cc}1 & 0\\v & R\end{array}\right]:v\in \mathbf{R}^{2\times1}\text{ and }R\in ...
2
votes
0answers
28 views

Properties of matrix Lie groups which are not shared by general Lie groups

I am reading Brian Hall's book 'Lie groups, Lie algberas, and Representations' and on p52, corollary 2.34 reads : " Every continuous homomorphism between two matrix Lie groups is smooth." I am ...
4
votes
2answers
133 views

free subgroups of $SL(2,\mathbb{R})$

In the example section of the wikipedia article on the the Ping Pong lemma, you can see how to construct a free subgroup of $SL(2,\mathbb{R})$ with two generators $$ a_1 = \begin{pmatrix} 1 & 2 ...
3
votes
1answer
58 views

Is it true that stabilizer in $O(n)$ of a rank $k$ matrix is isomorphic to $O(n-k)$?

Let $X\in M_{n, k}(\mathbb R)$ such that $\textrm{rank}(X)=k$ and,$$O(n)_X:=\{A\in O(n): AX=X\}.$$ Notice $O(n)_X$ is a subgroup of $O(n)$. Is it true that $O(n-k)\cong O(n)_X$? Here $O(n)=\{A\in ...
3
votes
1answer
122 views

Find the tangent space of $\mathrm{Aff}(n)$

Find the tangent space of $\mathrm{Aff}(n)$. see Proof: Tangent space of the general linear group is the set of all squared matrices $\mathrm{Aff}(n)$ is the set of all matrices of the form $$ ...
3
votes
1answer
115 views

Show that Aff(n) is a matrix Lie group?

Definition: A matrix lie group G is a group of matrices that is closed under nonsingular limits. That is, if A1, A2, A3,... is a convergent sequence of matrices in G, with limit A, and if detA is not ...
0
votes
1answer
75 views

Derived algebra of a lie algebra contained in an ideal

Let $\mathfrak{g}$ be a Lie algebra over $\mathbb{R}$ or $\mathbb{C}$. Assume $\mathfrak{i}$ is an ideal with $\mathfrak{g/i}$ abelian. Then the derived algebra $[\mathfrak{g},\mathfrak{g}]\subseteq ...
0
votes
0answers
56 views

Show that every matrix in SO(3) = $e^X$ for some skew symmetric matrix X.

Show that every matrix A in SO(3) = $e^X$ for some skew symmetric matrix X. I have to use this statement that I already derived. A$\begin{pmatrix} \cos x & -\sin x & 0 \\ ...
1
vote
0answers
95 views

Connected subgroups of SU(2) and SU(3)

I am reading 'Lie groups, Lie Algebras, and Representations : An Introduction' by Brian Hall and am unable to do the problem 17 in chapter 3. It says Show that every connected Lie subgroup of ...
0
votes
1answer
54 views

Find a homomorphism from $\mathbb{Z}$ to $SO(2,\mathbb{R})$

If we pick some element $x$ in the $SO(2,\mathbb{R})$, how do we write out the homomorphism explicitly?
3
votes
1answer
120 views

What are the one-parameter subgroups of GL?

Are the multiplicative one-parameter subgroups of the general linear group (i.e., morphisms $\lambda:\Bbbk^\times\to\mathrm{GL}_n\Bbbk$ of algebraic groups) completely classified? The obvious ...
11
votes
1answer
258 views

The Quaternions and $SO(4)$

I am interested in the map $\phi:S^3 \times S^3 \to GL_4(\mathbb{R})$ given as follows: Let $(p,q) \in S^3 \times S^3$. We identify $p$ and $q$ as real quaternions with unit norms and define ...
1
vote
1answer
191 views

Lie algebra of normal subgroup is an ideal

I want to prove that if $G$ is a connected Lie group, $H$ is a normal Lie subgroup of $G$, $\mathfrak{g}$ and $\mathfrak{h}$ their respective lie algebras, then $\mathfrak{h}$ is an ideal of ...
6
votes
2answers
300 views

Computing Sylow $p$-subgroups of classical groups

Let $p>4$ be prime, and let $G=GL_2(\mathbb{F}_p)$, $H=O_3(\mathbb{F}_p)$, and $K=Sp_4(\mathbb{F}_p)$. We know that $|G|=p(p-1)^2(p+1)$, so that a Sylow $p$-subgroup of $G$ is isomorphic to ...
2
votes
0answers
54 views

Dihedral and quaternion groups as subgroups of SO(n), SU(n), Spin(n), SO(n)$\times$SO(n), SU(n)$\times$SU(n)

This is a very simple question on whether these three discrete groups $D_4$,$Q_8$,$(\mathbb{Z}_2)^3$ are subgroups of certain Lie groups. More precisely, given discrete groups below (a), (b), (c): ...
5
votes
0answers
358 views

Discrete subgroups of SU(n) and SO(n).

Thank you very much for your concern. I am in physics background, any simpler but complete explanation would be helpful. I would like to know whether there is a complete understanding of discrete ...
4
votes
0answers
38 views

Relationship between representations of $\mathfrak{sl}_{2n}\mathbb{C}$ and $\mathfrak{sp}_{2n}\mathbb{C}$

If $V=\mathbb{C}^{2n}$ denotes the standard representation of $\mathfrak{sl}_{2n}\mathbb{C}$, what can we say about $\wedge^kV$ in terms of the standard representation $W$ of ...
2
votes
0answers
63 views

Character of half-spin representation

Let $S^\pm$ be the half-spin representations of $\mathfrak{so}_{2n}\mathbb{C}$. Fulton-Harris's Representation Theory says on page 378 that the character $D^\pm$ of $S^\pm$ is the sum $$\sum x_1^{\pm ...
0
votes
1answer
122 views

Classifying all rank 2 and 3 root systems

I am working with the representation theory of complex simple Lie algebras, and have a question: It is intuitively clear that the root systems $A_1\times A_1$, $A_2$, $B_2$, and $G_2$ comprise all ...
8
votes
1answer
131 views

Trivial summand of a representation's symmetric power

The following comes from Exercise 13.17 of Fulton and Harris's book, Representation Theory: A First Course. Let $V$ denote the standard representation of $\mathfrak{sl}_3\mathbb{C}$, with weights ...
4
votes
1answer
96 views

Characters of diagonalizable algebraic groups with no p-torsion

Let $G$ be a diagonalizable algebraic group and $X$ be the character group of $G$. Let $Y$ be a subgroup of $X$. We define $Y^{\perp}$ to be all the $x\in G$ such that $\chi(x)=1$ for all $\chi\in Y$. ...
1
vote
1answer
176 views

Determining which maps are isomorphisms

1) Let $G=G'=\{(a,b)\mid a,b \in \mathbb{R}, a, b \ne 0\}$ with group operation $(a_1,b_1)(a_2,b_2)=(a_1a_2,b_1b_2)$. Let $\phi (a,b) = (b^{-1}, ab^2)$. My solution: 1-1: Suppose ...
1
vote
1answer
75 views

Invariant form on Lie algebra

Does anyone have a reference for the following fact? Let $G \subset GL(n)$ be a compact Lie group. Then the form $$f(A)=-Tr(A^2)$$ defined for $A \in T_e G \subset \mathfrak{gl}(n)$ is positive ...
2
votes
1answer
308 views

Conjugate Representations

Are there any general results on when conjugate representations of a real Lie algebra are equivalent? I'm inclined to say that they are often not, but this is merely going on my case by case ...
1
vote
1answer
66 views

When can you build up all representations from the fundamental and antifundamental ones?

Under what conditions can you determine all representations of a Lie algebra from the fundamental and antifundamental ones using just the tensor product, direct sum and Clebsch-Gordan decomposition? I ...
1
vote
0answers
106 views

Conjugate Representations for $\mathfrak{sl}(2,\mathbb{C})$

Let $\mathfrak{sl}(2,\mathbb{C})$ be the complex Lie algebra of $SL(2,\mathbb{C})$ and $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ be its realification; that is $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ ...
2
votes
1answer
109 views

Weyl group, permutation group

Let $U(n)$ be the unitary group and $T$ its maximal torus (group of diagonal matrix) and $N(T)$ the normalizer of $T$ in $G$. Why $N(T)/T$ is the permutation group $S_{n}$?
2
votes
1answer
36 views

How can I show that $ASL_n(F)$ is acting 2-transitively?

One of my friends asked me to ask this question here. This is a question from his last exam: Let $$ASL_n(F)=\{T_{A,v}:V_n(F)\to V_n(F)\mid\exists A\in SL_n(F), \exists v\in V_n(F), ...