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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Abstract Lie algebra

For six dimensional Lie algebra with non-zero Lie brackets defined as follow: $[e_{1}, e_{3}] = -e_{1}, [e_{1}, e_{6}] = -e_{2}, [e_{2}, e_{3}] = -e_{2}, [e_{2}, e_{4}] = e_{1}, [e_{2}, e_{5}] = e_{2},...
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29 views

Importance of universal enveloping algebras and Poincare-Birkhoff-Witt theorem.

Let $L$ denote a Lie algebra and $U(L)$ denote its universal enveloping algebra. I am trying to see why universal enveloping algebra and PBW theorem are important. Precisely: Given any $L$-...
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16 views

Every Lie algebra contains a maximal proper Lie subalgebra

I am working though the proof of Proposition 6.2 in Erdmann's "Introduction to Lie Algebras". I can't verify that every Lie subalgebra of $L \subseteq \mathfrak{gl}(V)$ contains a maximal (proper) ...
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13 views

Let $F$ be a field and let $L = b(n,F)$ be the Lie algebra of $n×n$ upper triangular matrices and $V = {F^n}$ .

$\space$ Let $F$ be a field and let $L = b(n,F)$ be the Lie algebra of $n×n$ upper triangular matrices and $V = {F^n}$ . Let $e_{1} , ... , e_{n}$ be the standard basis of $F^n$. For $1 \le r \le n$ , ...
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28 views

Relationship between Casimir and index of a representation of a Lie algebra.

In several QFT textbooks (namely, those of Peskin and Shroeder and of Schwartz) there is presented an identity for representations of Lie algebras, $$ d(R) C_2(R) = T(R) d(G),$$ where $d(R)$ is the ...
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32 views

How to view a morphism of Lie algebras?

Let $G = \textrm{GL}_n k$, and let $\sigma: G \rightarrow G$ be an automorphism of algebraic groups. The Lie algebra $\mathfrak g$ of $G$ can be described in three ways: 1 . The space $T_e(G)$ ...
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52 views

Reference for category of Lie algebras?

Are there any references which deal with categorical aspects of Lie algebras? I'm looking for constructions like kernels, products, coproducts (limits and colimits in general) etc. My goal is to ...
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24 views

Show that $\left.\dfrac{d}{d s}\right|_{s=0}X_\eta (m)(f\circ \varphi_{\exp(s\xi )})=(X_\eta f)_{*,m}(X_\xi (m))$

Let $G$ be a Lie group, $M$ be a manifold, $\varphi:G\times M \rightarrow M$ smooth action of $G$ over $M$ where $\varphi(g,m)=g\cdot m$. For each $\xi \in \mathfrak{g}=T_e G$ we difine the ...
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94 views

Do I understand the Chevalley Restriction Theorem correctly?

Let $G$ be a complex semisimple Lie group with Lie algebra $\frak g$, and let $\frak h$ be a Cartan subalgebra with Weyl group $W$. The Chevalley Restriction Theorem states that the restriction map $\...
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33 views

Is every element contained in a Borel subalgebra?

Let $\frak g$ be a complex semisimple Lie algebra. Is every $X\in\frak g$ contained in some Borel subalgebra $\frak b$? Attempt: I know that a Borel subalgebra is by definition a maximal solvable ...
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10 views

How many cyclic conjugate class can $su(2)$ has

In the lie algebra of $su(2)$, we can easily find out three linear independent elements $t_1,t_2,t_3\in su(2)$ such that $[t_1,t_2]=t_3,[t_3,t_1]=t_2,[t_2,t_3]=t_1$ and this relation is preserved ...
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27 views

Compute the associated induced Lie algebra action $\text{d}\pi$

Let $G=\mathrm{SL}_2(\mathbb{C})$ and consider the action of $G$ on the space of smooth functions on column vectors $\mathbb{C^2}$ given by $\big(\pi(g)\phi\big)(v)=\phi\left({g^\top}\,v\right)$ for ...
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37 views

Let $G$ be a Lie group and $H$ be a closed subgroup of $G$. Show that if $H$ and homogeneous space $\frac{G}{H}$ are connected, then $G$ is connected.

Let $G$ ge a Lie group and $H$ be a closed subgroup of $G$. Show that if $H$ and homogenuous space $\frac{G}{H}$ are connected, then $G$ is connected. Remark: For this proof I use the following ...
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42 views

Why can we find a basis for the elements of the Lie algebra?

I am a physicist and we do Lie algebras pretty informally, so I hope my question makes any sense to a mathematician. There is one thing that I don't quite understand, which is why we can find a basis ...
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30 views

Finite dimensional, irreducible representations of the Lie superalgebra gl(1|1)

I am wondering how the finite dimensional, irreducible representations of the Lie superalgebra gl(1|1) are parametrized. I understand that they are all highest weight, and that the only non-trivial ...
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10 views

The Killing form of elements in the Cartan subalgebra

Let $L$ be a Lie algebra and $H\oplus\bigoplus_{\alpha\in\Phi}L_\alpha$ be its Cartan decomposition. Now if $x\in L_\alpha$ and $y\in L_\beta$ and $\alpha+\beta\not=0$ then the Killing form $\kappa(x,...
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8 views

Ad-nilpotency is preserved under conjugation

Let $G$ be a complex semisimple connected Lie group with Lie algebra $\mathfrak{g}$. An element $x \in \mathfrak{g}$ is called ad-nilpotent if the operator $\text{ad} \ x : \mathfrak{g} \to \mathfrak{...
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19 views

Calculating the Killing form of the classical algebras

I was reading in the book Parabolic Geometries (p.170-172). But I didn't get how he obtained the Killing form from the roots of each simple Lie algebra. For example: In case $sl(n,\mathbb C)$, given ...
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Lie algebra associated to Leibniz algebra

We know that for any Leibniz algebra $L$ we can associated its Lie algebra denoted by $L_{Lie}$. for example the ideal generated by $\{[x,x] | x\in L\}$ determines the non-Lie character of $L$. Is it ...
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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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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 \oplus\displaystyle\sum_{\...
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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 [L_\...
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26 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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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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29 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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37 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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21 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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63 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 V^{\...
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33 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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34 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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40 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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21 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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33 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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22 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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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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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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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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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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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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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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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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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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45 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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19 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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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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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 $...