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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38 views

Compact Lie group $G$ with Lie algebra $\frak g$ satisfying $gZg^{-1}=-Z$ for $Z\in\frak g$ and $g\in G$

Let $G$ be a compact Lie group with Lie algebra $\frak g$. Are there known conditions on $G$ guaranteeing the following property: $$ \hbox{For each $Z\in\frak g$ there exists an element $g\in G$ ...
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1answer
17 views

Parabolic subalgeba

Let $L$ be a Lie algebra and let $\Phi$ be a root system and $\Delta$ be a basis. Let $\Gamma\subset \Delta$. Define, $$P:=H\oplus\displaystyle\sum_{\alpha\in\Phi_+} L_\alpha ...
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0answers
16 views

Lie subalgebra generated by elements of $L_{\alpha}$ and $ L_{-\alpha}$

Let $L$ be a Lie algebra with four roots $\alpha,-\alpha,\beta,-\beta$. Let $K$ be the subalgebra generated by $L_\alpha,L_{-\alpha},L_\beta,L_{-\beta}$. Is $K=\{[L_\alpha,L_{-\alpha}]\oplus ...
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23 views

Pushforward of a Matrix Lie Algebra Bracket

For a matrix Lie group we know that the left inv. push forward is given by $$ (L_g)_* X = g X \quad\quad | X\in \mathfrak{g}, g \in G $$ With Lie bracket the commutator $$ [X,Y]_\mathfrak{g} = XY-YX ...
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1answer
15 views

The Lie algebra of the generalized unitary group $\{g \in GL_n(\mathbb{C}) : gS\bar{g}^t=S\}$ is $\{XS+S\overline{X}{}^t=0\}$

Let $ S \in M_n(\mathbb{C}) $ be a square matrix and let $ X$ be in the Lie algebra $\mathbb{\mu(S)} $ of the generalized unitary group, $$U(S):=\{g \in GL_n(\mathbb{C}); gS\bar{g}^t=S\} .$$ ...
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1answer
26 views

Adjoint matrix in $\mathbb{so_3}$

$\mathbb{so_3}$ has the following basis: $X_1=\begin{bmatrix} 0 & & \\ & &1 \\ & -1 & \end{bmatrix}$, s: $X_2=\begin{bmatrix} & & 1\\ & 0& \\ -1 & & ...
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1answer
30 views

Compute $ad_X$, $ad_Y$, and $ad_Z$ relative to a basis

For a lie algebra $\mathbb{g} $ we can define the adjoint representation as: $ ad: \mathbb{g} \rightarrow End(\mathbb{g}) $ as the map such that $ad_x(y)=[x, y] $ for all $\in \mathbb{g} $ I am ...
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22 views

Lie algebra of $SL_2(\mathbb{R})$ and show $\exp(X)=I+X$ where $I \in SL_2(\mathbb{R}) $ and $X \in sl_2(\mathbb{R})$

I am doing an undergraduate course on Representation Theory and am trying to solve these consecutive questions. The first two I am ok with (I just included them for context), but I could do with some ...
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20 views

Lie subalgebra generated by a subset of a basis of root system

Let $L$ be a semisimple Lie algebra, ad let $\Phi$ be a root system. Fix a fundamental root system $\Delta$ of $\Phi$ with corresponding to $\Phi^+$. I would like to understand the subalgebra ...
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1answer
61 views

Show that any representation of $\mathfrak{sl}(2,\mathbb C)$ is a subrepresentation of $V^{\otimes m} \oplus V^{\otimes {(m+1)}}$ for some $m$

Suppose $M$ be a finite-dimensional representation of $\mathfrak{sl}(2,\mathbb C)$, then there is a positive integer $m$ such that $M$ is isomorphic to a subrepresentation of $V^{\otimes m} \oplus ...
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1answer
28 views

G2 as algebra of endomorphisms preserving a trilinear form

I am trying to find some literature or papers about the topic in the title. I've read multiple times that the Lie-Algebra G2 can be described in such a way, but I've yet to find some good, ...
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1answer
25 views

Using Lie's Theorem to prove an ideal is nilpotent

Let L be a finite-dim'l Lie algebra over an algebraically closed field of characteristic zero, and I be a solvable ideal of L. Prove that the ideal [L,I] is nilpotent. My reasoning: Consider the ...
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1answer
36 views

Dual representations of fundamental representations of a Lie algebra.

Let $g$ be a Lie algebra. Let $V(\omega_i)$, $i=1,\ldots,n$, be the fundamental representations. Are the dual representations $V(\omega_i)^*$ highest weight representations? The dual representation ...
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1answer
19 views

Relations of $S^2 V$ and heighest weight representations of Lie algebras.

Let $V$ be the natural representation of $sl_n$. Then $V = V(\omega_1)$, where $\omega_1$ is the first fundamental weight. We have $\Lambda^2 V = V(\omega_2)$. Is $S^2 V = V(\lambda)$ for some weight ...
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0answers
29 views

Existence of a canonical map of quadratic forms

For $X=\mathbb C^N\oplus \mathbb C^N$ equipped with a real structure $J^2=1$ and symplectic structure $S$ satisfying $$J(z_1,z_2)=(\bar z_2,\bar z_1),~~~~~S(z_1,z_2)=(z_1,-z_2)$$ we see that $X$ has ...
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1answer
17 views

Maximal nilpotent and solvable Lie subalgebras

If $\mathfrak g$ is a finite dimensional complex semi-simple Lie algebra with maximal toral subalgebra $\frak h$.If $(E, ( , ),\Phi )$ is the corresponding root system. Fix a fundamental system $R$ of ...
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0answers
20 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
31 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
25 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
36 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
147 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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33 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 ...
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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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0answers
21 views

How to show the following vector field is not left invariant

Suppose we have a Lie group G=(R,+), which is just the real line with addition as the group operation. Is the vector field $x\partial_x$, where $x\in G$, is not left invariant? How to show that? ...
3
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3answers
45 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& ...
3
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1answer
41 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 ...
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2answers
36 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 ...
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15 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 ...
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0answers
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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0answers
42 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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0answers
33 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
34 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: ...
2
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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
42 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( ...
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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 ...
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0answers
66 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, ...
2
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2answers
29 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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47 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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0answers
17 views

Does taking the dual preserve the isomorphism?

For example, I have shown that $(\mathfrak{su}(2))^* \cong (\mathfrak{su}(1,1))^*,$ but also that $\mathfrak{su}(2) \ncong \mathfrak{su}(1,1).$ Is this a general property of taking duals, that the ...
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1answer
36 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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0answers
38 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 ...
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0answers
23 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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0answers
23 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
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 ...