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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21 views

Question about injectivity of exponential mapping between SE(3) and se(3)

If we denote $X, Y \in se(3)$, and they have this relationship $$e^X = e^Y$$ is it safe to assume that $X = Y$ for every element? If it is not, may I know the case when it is not? Intuitively, the ...
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1answer
32 views

When Killing form equals a constant times the trace

How to find an element $0\not =a\in \mathbb C$ such that $\kappa_L (x,y)=aTr(xy)$ for all $x,y\in L$. Where $L$ is: $A_l$ $B_l$ $C_l$ $D_l,\ \ l>2$. Why such an $a$ is unique?
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1answer
30 views

The Lie Algebra of Invertible Upper Triangular Matrices

From Wikipedia: The Lie algebra of the Lie group of invertible upper triangular matrices is the set of all upper triangular matrices, not necessarily invertible, and is a solvable Lie algebra. I ...
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1answer
45 views

Bracket of Lie algebra-valued differential form

In this wikipedia article: https://en.wikipedia.org/wiki/Lie_algebra-valued_differential_form the bracket of Lie algebra-valued forms is defined. At one point it mentions that it is the bilinear ...
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1answer
152 views

Are ideals of the Lie algebra invariant under the adjoint action?

Let $G$ be a connected algebraic group over a field of characteristic $p \geq 0$ and let $H < G$ be a connected closed subgroup. If the lie algebra $\mathfrak{h}$ of $H$ is an ideal of the Lie ...
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16 views

Does $r \in \Lambda^2 g$ imply that $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] \in \Lambda^3 g$?

Let $g$ be a Lie algebra. Does $r \in \Lambda^2 g$ imply that $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] \in \Lambda^3 g$? Thank you very much.
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35 views

Vector Fields and basis

Let $M$ be a differentiable manifold; $p \in M$ ; $\sigma$ be a chart at $p$ with $\sigma(p)= (x^{i}),i=1,2,\cdots n$. $T_{p}(M)$ the tangent space at $p$ has basis $\{ \frac{\partial}{\partial x^{i}}|...
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1answer
47 views

Why does $\frac{d}{dt}e^{X+tY} |_{t=0}$ depend linearly on $Y$ with $X$ fixed?

I'm studying the proof of Baker-Campbell-Hausdorff formula from Brian Hall's book Lie Groups, Lie Algebras and Representations. I am stuck at this part: I don't get why continuity of exp implies ...
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3answers
65 views

Direct calculation of the tangent space of $SO(3)$

Let $SO(3)$={$RR^T=I$, $det(R)=1$}, I need to show that a base of the tangent space in the identity is given by: $$E_i=\frac{d}{dt}\exp(tL_i)|_{t=0}$$ where $$L_1= \left(\begin{matrix} 0& 1& 0\...
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1answer
43 views

finding high weight vector in Verma module

Let $\frak{g}$ be a (semi-)simple lie algebra. Let $\lambda$ be a dominant integral weight. Denote $L(\lambda)$ to be the irreducible representation of highest weight $\lambda$. From BGG resolution, ...
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1answer
18 views

How to prove this lemma about Weyl group?

Let $\mathscr{W}$ be the Weyl group of a root system $\Phi$ with basis $\Delta$. If $\sigma\in \mathscr{W}$, $\sigma = \sigma_{\alpha_1} .. \sigma_{\alpha_t}$ where $\alpha_1, ...,\alpha_t \in \Delta$...
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2answers
41 views

Existence of Ad-invariant bilinear form gives a certain Lie algebra homomorphism

Let $G_1 \subset G$ be Lie groups and $\mathfrak{g}_1, \ \mathfrak{g}$ the corresponding Lie algebras. Assume that there is a non-degenerate bilinear form $\langle \cdot, \cdot \rangle$ on $\mathfrak{...
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16 views

why $ t\oplus {\sqrt{-1}t}$ is a Cartan subalgebra of $\mathfrak{g}$?

Let $G$ be the complexification of a connected, simply-connected compact Lie group $K$ and $\mathfrak{g}$ be a Lie algebra of $G$. If $t$ is a maximal abelian subalgebra of the Lie algebra $\mathfrak{...
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13 views

Zero action on a Lie algebra quotient submodule

Let $L $ be a Lie algebra and $I$ be an ideal in $L$. If $M,N$ are $L$-module with $N\subset M$. If $M/N$ is an $L/I$-module and $K/N$ is am $L/I$-submodule of $M/N$ with $(L/I)\cdot (K/N)=0$. Does ...
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43 views

Does a $k$-derivation of $k[G]$ into $k$ induce an element of $k[G]$?

Let $G$ be a linear algebraic group over an algebraically closed field $k$, and let $e$ be the identity of $G$. Then $k$ is a $k[G]$-module via the action $f \cdot a = f(e)a$. The tangent space of $...
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1answer
31 views

Associated Lie algebras of p-groups of maximal class

I was reading a paper the other day on Lie algebras of maximal class and they keep saying that some results are taken from p-groups theory. So my question is how do you get the associated Lie ...
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34 views

Properties of the complexification functor for Lie algebras

The complexification of Lie algebras determines a functor from real Lie algebras to complex Lie algebras, whose right adjoint is the restriction of scalars functor. Thus, we know that complexification ...
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1answer
35 views

Apparent Contradiction to Weyl's Theorem

Let $L$ be $sl(2)$, i.e., $L=span\{h,e,f\}$, where $[h,e]=2e$,$[h,f]=-2f$,$[e,f]=h$. This is semi-simple. Suppose I create a module $V=span\{v_1,v_2,v_3\}$ and define actions as follows: $$h(v_1)=v_1,...
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1answer
24 views

If $H$ is a closed subgroup of $G$, is the Lie algebra of $H$ contained in the Lie algebra of $G$?

Let $H \subseteq G$ be connected linear algebraic groups with $H$ closed in $G$, and let $e$ be the identity of $G$. The Lie algebra of $G$ is the tangent space $T_eG$ of $G$ at $e$, which we can ...
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1answer
45 views

How calculate the cartan matrix of the twisted quantum affine algebras?

the cartan matrix of the type $A_{2}^{(2)}$, $A_{2r-1}^{(2)}$, $A_{2r-1}^{(2)}$, $D_{r+1}^{(2)}$, $E_{6}^{(2)}$. I know the cartan matrix of the type $A_{2}^{(2)}$ is \begin{align} \left( \begin{...
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37 views

Invariance of a Vector Field under the action of a Group

I've got a one-parameter group given by \begin{equation} \theta_{t}\left(x,y,z\right)=\left(e^{t}x,e^{t}y,e^{t}z\right) \end{equation} I already have th infinitesimal generator vector field $X=x\...
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0answers
68 views

Weyl group of complex Lie group

Let $G$ be a compact connected Lie group with maximal torus $T$. The Weyl group is defined by $$W:=N_G(T)/T.$$ Now, $G$ has a complexification $G_{\Bbb C}$ with maximal torus $T_{\Bbb C}$ which is the ...
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0answers
27 views

$\operatorname{Hom}(L,L)$ as an $L$-module

Let $L$ be a Lie algebra and let $V:=\operatorname{Hom}(L,L)$ i.e the set of all linear transformations from $L$ to itself. If we want to make $V$ as an $L$-module by setting $(x · \phi)(y) = [x, \phi(...
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2answers
32 views

Direct sum as vectors but not Lie algebra direct sum

Let $L$ be a Lie algebra and let $I$ be an ideal in $L$ and $K$ be a subalgebra in $L$ such that $L=I\oplus K $. Why this sum is direct as vector subspaces but not Lie algebra direct sum? Can't we ...
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1answer
34 views

Prove that $(\mathfrak{su}(2))^* \cong \mathfrak{sb}(2)$

Let $\mathfrak{su}(2)$ be the Lie algebra with basis elements $$ e_1=\begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} , \quad e_2=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} , ...
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48 views

Find the extended form of the group generated by an operator?

I tried to find the extended form of the group generated by the following operators. (I): The first operator $$A=z\frac{\partial }{\partial z}+1$$ To find the extended form of the group ...
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1answer
37 views

A quotient module of a Lie algebra

Let $L$ be a Lie algebra. If $A$ and $B$ are $L$-submodules of an $L$-module $V$, such that $A\subset B$ and $I\cdot B\subset A$ for some ideal $A$ in $L$. I want to understand why this implies that ...
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40 views

Adjoint orbits are described by minimal polynomials

I've read that if $\frak g$ is a complex semisimple Lie algebra and $G$ a Lie group with Lie algebra $\frak g$, and $H\in{\frak g}$ is regular semisimple, then the adjoint orbit $${\cal O}_H=\{{\rm Ad}...
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24 views

Proof concerning the Jacobi identity

I'm concerned with a (supposedly) simple identity from Guillemin and Sternberg's book Supersymmetry and Equivariant de Rham Theory. However, it seems false to me, and I would like to have some ...
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25 views

Automorphism of semisimple Lie algebra corresponding to a simple reflection

Let $\mathfrak{g}$ be a complex, finite-dimensional Lie algebra. Let $\mathfrak{h}\subset \mathfrak{g}$, $W$ and $\Pi$ be a Cartan subalgebra, its Weyl group and the set of all simple roots, ...
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1answer
36 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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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
29 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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25 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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32 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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1answer
26 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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34 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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57 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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36 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
51 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 r\sum\...
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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 ...
5
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71 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}) + yR(\...
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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? $$d_ef\left([\xi,...
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23 views

Lie group of differential operators

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