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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33 views

Eigenvalues of Cartan matrix D_n

please consider the following Cartan matrix (it corresponds to $D_n$ $-$ zeros are replaced by "."'s for better view) $ C=C_{D_n}=\begin{bmatrix} % dd 2 & . & -1 & . & \cdots ...
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1answer
23 views

Showing $M_\alpha$ subalgebra of semisimple complex Lie algebra $L$ of dimension 3

I would like to show that $M_\alpha = \langle x, y, [x, y] \rangle_\mathbb{C}$ is a subalgebra of a semisimple complex Lie algebra $L$ such that $\dim_\mathbb{C} M_\alpha = 3$. $H$ is the Cartan ...
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0answers
19 views

Number of SU(2) that can be embedded in SU(n)

Consider the Lie algebra su(3). Its generators $\lambda_i$ span 3-1 = 2 different Cartan subalgebra, which can be used to form two sets of ladder operators for each generator $H_i$ of the Cartan ...
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1answer
24 views

Manin Triple and Lie Bialgebra correspondence?

I know that there is meant to be a one-to-one correspondence between Manin triples $(\mathfrak{p},\mathfrak{p_+},\mathfrak{p_-})$ and Lie bialgebra structures on $\mathfrak{p_+},$ but I cannot seem to ...
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24 views

Show $\phi$ is a isomorphism as a lie algebra homomorphism

Show $\phi$ is a isomorphism as a lie algebra homomorphism $\phi: \textbf{su}_2 \bigotimes_{\mathbb{R}} \mathbb{C}\rightarrow sl_2(\mathbb{C})$ and $\phi: a(I \bigotimes 1)+b(J \bigotimes 1)+c(K ...
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31 views

Lie Bracket and Multiplication

I read here that the Lie bracket produces the notion of multiplication. Why is the Lie bracket for $GL(n)$ defined as $[A,B] = AB - BA $ and how is this like a "product" of the two matrices A and B?
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25 views

Root system independent of chosen Cartan algebra

I have read on "Lectures on Lie groups and Lie algebras" (by Carter, Segal, Macdonald) that Cartan subalgebras are related by some automorphism of the Lie algebra and this is proved using a density ...
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33 views

Symmetric Polynomials in Geometry

I'm interested in symmetric polynomials. Could you name some nice examples in Differential Geometry where they are clearly useful? I would also be interested in algebraic examples if connected with ...
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11 views

Proving Killing vectors for flat FRLW metric in Cosmology

So, I'll jump right into it. I have been given the flat FRLW metric: $$ g_{ab}=diag{(-1,a^{2}(t),a^{2}(t),a^{2}(t))} $$ And the following Killing vector: $$ \xi^{c}_{1}=\pmatrix{0\\1\\0\\0} $$ I ...
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1answer
19 views

Faithful and irreducible representation of the lie algebra of an algebraic group

Given a representation of an algebraic group $$\Gamma: G \to GL(V)$$ If we take the differential of $\Gamma$ at $e$, we get $$\Gamma^*: Lie(G) \to gl(V)$$ Suppose that $\Gamma^*$ turns out to be an ...
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56 views

Differential of a representation of a linear algebraic group

I asked a question if a representation of the lie algebra of a simply connected algebraic group G induces a representation of the group itself here: \link {Representation of the lie algebra of a ...
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26 views

Understanding Weyl character formula and highest weight integrable representations

Weyl character formula is $\chi=\frac{\sum_{w \in W} \epsilon(w) \exp{(w (\lambda + \rho}))}{\sum_{w \in W} \epsilon(w) \exp{(w ( \rho}))}$ So I understand what is $\epsilon(w)$ but I don't understand ...
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29 views

How many subalgebras are there in $sl_3$?

The Lie algebra $sl_3$ is 8 dimensional and $B=\{h_1, h_2, e_1, e_2, [e_1, e_2], f_1, f_2, [f_1, f_2]\}$ is a basis of $sl_3$. For every $x \in B$, $\text{Span}\{x\}$ is a one-dimensional subalgebra ...
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50 views

Do we have $\{g x g^{-1}: x \in \mathfrak{g}\} = \mathfrak{g}$?

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Fix $g \in G$. Do we have $\{gxg^{-1}: x \in \mathfrak{g}\} = \mathfrak{g}$? Thank you very much. Edit: I think that the answer is yes. We ...
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1answer
50 views

Strange parametrization of SO(3)

I have this parametrization of the sphere that seems quite a mess \begin{equation} R_{ij}=\cos r\left(\delta_{ij}-\hat{\theta}_{i}\hat{\theta}_{j}\right)+\hat{\theta}_{i}\hat{\theta}_{j}+\sin ...
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1answer
14 views

The inner product in Cartan matrix

Let $\Delta=\{e_i-e_{i+1} \}_{i=1}^{i=n}$ be a simple root system of $sl(n,\mathbb C)$, where $e_i\in H^*$ be such that $e_i(X)=$ the $i^{th}$ entry of $X$, and $H\leq sl(n,\mathbb C)$ consists of ...
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66 views

Pauli Matrices as Representations of Reflection Operators?

How does one derive, using, say, the Householder transformation $$ R(r) = (I - nn^*)(r),$$ the reflection representation of a vector $$ R(r) = R(x\hat{i} + y\hat{j} + z\hat{k}) = xR(\hat{i}) + ...
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1answer
19 views

For L = the lie algebra of 2x2 upper triangular matrices over the C, is ad L = Der L?

I am deeply confused about this. I have seen a proof of the fact: L semisimple over C implies ad L = Der L but i dont know if the converse is true or false.
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23 views

Differential of a Lie group homomorphism

If $f : G\to H$ is a Lie group homomorphism, what can we say about its differential $d_ef : \mathcal{G}\to\mathcal{H}$? Is it a Lie algebra morphism or anti-morphism? ...
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22 views

Lie group of differential operators

I have the following three partial differential operators $$A=y \frac{\partial}{\partial y}$$ $$B=y^{-1}(z\frac{\partial}{\partial z}+y\frac{\partial}{\partial y}+c-1)$$ ...
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1answer
21 views

Unitary dual of $\{0\}$ and $\mathbb R$

How to prove that, the unitary dual of $\{0\}$ and $\mathbb R$ are the trivial identity representation $id$ and the representation $\chi_x (y) = e^{i x y}; y\in \mathbb R$, respectively. Thank you ...
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23 views

One-dimensional L-submodules

If $L$ is a Lie algebra over $\mathbb C$. Consider the representation $\pi: L\rightarrow gl(V)$ where $dim\ V=1$. Can this representation be irreducible?
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52 views

The category of Lie algebra representations

A representation of a Lie algebra $\mathfrak{g}$ on a vector space $V$ is a homomorphism of Lie algebras $\mathfrak{g} \to \mathfrak{gl}(V)$. We define morphisms between representations as ...
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1answer
31 views

Existence of Lie algebra with dim =3 or ≥ 5 with [g,g]=g

How to show that: There is a Lie algebra $\mathfrak g$ of dimension $k = 3$ or $k≥ 5$ iff $\frak g=[g,g]$. Also, why it is possible to choose $\frak g$ such that its center is $0$. However, for the ...
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1answer
29 views

Cartan subalgebra and the decomposition of eigenspaces

Let $\frak g$ be a semisimple Lie algebra over a finite dimensional field $F$, and let $\frak h$ be a Cartan subalgebra in $\frak g$. I need actually some explanation on why $$\mathfrak g = ...
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1answer
27 views

Semi-direct product Lie algebra

Given Lie algebras $S$ and $I$ and a Lie homomorphism $\theta:S\to Der I$, we have the semidirect product to be the space $S\oplus I$ with operation $$ ...
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1answer
49 views

Matrix representation of Heisenberg group

Equiped with the law $(a,b,c)\circ (a',b',c') = (a + a', b + b', c + c' + ab'),$ the matrix representation of the Heisenberg group $H^3$ is given by $$ \begin{pmatrix} 1 & a & c\\ 0 & ...
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25 views

Lie group action and Lie algebra action.

Let $G$ be a Lie group and $g$ its Lie algebra. Let $r \in g \otimes g$ and $b \in g$. Consider the adjoint action $g \times g \otimes g \to g \otimes g $ given by $(b, x, y) \mapsto b.(x \otimes y) = ...
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39 views

Why do Clifford bivectors represent the orthogonal Lie algebra?

It takes a long, painful, but straightforward calculation to see that the commutators of grade one elements $[\mathbf e_i,\mathbf e_j]$ of a Clifford algebra $\mathrm{Cl}(p,q)$ have exactly the same ...
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2answers
67 views

Why do we require that a simple Lie algebra be non-abelian?

We say that a Lie $k$-algebra is simple if it is a simple object in the category of Lie algebras, and also nonabelian. The only simple object which we do not consider to be a simple Lie algebra under ...
4
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1answer
67 views

Equivalent definitions of a root system.

For studying root systems many authors start from a vector space $V$ over $\mathbb{R}$ with a positive definite scalar product $(\cdot,\cdot)$, in which a reflection $\sigma_\alpha$ is a linear ...
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25 views

Example of a non semisimple Lie Algebra over $\mathbb{C} $ such that $ L' = L$ [duplicate]

I wanted to show that if $L$ is a finite dimension semisimple Lie Algebra over $\mathbb{C}$, then $L' = L$ ($L' = [L,L]$) but the converse is not true. I could prove the statement but am unable ...
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2answers
37 views

What can we say about $Aut(G)$ for an arbitrary Lie group $G$?

Let $G$ be a Lie group, $\mathfrak g$ be its Lie algebra, and $Aut(G)$ be the group of its smooth automorphism. Then, my questions are: (1) Is $Aut(G)$ again a smooth manifold? and particularly a Lie ...
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0answers
57 views

Solvable Lie algebra with non-characteristic nilradical

It is well known that the nilradical of a finite-dimensional Lie algebra over a field of characteristic p > 0 need not be characteristic (that is, invariant under all derivations of the algebra), but ...
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1answer
32 views

Are surjective Lie group homomorphisms which induce isomorphisms of Lie algebras covering maps?

Question: Suppose that $\varphi:G\to H$ is a surjective Lie group homomorphism whose differential $\varphi_*:{\frak g}\to{\frak h}$ is a Lie algebra isomorphism. Is $\varphi$ necessarily a smooth ...
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19 views

Splitting of short exact sequence. with the existence of non-deg. bilinearform

Let $\mathfrak{h}$ be a subalgebra of $\mathfrak{g}$,and $j \colon \mathfrak{h} \to \mathfrak{g}$ be the injective Liealgebra-Homomorphismus. Assuming now, that we have a non-degenerate bilinearform ...
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1answer
23 views

Non-nilpotent and not semisimple algebra with maximal toral subalgebra = 0

I am looking for 3 dimensional non-nilpotent Lie algebra whose only toral subalgebra is $0$. In $sl_2$ the element $\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ is diagonalizable so the ...
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2answers
30 views

Abelian subalgebra doesn't imply toral

Let $L $ be a Lie algebra. A subalgebra $H$ of $L$ such that $ad_H:\frak g → g$ is diagonalizable for every $h ∈ H$ is called toral. Now, every toral subalgebra is abelian. But, what is an example ...
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1answer
22 views

Complexification of maximal compact subgroup of $GL(2,\mathbb{R})$.

Given the Lie group $G=GL(2,\mathbb{R})$, we have that $K=O(2)$ is a maximal compact subgroup of $G$. I am trying to describe the complexification $K_\mathbb{C}$ of $K$. The Lie algebra $k_0$ of $K$ ...
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1answer
88 views

A tensor identity - $\text{Hom}_{R}(A,B \otimes_S C) \cong \text{Hom}_{R}(A,B)\otimes _SC$

Let $R,S$ be associative algebras over $\mathbb{C}$. Let $A$, $B$ and $C$ be, a left $R$-module, a $(R,S)$-bimodule, a left $S$-module, respectively. Assume that $B\otimes_S C$ is finite-dimensional. ...
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1answer
28 views

Different methods to compute a unitary representation

Given a nilpotent Lie group $G$ (for example the Heisenberg group), what is the most effective method to calculate their unitary representation: The orbit method due to Kirillov; or The induction ...
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1answer
25 views

Lie group homomorphism with injective Lie algebra homomorphism

Question: Suppose that $\varphi:G\to H$ is a Lie group homomorphism such that $G$ is simply connected and $\varphi_*:{\frak g}\to{\frak h}$ is injective. Is $\varphi$ injective? Since Lie group ...
2
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1answer
47 views

Poisson bracket makes $C^\infty(M)$ into a Lie algebra

Let $M$ be a symplectic manifold with symplectic form $\omega$. Define the Poisson bracket of two smooth functions $f$, $g$ by $\{f, g\} := \omega(X_f, X_g)$. How do I see that $X_{\{f, g\}} = [X_f, ...
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1answer
114 views

Flows and Lie brackets, $\beta$ not a priori smooth at $t = 0$

Let $X$ and $Y$ be smooth vector fields on $M$ generating flows $\phi_t$ and $\psi_t$ respectively. For $p \in M$ define$$\beta(t) := \psi_{-\sqrt{t}} \phi_{-\sqrt{t}} \psi_{\sqrt{t}} ...
3
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1answer
118 views

Identity for bracket operator in tangent space at identity

Let $G$ be a Lie group and $X,Y\in T_eG$. Show that $$[X,Y]=\left.\frac{\partial}{\partial s}\right\vert_{s=0}\left.\frac{\partial}{\partial t}\right\vert_{t=0}\exp(sX)\exp(tY)\exp(-sX)\exp(-tY).$$ ...
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39 views

Finite dimensional representation of $SL_{2}$

Let ($\pi , V$) be a finite dimensional representation of $SL_{2}$. Also, let $\alpha$ be highest weight vector. I want to show that for any $m>0$, then the following holds: ...
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30 views

Classify all $2 \times 2$ and $3 \times 3$ matrices in $sl_2$ and $sl_3$ respectively.

I would like to classify all $2 \times 2$ and $3 \times 3$ matrices in $sl_2$ and $sl_3$ up to conjugation respectively. We take the ground field to be $\mathbb{C}$. Let $g = sl_2$. A matrix in ...
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0answers
20 views

How to show that $( \Lambda^3 g)^{g} = \mathbb{C}Z$?

Let $g$ be a simple Lie algebra over $\mathbb{C}$. Let $\Omega$ be the Casimir element of $g \otimes g$ associated to a non-degenerated invariant form on $g$. How to show that $( \Lambda^3 g)^{g} = ...
3
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1answer
22 views

Express cofundamental weights using coroots.

In type $A_2$ root system, we have $\alpha_1 = 2 \omega_1 - \omega_2$, $\alpha_2 = - \omega_1 + 2 \omega_2$. How to express cofundamental weights $\omega_1^{\vee}, \omega_2^{\vee}$ using coroots ...
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24 views

Identifying $\bar{D^3}$ with $S^3$ and then $\mathbb{R}P^3$ and $SO(3)$

After working it out, I found that we can identify the closed 2-disk, $\bar{D^2}$ with the 2-sphere, $S^2$. Let $\bar{D^2}$ have radius $\pi$. Then $\partial \bar{D^2}$ is identified with the south ...