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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Question on inner product on space of representations of compact Lie groups

Let $K$ be a compact connected Lie group, wiewed as subgroup of unipotent matrices. Let $G=\mathfrak{k}^\mathbb C$ be the complexification with Lie algebra $\mathfrak{g}=\mathfrak{k}\oplus ...
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41 views

What are the root systems for the n-dimensional torus?

My question may seem silly at first, but currently I am not able to work out the question of finding all roots for the n-dimensional torus. At first, it seemed obvious to me that there are no roots at ...
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29 views

invariant polynomial on a lie algebra $\mathfrak{g}$

This question (maybe an easy one) arose when I was reading Humphrey's book "an introduction to Lie algebra and its representations". Suppose $\mathfrak{g}$ is a complex semisimple lie algebra, $V$ ...
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23 views

Hermit reciprocity, $\mathfrak{sl}_2(\mathbb{C})$

Let $V$ be the standard $2$-dimensional representation of $\mathfrak{sl}_2(\mathbb{C})$. Hermit reciprocity states that $S^n(S^mV)\simeq S^m(S^nV)$. Can anybody give me a hint to prove it or give a ...
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37 views

Derived series of a Lie algebra

I've been studying semisimple Lie algebras and solvability and was wondering if someone could explain to me the meaning of the derived series of a Lie algebra L and this part: $$L^{(1)}=[LL]$$ I don't ...
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47 views

Lie bracket as defining element for transformations

Why is it precisely the Lie bracket that encodes the information about a given transformation? A Lie algebra is defined by its commutator. Using the exponential map one ends up with a given ...
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160 views

Adjoint action of semi-direct product

Let $G$ and $H$ be Lie groups with associated Lie algebras $\mathfrak{g}:=\text{Lie}(G)$ and $\mathfrak{h}:=\text{Lie}(H)$ and adjoint actions $\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ ...
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21 views

How to classify all $\theta$-stable Cartan sub algebras?

Let $G$ be a linear connected semisimple Lie group, $\mathfrak g$ its Lie algebra. With respect to the Cartan involution $$ \theta:X\mapsto -\overline{X}^t, $$ one has $\mathfrak{g}=\mathfrak{k}\oplus ...
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43 views

References to Lie algebras representaions

Could you give me a reference to a brief introduction to representations of Lie algebras, especially $\mathrm{sl}_2(\mathbb{C})$. I mean some basic Verma modules, Weyl groups etc.
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51 views

Adjoint Representation of Lorentz Group

I'm thinking about the image under the adjoint representation $\mathrm{Ad}$ of the proper (identity connected component) Lorentz group $SO^+(1,3)$. Since this group has a trivial centre (it contains, ...
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33 views

What is a Complete Set of Weights of a Representation of a Lie Subalgebra?

In relation to Lie Group and Lie Algebra theory, I am studying about the weights of representations. I have come across the terminology "a complete string of weights" in my lecture course, but it is ...
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36 views

Component of a pushover vector by one-parameter transformation

I am curious about a step on the proof that shows Lie derivative of a vector field is equivalent Lie bracket. Following comes from Nakahara. We define integral curves by vector field X and Y as ...
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17 views

Symetric powers of $sl_2$ representations

I'd like to understand some special things about representations of $sl_2$ (which is considered as a Lie algebra over $\mathbb{C}$). First, it can be shown that for each $n\in \mathbb{N}$ there is ...
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65 views

Automorphisms of a split Lie algebra — strange proof in Bourbaki

This is about ch. VIII § 5 no. 3 Proposition 5 in Bourbaki's book on Groupes et algèbres de Lie (unchanged on p. 109 in the Hermann 1975 or Springer 2006 edition). The assertion is that for a ...
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27 views

Lie Algebra of Reduced Heisenberg Group Identities

I am having problems trying to understand a statement by Howe in his paper "On the role of the Heisenberg group in harmonic analysis". Here is the setting: Howe defined the (reduced) Heisenber group ...
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40 views

How could we decompose anticommutator of representation matrices for a Lie algebra?

For commutator, we know that $[T^a,T^b]=if^{abc}T^c$, where $f^{abc}$ is the structure constant. But is there a similar formula for $\{T^a, T^b\}$? Thank you.
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32 views

Why is the restricted nullcone a variety?

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $(\mathfrak{g},[\cdot,\cdot],(\cdot)^{[p]})$ be a finite-dimensional restricted Lie algebra. Define the restricted ...
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46 views

Basis of Witt algebra

The Witt algebra $W(n,m)$ is defined as the set of element $\{\sum f_j D_j$ such that $ f_j ∈ A(n,m)\}$ with usual Lie bracket. I am a bit confused about basis for $W(n,m)$? What is the meaning of ...
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35 views

Integration on associated vector bundle

Let $G$ be a compact lie group and $\mathfrak{g}$ be its Lie algebra then we can construct the integral on $G\times \mathfrak{g}$ by $$\int_G\int_{\mathfrak{g}}f(x,Y)dxdY$$ Where $x\in G$ and $Y\in ...
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65 views

Structure constants of Lie algebra from system of linear equations

The Jacobi identity in terms of the structure constants $c_{p,q}^r$ of an $N$-dimensional Lie algebra with $p,q,r=1,\ldots,N$ reads $$ J_{i,j,k}^l \equiv c_{i,j}^m \, c_{k,m}^l + ...
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19 views

Show that $\langle[U,X],V\rangle = -\langle U,[V,X]\rangle$ for bi-invariant metric in Lie group

I know that $\langle U,V \rangle = \langle dR_{x_{t(e)}}U, dR_{x_{t(e)}}V \rangle$ and $\langle U,V \rangle = \langle dL_{x_{t(e)}}U, dL_{x_{t(e)}}V \rangle$ because it is bi-invariant. How do I ...
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42 views

Reference request: exact sequences of Lie algebras

I have a reference request: where can I read more about the following? Consider the short exact sequence $0\rightarrow \mathfrak{n}^- \rightarrow \mathfrak{gl}_n\rightarrow \mathfrak{b}\rightarrow ...
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90 views

Which is the Weyl group of $U(n)$

Consider the unitary group $U(n)$. How does one compute its Weyl group? Is it the same as the Weyl group of $SU(n)$ since $U(n)\simeq SU(n)\times U(1)$?
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49 views

Exercise in Lie algebra course

Let $A$ and $B$ be subalgebras of a Lie algebra $L$ such that $B\subset N_L(A)$. (a) Verify that the space $A+B$ is a subalgebra of $L$. (b) Verify that $A\triangleleft(A+B)$ and $(A\cap ...
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28 views

Simple question: Lie algebra and p-groups

Assume $p$ is a prime and $\pi$ is the set of primes dividing $(p-1)!$. $\mathbb{Q}_{\pi}$ is the set of all rational numbers with $\pi$-numbers as denominators. A $\pi$-number is a product of ...
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115 views

Questions about affine Weyl group and extended affine Weyl group for SL2.

Let $G=SL_2$. Then the Weyl group is generated by $s_1$. On page 3 of the lecture notes, it is said that the affine Weyl group is generated by $s_0, s_1$. (1) The element $s_0s_1$ can be identified ...
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51 views

Lie algebra of the unipotent radical of a standard parabolic subgroup in $GL_n$

Let $k$ be a field, and consider the algebraic group $G=GL_n(k)$. For any partition $n_1+n_2+\ldots+n_m=n$, we have a parabolic subgroup of the form ...
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33 views

Lie algebra homomorphism preserves Jordan form

Fact : $\phi : L_1\rightarrow L_2$ is $surjective$ Lie algebra homomorphism. If $h\in L_1$ and ${\rm ad}_h$ is diagonalizable then ${\rm ad}_{\phi(h)}$ is diagonalizable ...
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42 views

Example of $3$-dimensional Lie algebra

I have a question on $3$-dimensional Lie algebra $L$ over ${\bf C}$ (cf. Erdmann and Wildon's book) Assume that $$ L=(x,y,z),\ L'=(y,z)$$ Then the book states that there exits two kinds of $L$ : ...
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28 views

Base of a root system

Let $R \subset V$ be a reduced root system, and $R' \subset R$. Assume that: (i) $\alpha \in R' \ \to \ - \alpha \notin R'$, (ii) $ \alpha, \beta \in R'$ and $\alpha + \beta \in R$ implies $\alpha ...
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88 views

How do I find the Cartan subalgebra?

I know the definition of a Cartan subalgebra, but how do I actually find it explicitly for a particular Lie algebra over the complex numbers?
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51 views

The Lie subgroup of the compact Lie group

$G$ is a compact connected Lie group with Lie algebra $g$ whose center is $h$. Let $h^{\bot}$ be the orthogonal complement of $h$ where the inner product is chosen to be invariant under the adjoint ...
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44 views

Intertwiner for $U(n-1) \subset U(n)$

I'm using the notation of Vilenkin and Klimyk, ''Part3: Representations of Lie Groups and Special Functions''', chapter 18. Given an irreducible representation $T_m$ of the complex Lie algebra $U(n)$ ...
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30 views

Weight space for a finite-dimensional $\mathfrak{g}-$module $M$.

Let $\mathfrak{g}$ a semisimple Lie algebra, $M$ finite-dimensional $\mathfrak{g}-$module, $\mu\in\mathfrak{h}^*_{\mathbb{Z}}$ and $s_i$ simple reflection such that ...
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32 views

Nil and nilpotent restricted lie algebras

Let $k$ be a field of characteristic $p$, and let $L$ be a restricted Lie algebra over $k$. Thus $L$ is a lie algebra together with a map $(-)^{[p]}:L\to L$ satisfying the three axioms found here. ...
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27 views

Finding an orthonormal basis for a gl(3) module

I'm trying to find an orthonormal basis for gl(3)-module V(ε1-ε3), where ε1-ε3 is the weight (1,0,-1) of the highest-weight vector. Using Gelfand-Tsetlin (/Zetlin/Zeitlin) patterns, I'm at the point ...
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119 views

Connected subgroups of SU(2) and SU(3)

I am reading 'Lie groups, Lie Algebras, and Representations : An Introduction' by Brian Hall and am unable to do the problem 17 in chapter 3. It says Show that every connected Lie subgroup of ...
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53 views

Casimir Invariants within the universal enveloping algebra

I've been asked to determine the eigenvalue of the Casimir invariant $I_2$ on any irreducible module with highest weight $\lambda = (\lambda_1, \lambda_2, ..., \lambda_n)$, where; $$I_m = ...
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107 views

Sources for learning Lie groups and symplectic geometry for Quantum optics

I am asking this question on behalf of my junior who has recently joined in the graduate programme. As suggested by my boss, the student wants to work on quantum optics from a symplectic geometric ...
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87 views

Concrete example of a particular 3-dimensional Lie algebra

I'm reading over the classification of 3-dimensional complex Lie algebras, and have come to the classification of a particular Lie algebra spanned by $\{x,y,z\}$ satisfying the relations $$[x,y] = y, ...
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78 views

homomorphisms between $Gl(n, \mathbb{R})$ and $\mathbb{R}$

I'm tryng to find all differential homomorphism between $Gl(n, \mathbb{R})$ and $\mathbb{R}$ (viwed as Lie groups). My first thought was find the homomorphis between their the Lie algebras, and then ...
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101 views

Dimension of the root spaces of a semisimple complex Lie algebra

I have problems in understanding the proof that the root spaces of a semisimple Lie algebra are all 1-dimensional and that the only multiples of a root $\alpha \in \Phi$ which occur in $\Phi$ are $\pm ...
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82 views

Cartan subalgebra

Let $g$ be a real semisimple Lie algebra with Cartan decoposition $(l,p)$. How can we show that a Cartan subspace $a$ of $p$ (Cartan subspace of $p =$ maximal element in a set that consists of all Lie ...
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Lorentz group and eigenvalues

For generators of the Lorentz group ($\hat {R}_{k}$ corresponds to the generators of 3-rotations, $\hat {L}_{k}$ corresponds to the generators of the boosts) we have the following algebra: $$ [\hat ...
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writting a code for finding the Kostant partition function

How to write a code in sage for finding the Kostant partition function for the elements of root lattice of rank 1 affine lie algebra $A_{1}^{(1)}$ which is defined as follows: $K(\beta)$ = the ...
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58 views

Heisenberg algebra and other Lie algberas

Is there a sub Lie algebra $K$ such that for an ideal $M$ of a heisenberg algebra $H$, $H=K+M$ and $K\cap M=0$ ($M$ has a complement in $H$)? Is there a class of Lie algebras such every ideal $M$ ...
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31 views

If Lie(H) preserves a subspace, must H also preserve that subspace?

Assume $H \subset G$ is a closed connected subgroup of a linear algebraic group over an arbitrary field (both assumed to be smooth). Assume $G$ acts linearly on the (finite dimensional) vector space ...
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90 views

A complex lie algebra is the direct sum of simple ideals iff it is semisimple

So I am wanting to show that a complex lie algebra is the direct sum of simple ideals iff it is semisimple. In fact I have already proved <= It remains for me to prove => $\textbf{Currently I ...
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34 views

Given $L$ a complex finite dimensional Lie algebra. Then suppose $L$ is solvable. Show $L^{(1)}$ is nilpotent.

Given $L$ a complex finite dimensional Lie algebra. Then suppose $L$ is solvable. Show $L^{(1)}$ is nilpotent. Okay, so I have the existence of a flag of ideas in $L$. Can I deduce from this that ...
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230 views

Lie algebra of Euclidean group

From the book "Spinning Tops" by Audin, she claims that $$\mathfrak{so}(3)[\epsilon]/\epsilon^2$$ with coefficientwise Lie bracket is a Lie algebra of a Lie group that is $TSO(3)$ (group action not ...