For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

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Adjoint action on Lie algebra su(2) ($A \in SU(2), X \in \mathfrak{su(2)} \Rightarrow AXA^{-1}\in \mathfrak{su(2)}$)

I am trying to understand ho $SU(2)/\{\pm I\} \cong SO(3)$ (see: how to show $SU(2)/\mathbb{Z}_2\cong SO(3)$) but i am not sure about the adjoint action. In especially, as I understand, the adjoint ...
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1answer
45 views

$su(2) $ and $ sl(2;R)$ are not isomorphic? [duplicate]

As real Lie algebras, both are three-dimensional. The basis of $su(2)$ is $$ \left( \begin{matrix} i & 0 \\ 0 & -i \end{matrix} \right), \left( \begin{matrix} 0 & 1 \\ -1 & 0 ...
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28 views

Lie functor produces an antihomomorphism in Lavendhomme's synthetic differential geometry text?

Classically the Lie functor maps a Lie group homomorphism to a Lie algebra homomorphism. But in Proposition 15 on page 249 in Basic Concepts of Synthetic Differential Geometry, Lavendhomme states that ...
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2answers
67 views

Lie algebra of the automorphism group of a Lie group?

Let $G$ be a Lie group and $\text{Aut}(G)$ the group of all Lie group automorphisms of $G$. If $\text{Aut}(G)$ can be interpreted to be a Lie group (for example, in the context of synthetic ...
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35 views

Can the parameter $t$ in the exponential map $e^{tX}$ be complex?

From a Lie algebra to a Lie group, can the parameter $t$ in the exponential map $t\rightarrow e^{tX}$ be complex? If the Lie algebra is a complex one, this is legal, right?
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21 views

Lie transformation applied to Poisson brackets

Given the following Lie transformation: $$ \exp(\lbrace H, \cdot \rbrace):=\sum_{n=0}^{\infty} \frac{(\lbrace H, \cdot \rbrace)^n}{n!} $$ and apply it to a Poisson Bracket $\lbrace g_1, g_2 ...
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30 views

Action of $sl(2,\mathbb{C})$ on Dual of Polynomials does not Exponentiate

Let $V$ be the space of holomorphic polynomial functions in two complex variables $\xi,\eta$ and let $V^\ast$ be its dual space with subspace $W$ of linear functionals of the form $Df(1,0)$ where $D$ ...
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1answer
45 views

Killing fields and symmetries

In my answer to Why are Killing fields relevant in physics? I wrote: The flow of a Killing field can drag the whole manifold. Since Killing fields are divergence free and thus their flows have no ...
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43 views

Is the exponential map of complex spin group surjective?

The complex spin group $Spin(n,C)$ is defined as the double cover of $SO(n,C)$. If the the exponential map is surjective, it will give a parametrization of this Lie group. Is it true for this ...
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27 views

Definition for Shimura datum

The following definition for $\textbf{shimura datum}$ is due to wikipedia. Let $S=\mathrm{Res}_\mathbb{R}^\mathbb{C}G_m$ be the Weil restriction of the multiplicative group from complex field ...
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1answer
60 views

Derivations of $k[x]/(x^2)$

I am trying to find all the derivations of $k[x]/(x^2)$. Let us consider only outer (i.e. not inner) derivations. It is easy to see that the map given by $$ \delta: r + sx \mapsto sx $$ is a ...
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52 views

Proving $e^{-i\left( \frac{\theta}{2}n\cdot \sigma\right) }=\cos\left(\frac{\theta}{2}\right) - i n \cdot \sigma \sin\left(\frac{\theta}{2}\right) $

I believe I've almost proved that $e^{-i\left( \frac{\theta}{2} n \cdot \sigma\right) } = \cos\left(\frac{\theta}{2}\right) - i n \cdot \sigma \sin\left(\frac{\theta}{2}\right)$ such that $ n ...
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1answer
24 views

Prove identity map of a Lie algebra is unique Cartan involution when Killing form is negative definite

First some Definitions for convenience: Let ${\mathfrak {g}}$ be a real semisimple Lie algebra and let $B(\cdot ,\cdot )$ be its Killing form. An involution on ${\mathfrak {g}}$ is a automorphism ...
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1answer
58 views

The set of ideals of a solvable lie algebra L is a chain?

I know that every finite dimensional lie algebra over a field $\mathcal{F}$ has a unique maximal solvable ideal, all subalgebras of a solvable lie algebra are also solvable, and a sum of solvable lie ...
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1answer
28 views

How to find roots of a Lie algebra from simple roots using root strings?

Suppose you've been given the simple roots of a Lie algebra. When finding the remaining roots, do you need to check the root string of all the roots through all the other roots, just simple roots ...
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21 views

Cartan type of real forms of semisimple Lie algebras

In lists of Cartan's classification of the real forms of semisimple complex Lie algebras, the (isomorphism classes of) real forms of the families of classical complex semisimple Liealgebras A,B,C,D ...
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13 views

Flag varieties from the representation of a solveable Lie algebra

I've been reading Lie Algebras, and I've come across this problem: "Let $\mathfrak{g}$ be a solveable Lie Algebra over $\mathbb{R}$. $V$ a vector space over $\mathbb{R}$, and $\rho$ a representation ...
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1answer
30 views

Radical of a Lie algebra bracket itself

Let $\mathfrak{g}$ be a Lie Algebra over $k$, $\mathfrak{n}$ its radical. Why is $[\mathfrak{n},\mathfrak{g}]$ the smallest of its ideals $\mathfrak{a}$ such that $\mathfrak{g}/\mathfrak{a}$ is ...
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38 views

Nilpotent Lie Algebra with determinant 0 [duplicate]

If I have a nilpotent Lie Algebra $\mathfrak{g}$ and a representation $\rho(X)$ in a vector space $V$ such that $det \rho(X) = 0 $ for all $X \in \mathfrak{g}$, then how do I show that there is a ...
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Quotient of the abelian Lie group $(\mathbb{C}, +)$ by a full rank lattice

How can I show that a quotient of the abelian Lie group $(\mathbb{C}, +)$ by a full rank lattice has no faithful finite-dimensional linear representation as a complex Lie group? I was thinking of ...
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1answer
24 views

Every finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{R})$ is semisimple.

Can I deduce that every finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{R})$ is semisimple from the fact that every finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{C})$ is ...
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1answer
30 views

Isomorphism of Lie algebras $\mathfrak{g}\simeq \mathfrak{X}^v(P)$?

Let $\pi:P\longrightarrow M$ be a $G$-principal bundle. For $p\in P$, $V_p$ denotes the space of tangent vertical vectors, that is, $V_p:=T_pP_{\pi(p)}$. The space of vertical vector fields will be ...
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1answer
62 views

Let $\rho$ f.d. rep of a nilpotent Lie algebra such that $\rm{det} \rho(X) = 0$, $\forall X$. Then $\exists v \neq 0$: $\rho(X)v = 0, \forall X$.

Could you help me with this question? Let $\mathcal{g}$ be a nilpotent Lie algebra over $k$ and $\rho$ a representation of $\mathcal{g}$ in a finite-dimensional nonzero vector space $V$ over $k$. ...
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41 views

Multiplicities in weight diagram of representations of $\mathfrak{sl}(3,\mathbb{C})$

In the weight diagram of an irreducible (finite dimensional, complex) representation of $\mathfrak{sl}(3,\mathbb{C})$, there are 'rings' of weights in the shapes of triangles or hexagons. Is there an ...
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1answer
42 views

Lie group question: If $\gamma^{-1}\dot{\gamma}\in\mathfrak{g}$ everywhere, does $\gamma(t)\in G$?

Let $G$ be a Lie subgroup of $GL(n,\Bbb R)$ and $\mathfrak{g}\subseteq M(n,\Bbb R)$ its Lie algebra. Suppose that we have a smooth curve $$\gamma:\Bbb R\to G$$ with $\gamma(0)=I$. Then, it induces a ...
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1answer
86 views

$\alpha$ is a root $\implies -\alpha$ is a root

Let $\mathfrak{h}$ be a Cartan subalgebra of Lie algebra $\mathfrak{g}$. I want to prove: $\alpha\in \mathfrak{h}^*$ be a root of $\frak g$, $\implies$ so is $-\alpha$. Let $\mathfrak{h}$ ...
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1answer
30 views

Representation of indefinite Kac-Moody algebras

The Kac-Moody algebras are divided in three very distinct classes: finite-dimensional, affine and indefinite type. For the first class the finite-dimensional representation theory is very known. For ...
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1answer
38 views

Bilinear forms on Lie algebras that vanish on commuting elements

Let $\mathfrak g$ be a finite dimensional real Lie algebra. If a bilinear map $A:\mathfrak g\times\mathfrak g\to\mathbb R$ vanishes on commuting elements (i.e. $[U,V]=0\implies A(U,V)=0$), is there a ...
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45 views

Universal Enveloping Algebra of $\mathfrak{gl}(n,\Bbb R)$

I am just learning about universal enveloping algebras, and I am wondering about the following. Question: Is the universal enveloping algebras of $\mathfrak{gl}(n,\Bbb R)$ just ...
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1answer
40 views

Distinct real N tuples can be dotted by another to give distinct real numbers

Given $ x_1,x_2, ..., x_n$ distinct real $N$ tuples. Show that there exists a $N$ tuple $a$ such that $(x_i . a)^n_{i=1}$ are all distinct where . is the real dot product. Thoughts: I tried proving ...
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1answer
47 views

An algebra with an antisymmetric product that is not a Lie algebra?

Let $F$ be a field and $A$ an $F$-algebra with an antisymmetric product $\cdot$. In other words, for all $v,w\in A$ we have $$v\cdot w=-w\cdot v.$$ Examples of such algebraic object include all Lie ...
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Why is $[\mathfrak{q}, \mathfrak{g}]$ the smallest ideal $\mathfrak{a}$ of $\mathfrak{g}$ such that $\mathfrak{g}/\mathfrak{a}$ is reductive. [duplicate]

Does anyone know how I would go about answering this question? Any feedback is appreciated. I'm not too sure where to start. Thanks. Let $\mathfrak{g}$ be a Lie algebra over $k$ and $\mathfrak{q}$ ...
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23 views

Showing $\ker(\phi)$ is a $\mathfrak{g}$-invariant subspace of $V$.

Let $\phi:V\to W$ be a linear map between the irreducible $\mathfrak{g}$-modules $V,W$. I want to show that $\ker(\phi)$ is a $\mathfrak{g}$-invariant subspace of $V$ and $\text{im}(\phi)$ is a ...
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7 views

Proof verification: $\mathfrak{g}$-module $V$ is irreducible if and only if $V\subseteq \mathfrak{g}(v)$ for each nonzero $v\in V$.

I want to prove that: The $\mathfrak{g}$-module $V$ is irreducible if and only if $V\subseteq \mathfrak{g}(v)$ for each nonzero $v\in V$. Since $\mathfrak{g}(v)$ for any non-zero $v\in V$ generates ...
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1answer
43 views

Why simply connected solvable analytic groups have no nontrivial compact subgroups?

Why do simply connected solvable analytic groups have no nontrivial compact subgroups? I'll appreciate any help on this question.
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26 views

Prove that a subgroup in a Lie group is homogeneous

Let $\mathbb E:=(\mathbb R^4, \cdot)$ be a Carnot group whose Lie algebra is given by $\mathfrak g=V_1\oplus V_2 \oplus V_3$, where $V_1=span\{X_1,X_2\},$ $V_2=span\{X_3\},$ $V_3=span\{X_4\}$, the ...
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Moment map in general

Let the Lie group $G$ act on the smooth manifold $X$ with the map $(g,x)\to gx$. In any point $x\in X$, the differential of this map induces a linear map: $$ \mu:T_e G \to T_xX\;, $$ and globally, if ...
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1answer
26 views

Four dimensional nilpotent real Lie algebras

I've heard that there are finitely many isomorphism classes of four dimensional nilpotent real Lie algebras. I could find two of them, the abelian algebra and the algebra with four generators ...
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22 views

Is there a general formula for the following Lie algebra quantity?

Consider the generators of $SO(n)$, written as $M_{\mu\nu} = - M_{\nu\mu}$ and they satisfy $$ \left[ M_{\mu\nu} , M_{\rho\sigma} \right] = i \left( \eta_{\nu\rho} M_{\mu\sigma} + \eta_{\mu\sigma} ...
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32 views

Determining if a (Lie algebra) central extension is trivial.

Given a central extension for a given Lie algebra, is there any simple way to check that it is/isn't isomorphic to the trivial extension ("simple" meaning, not as tedious [and daunting, for an algebra ...
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38 views

Primitive of the matrix elements of irreducible representations of Lie groups

I am interested in the matrix coefficients $U_{ij}(g)$ of unitary irreducible representations of a Lie group $G$. In my case, these coefficients arise from the Peter-Weyl theorem. I would like to ...
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26 views

Tensor product of $Spin(2k)$ representations

I am trying to find the tensor product of spinor representations of $SO(2k)$. Labels are given as $$(n+I/2,I/2,\ldots,I/2,s)\otimes(I/2,\ldots,I/2).$$ Where $I$ and $n$ positive integers. How can ...
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29 views

Map to a Lie group as exponential of a map to the Lie algebra

Let $U$ be an open subset of $\mathbb{R}^n$, $G$ a Lie group and $f:U\rightarrow G$ a smooth surjective map. Under which conditions there is a smooth function $\phi:U\rightarrow\mathfrak{g}$ such ...
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1answer
27 views

Lie Algebras and Lie Groups by Serre, Exercise mistake.

In the book Lie Algebra and Lie Groups by Serre, there is an exercise in Chapter three that reads as follows: Exercise(Bergman). Prove that $U(\mathfrak{g})=k$ $\iff$ $\mathfrak{g}=0$. (Hint. Use ...
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1answer
56 views

Question about the definition of the adjoint representation of Lie groups

Let $\mathfrak{g}$ denote the Lie algebra of a Lie group $G\leq GL(n)$. The adjoint representation of $G$ is defined as the function $Ad_g:\mathfrak{g}\rightarrow\mathfrak{g}$ that maps each ...
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28 views

Eigenvalues of skew-symmetric matrix $(x_i-x_j)_{ij}$

Given a set of positive reals $x_1,\ldots, x_n$, construct a skew-symmetric matrix $A=(a_{ij})=x_i-x_j$, in matrix form, ...
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36 views

In what sense do roots span a vector space?

If I am in two dimensional space, the meaning I have for the span is the usual one from linear algebra. But I do not know what it means to say the roots in a root system, R, span the inner product ...
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42 views

Nonstandard analysis, Lie groups and universal enveloping algebras

The idea of nonstandard analysis is to combine finite quantities with infinitesimals. And, back in the day, Lie algebras were roughly considered the "infinitesimal elements" of Lie groups. Say we want ...
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0answers
47 views

Root system of an abelian lie subalgebra.

Let $L$ be a lie algebra and $H$ an abelian subalgebra of $L$ such that each element of $h \in H$ is diagonalizable under the adjoint representation. So there exists a basis of common eigenvectors for ...
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1answer
27 views

Root spaces for symplectic Lie algebra $\mathfrak{s} \mathfrak{p}_n$

Consider the symplectic Lie algebra $\mathfrak{s} \mathfrak{p}_n$ over a field $K$. I know that the root system is given by $C_n=\{\pm 2e_j, \pm e_j \pm e_k:j,k=1 \cdots n, j \neq k\} $ where ...