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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Contraction of representations of universal enveloping algebra

$\quad$ (Following, e.g. SBBM) Given a Lie algebra contraction $\mathfrak{g}\xrightarrow{t(\epsilon)}\mathfrak{g}_0$, one can contract a family $\{\rho_{\epsilon}:\mathfrak{g}\rightarrow ...
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Adjoint action on quotient space of Lie algebras and vector fields on quotient group

Let $G$ be a Lie group and $H$ a closed subgroup. Then $G/H$ has a unique structure of a smooth manifold with canonical projection $p: G \to G/H$. If $\mathfrak g = T_e(G), \mathfrak h = T_e(H)$ are ...
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24 views

Schubert cell decomposition and full flags

I am looking for a self-contained basic theory of Schubert-cells through finding the decomposition of the full flag $Fl_3(\mathbb C^3)$.
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Deriving SO(3) multiplication rule from so(3) commutation rules

I've heard it said that the commutation relations of the generators of a Lie algebra determine the multiplication laws of the Lie group elements. I would like to prove this statement for $SO(3)$. I ...
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29 views

Understanding Cartan clasification

In class we defined Cartan subalgebra h (of g) as maximally abelian subalgebra containing only ad diagonalizable elements. ad is adjoint map $ad_{H_i}(E) =[H_i, E]$. I have a couple of questions ...
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Free (nilpotent) Lie algebras

If $F$ is a free Lie algebras of finite rank $n\gt 1$. When $\dim (F) $ and $\dim (F/\gamma_{n+1}(F))$ is infinity, in where $\gamma_n (X)$ is the $n-$term of lower central series $X$. Thanks for ...
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32 views

Two questions on roots of finite, simple, complex lie algebra

Why are there at most two root lengths for a finite, simple, complex lie algebra? I know it is from the constraint that the $2(\alpha,\beta)/(\alpha,\alpha)$ is integer, but what is the argument? ...
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How to determining the roots and step operators of a L(SO(3)

I've come across a question in my "Lie Group and Lie Algebras for Physicists" course that asks me to determine the a basis for the Cartan subalgebra of $L(SO(3))$ and "hence find the roots and write ...
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33 views

What's wrong with this derivation about sympletic matrices?

Let $M$ be a $2\times 2$ matrix, the definition of sympletic that I have is that $M$ is sympletic if $$MJM^T = J,$$ being $J$ the matrix $$J = \begin{pmatrix}0 & -1 \\1 & 0\end{pmatrix}.$$ ...
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26 views

Do we have a $g \otimes g'$-action on $V \otimes V'$?

Let $g, g'$ be Lie algebras. Let $V$ (resp. $V'$) be a $g$-module (resp. $g'$-module). Do we have a $g \otimes g'$-action on $V \otimes V'$? In particular, when $g=g'$ and $V = V'$, do we have a $g ...
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36 views

Do we need transpose in the definition of a dual representation?

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. There is an action of $G$ on itself given by left multiplication: $G \times G \to G$, $(f,g) \mapsto fg$, $f, g \in G$. There is a ...
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Trivial representation of a lie Algebra?

Can someone explain why if $\rho:L \rightarrow \text{End}(\mathbb{C})$ is a lie algebra representation then it must be that $\rho(x)=0\ \forall \ x\in L$.
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41 views

What is the diagonal $\mathfrak{g}$-action on $V \otimes V^*$?

Let $\mathfrak{g}$ be a Lie algebra and $V$ a left $\mathfrak{g}$-module. Then the dual vector space $V^*$ is a right $\mathfrak{g}$-module with right $\mathfrak{g}$-action given by $(f.g)(v) = ...
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29 views

Affine Kac-Moody Group Isometry of a Manifold

An isometry of a Riemannian manifold is an infinitesimal displacement generated by a Killing vector field $V=\zeta^aV_a=\zeta^aV_a^i\frac{\partial}{\partial x^i}$. If the isometry corresponds to the ...
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Chirality in Lie groups and Lie algebras

Is there an example of a chiral Lie group?In particular, is it true to say that the map $g\mapsto g^{-1}$ is orientation reversing for odd dimensional Lie groups? Moreover is there a concept of ...
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79 views

Representation of ${\frak gl}(n, \Bbb R)$ in ${\cal C}^\infty(\Bbb R^n, \Bbb R^n)$.

I'm trying to check that $\rho\colon{\frak gl}(n, \Bbb R) \to {\frak gl}({\cal C}^\infty(\Bbb R^n, \Bbb R^n))$, given by $\rho(A)\colon {\cal C}^\infty(\Bbb R^n, \Bbb R^n)\to {\cal C}^\infty(\Bbb R^n, ...
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22 views

sympletic form of the hyperboloid

I'm working with the Hyperboloid of two leefs $H_2$ as a coadjoint orbit of $\mathfrak{sl}^*(2, R)$. I know that $H_2$ is a symplectic manifold by the following theorem: Given a Lie group G and ...
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19 views

SU(3) tensor methods in representations [duplicate]

I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ...
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30 views

Tensor formula in SU(3) representations

I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ...
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25 views

Is the log of closed connected subgroups a vector space?

Let $G$ be a simply connected nilpotent Lie group (so that the exponential map $\exp:{\mathfrak g}\to G$ from the Lie agebra ${\mathfrak g}$ to $G$ is a diffeomorphism, and hence so is the inverse ...
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Deriving the structure constants of the SO(n) group

The commutation relations for the $\mathfrak{so(n)}$ Lie algebra is: $$([A_{ij},A_{mn}])_{st} = -i(A_{j[m}\delta_{n]i}-A_{i[m}\delta_{n]j})_{st}$$ where the generators $(A_{ab})_{st}$ of the ...
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Question 2 in Exercises section 5.8 in the book of Brian Hall's Lie groups,… : an elementary introduction.

Let $\pi$ be an irreducible representation of $\mathfrak{sl}(3,\mathbb{C})$ and let $\pi^*$ be the dual representation of $\pi$ defined by $\pi^*(X) = - \pi(X)^T$, where $T$ stands for transpose. Show ...
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Differential equation and Lie algebra

If I have this differential equation: $$ \frac{d\vec{x}}{dt} = F(\vec{x}) $$ and when $F = A$ is a matrix we can have the solution: $$ \vec{x}(t) = e^{At} \; \vec{x} $$ But what if $F = \mathfrak{g} $ ...
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32 views

Relationship between representations of Lie groups and Lie algebras [closed]

Consider all the finite-dimensional irreducible representations of a group. For each finite-dimensional irreducible representation of a group, is there one and only one corresponding representation ...
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44 views

A canonical isomorphism of Lie algebras

Let $g$ be a Lie algebra/$\mathbb{C}$. I would like to investigate the existence of a canonical Lie algebra isomorphism/$\mathbb{C}$ of the form $g\otimes_{\mathbb{R}}\mathbb{C}\rightarrow g\oplus g$. ...
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18 views

Deriving the commutation relations of the so(n) Lie algebra

The generators $(A_{ab})_{st}$ of the $so(n)$ Lie algebra are given by: $$(A_{ab})_{st} = -i(\delta_{as}\delta_{bt}-\delta_{at}\delta_{bs}) = -i\delta_{s[a}\delta_{b]t}$$, where $a,b$ label the ...
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95 views

The second integral of the Killing form

Let $G$ be a lie group. Assume that $B$ is the Killing form of its Lie algebra $T_{e}G$. So $B$ is counted as a symmetric $2$-form on $G$ by translation. Is there a smooth function $f$ on $G$ ...
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28 views

Left invariant metrics on a Lie group coming from Lie algebras

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. If we have an inner product on $\mathfrak{g}$ then we can create a left invariant metric $d_G$ on $G$ by translations. On the other hand ...
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$\mathrm{ad}$-invariant splitting of Lie algebras.

I am reading a text on which the following situation appears: Let $G$ be a a Lie group and $H$ a compact Lie subgroup. Let $\mathfrak{g}$ and $\mathfrak{h}$ be their Lie algebras respectively. The ...
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Lie algebra of the invertible morphisms of a Lie algebra.

I am confused with some facts of Lie groups and Lie algrebras. If a have a Lie algebra $\mathfrak{g}$ and take its set of invertible morphisms $GL(\mathfrak{g})$ it is clear to me that this is a ...
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Hermitian metric from Killing form

Let $G$ be a semisimple Lie group. Its Killing form is a nondegenerate inner product on the tangent space to $G$ at the identity, and this form can be naturally extended to a metric on the whole of ...
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The Levi-Civita connection on $S^3$ and $SU(2)$

The fundamental theorem of Riemannian geometry implies that there is a unique symmetric (i.e., $\Gamma^{a}_{bc}=\Gamma^{a}_{cb}$, using a coordinate basis) connection on the three-sphere, $S^3$ which ...
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How to derive second order casimir invariant for u(N)

I am told that for semi-simple lie algebra, second order casimir invariant is defined to be: $I_2=g^{\mu\nu}\rho(e_\mu)\rho(e_\nu)$, where summation over repeated indices is used. $g^{\mu\nu}$ is the ...
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96 views

Basis of $SO(4)$ group

I have a rotation matrix $$ R_{(\phi)} = \left( \begin{matrix} \cos (\phi) & \sin (\phi) & 0 & 0 \\ -\sin(\phi) & \cos(\phi) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & ...
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How to find a highest weight vector in a tensor product of two representations of $sl_2$.

Let $V_\lambda, V_{\mu}$ be two representations of $sl_2$. How to find a highest weight vector in a tensor product of two representations of $sl_2$? Thank you very much.
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Lie algebras of same dimension are isomorphic

I just began studying Lie algebras and I'm trying to prove that two Lie algebras $\frak g$ and $\frak h$ are isomorphic if and only if they have the same dimension. If they're isomorphic, then ...
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48 views

Infinitesimal Generator of Local Group of Transformation

I am stuck on concept of infinitesimal generator. I am reading Olver and i quote definitions from there Given a local group of transformation G acting on Manifold M via $g.x= \Psi(g,x)$ for $(g,x)\in ...
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Tensor products and decomposition of $SU(3)$ representations

For each finite irreducible representation of Lie algebra $su(3)$ one knows that it is characterized by highest weight $(\lambda_1, \lambda_2)$ with integral entries. In this notation, $(1,0)$ is ...
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25 views

Are different representations (with the same dimension) of a Lie Algebra share same weights?

Are non-equivalent irreps with dimension n of a lie algebra share the same weight set? E.g. in su(2), given a dimension of the irrep, the weight set is the same no matter what exactly the irrep is. Is ...
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Is the category of all finite dimensional modules of $g$ equivalent to the category of all finite dimensional modules of $U_q(g)$?

Let $g$ be a complex simple Lie algebra and $U_q(g)$ the corresponding quantum group. Is the category of all finite dimensional modules of $g$ equivalent to the category of all finite dimensional ...
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23 views

Is there an analogue/extension of Baker-Campbell-Hausdorff formula for the conjugate?

Let $G$ be a Lie group (I only care about finite dimensional connected simply connected nilpotent groups, if that makes the answer easier). Let $\mathfrak g$ be its Lie algebra and let ...
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Index of Hessian of Composition of maps

Given a Lie group $G$ and a pair of smooth maps: $f:\mathbb{R}^n \rightarrow G \\ g: G \rightarrow [0,1]$ with $g$ possessing a single global optima, but potentially many saddle points. consider ...
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An apparent contradiction in the simple Lie algebra $E_8$

The following is the Dynkin diagram for simple Lie algebra $E_8$ My question is the following: It is clear that $e_i+e_j$ for $i \neq j$ is a positive root. Let $\alpha _8$ be the fundamental ...
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90 views

Vectorfields satisfying SO(3) algebra 2 dimensional?

Let $M$ be an $n$-dimensional smooth connected manifold and $V_1$, $V_2$ and $V_3$ vectorfields whose commutators satisfy the $SO(3)$ algebra: \begin{equation} [V_1,V_2]=V_3 \end{equation} ...
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How to calculate the Lie algebra of a neural network?

Define $F$ as the standard multi-layer feed-forward perceptron: \begin{equation} F(\mathbf{x}) = \Theta( W_1 \circ \Theta( W_2 \circ .... W_L(\mathbf{x}))) \end{equation} where $\Theta$ is the sigmoid ...
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36 views

Proof of Weyl's Theorem [Humphreys]

I don't really understand the first paragraph of the proof of Weyl's Theorem in Humphreys' Lie Algebra book (p. 28). My problem is, that first of all I don't see, why (or in which sense) the exact ...
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Symmetry of Kahler metric on based loop group

The based loop group, $\Omega G$, is known to admit a Kaehler metric, given as \begin{equation} g(X,Y)=2\sum_{k>0}k\textrm{Tr}(X_{-k}Y_k), \end{equation} this is given in page 150 of Segal and ...
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Commutativity of a Lie algebra $\Rightarrow$ the Lie group is abelian

Let $G$ be a Lie group, $\mathfrak{g}$ it's Lie algebra. Assume $[x,y]=0 \, \, \forall x,y \in \mathfrak{g}$. Is it true that $G$ is abelian? Remarks: (1) The other direction ($G$ abelian ...
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31 views

Exactness of Lie algebra exact sequence

If $G\rightarrow H\rightarrow K$ is an exact sequence of Lie groups, then I want to show that the induced sequence $\mathfrak{g}\rightarrow\mathfrak{h}\rightarrow\mathfrak{k}$ in Lie algebras is ...
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Prove that two Lie algebras are equal

Suppose to have a Lie group $\mathbb G$ whose Lie algebra $g$ admits a stratification $g=V_1\bigoplus V_2$ with $\text{dim} V_1=m$ and $g=Lie(V_1)$, i.e. $g$ is generated by $V_1$. In other words, ...