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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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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31 views

Question about weights of $\mathfrak{sl}_2 \mathbf{C}$

On p. 148 of Fulton and Harris' book "Representation Theory: A First Course", they write that "Moreover, by the same token, the $V_\alpha$ that appear must form an unbroken string of numbers of the ...
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38 views

Question to Roger Carter's “Lie Algebras of Finite and Affine Type”

In the proof of Proposition 7.31 in Roger Carter's Lie Algebras of Finite and Affine Type, Carter notes that the sets $H_\mu$ and $H_\alpha$ are distinct. Can someone find a good argument why that is ...
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42 views

Dynkin Diagram $SU(n)$

The goal is to give the Dynkin diagram of $SU(n)$. One can show that the complexification of the Lie algebra $\mathfrak{g}$ of $G$ is given by $\mathfrak{G}_{\mathbb{C}}=\mathfrak{sl}(n,\mathbb{C})$ ...
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Naive question about the group $SU(n)$?

As usual, let $SU(n)$ represent the set of all the $n\times n$ unitary matrices with determinant $1$. It's easy to show that any matrix $U$ takes the form $U=e^{iA}$ ($A$ is a $n\times n$ traceless ...
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Action of $H$ in representations of $\mathrm{sl}_2$

Let $X,Y,H$ be the standard base for the Lie algebra $\mathrm{sl}_2({\mathbb{C}})$, i.e. $H=\begin{pmatrix} 1 & 0\\ 0 &-1\end{pmatrix}$, $X=\begin{pmatrix} 0 & 1\\ 0 & 0\end{pmatrix}$, ...
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38 views

A maximal subalgebra of $E_6$ !?

I'm puzzeled by the following sentence in one of Baez's posts: The Lie algebra $E_6$ has a subalgebra of maximal rank isomorphic to $\mathfrak{so}(10)\oplus \mathfrak{u}(1)$. However, I thought ...
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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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The diffential of commutator map in a Lie group

Leb $G$ be a Lie group and $f:G\times G\rightarrow G$ be the commutator map $:(x,y)\mapsto xyx^{-1}y^{-1}$. How to obtain the Lie bracket in the associated Lie algebra of $G$ from the derivatives of ...
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Is there a definition of a dual Lie algebra?

Let $L$ be a Lie algebra. For vector spaces, modules, Banach spaces, etc. we have the notion of a dual. Question: Is it possible to define naturally a Lie algebra $L^*$ that is in some sense dual to ...
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Where does the name “toral” come from?

Where does the name "toral" come from in "toral subalgebra"? I know a little (very little) Lie groups theory, so I guess it could be related to a Lie group whose Lie algebra is the toral one. Is ...
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Are the two definitions of the BGG category equivalent?

Let $\mathfrak{g}$ be a finite dimensional complex semisimple lie algebra, the BGG category $\mathcal{O}$ is defined as the set of $\mathfrak{g}-$ module $M$ such that $M$ is finitely generated; ...
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$[\sigma,F]=0$ if and only if $(z\in F^{(-1)}(0)\Rightarrow \sigma(z)=z)$?

Let $F$ be an element of a Lie algebra $\frak{g}$ with Lie bracket $[.,.]$ whose elements are maps from $\mathbb{C}$ to itself. Suppose $\sigma$ is a field automorphism of $\mathbb{C}$ such that ...
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36 views

invariants of a Lie algebra

What does it mean by "constructing invariants" in algebraic topology or algebra in general? How to define a "invariant" in algebra? What does it mean by the "invariant of a Lie algebra"?
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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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40 views

reduced space of coadjoint orbit

Let $G$ be a compact Lie group and $\lambda\in \frak{g}^*$$=(Lie G)^*$ and $O_\lambda$ be the Coadjoint orbit through $\lambda\in \frak{g}^*$ and $\mu:O_\lambda\to\frak{g}^*$ be the moment map, ...
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Representations of the Special Orthogonal Group in Three Dimensions.

This will perhaps be an unenlightening question, but here I go. Hopefully someone can varify my thoughts. $\\$ Considering Lie Group Theory and Representation Theory, for the case of the $SO(3)$, ...
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36 views

Skew-Symmetric after base change symmetric?

Are there invertible matrices $A,B \in \textrm{GL}(\mathbb{C}^3)$ such that for every skew-symmetric matrix $S \in \textrm{Mat}_{3 \times 3} (\mathbb{C})$ the matrix $A \cdot S \cdot B$ is symmetric? ...
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59 views

Gradient of a real-valued function on SO(3)

I have struggling with a problem of evaluating the gradient of a cost function on the Lie group of rotations: SO(3). The cost is the following: \begin{equation} ...
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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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Conjugacy of simple system in a root system

I'll set up the problem, then ask the question. Let $V$ be a finite dimension vector space over $\mathbb{R}$ and let $\Phi$ be a root system in $V$, i.e. (1) $\Phi \cap \mathbb{R} \alpha = ...
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36 views

Representations of two-dimensional Lie algebra

It is widely known that there is only one $2$-dimensional non-abelian Lie algebra: it can be generated by two vectors $e_1$ and $e_2$ such that $[e_1,e_2]=e_1$. Let us lenote it by $L$. The question ...
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42 views

Weight space of a representation of ${\frak sl}(2,\mathbb C)$

Suppose $(\pi,V)$ is a finite representation of $SU(2)$. Then there's an induced representation $(\pi_*,V)$ of the complexified ${\frak su}^\mathbb C(2) = {\frak sl}(2,\mathbb C)$. Show that the ...
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kac moody algebra and pde

I study PDE via Lie groups method, I also very much into Lie theory, including the infinite dimensional version. Recently I come across some infinite dimensional Lie algebra so-called Kac Moody ...
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Construction of lie algebra $g(A)$ in Victor Kac's book

In Kac's book "infinite dimensional lie algebras" Chapter I, he constructed a infinite lie algebra $g(A)$ starts from any $n\times n$ complex matrix $A$ as follows: Let $\mathfrak{b}$ be a vector ...
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82 views

Lie group and Lie algebra automorphisms

Assume that $G$ is a connected Lie group and that $\alpha:G\rightarrow G$ is an automorphism of $G$. Furthermore let $\alpha_*:\mathfrak{g}\rightarrow\mathfrak{g}$ be the corresponding tangent map at ...
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1answer
43 views

Invariant Non degenerate symmetric bilinear forms on semisimple lie algebras?

We know every finite-dimensional semisimple lie algebra can be written as direct sum of simple lie algebras. Also, everybody knows all invariant symmetric bilinear and non degenerate forms on simple ...
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Actions of Weyl group

I get a feeling what I am going to ask is very standard and classic, but I am not able to find any reference. Any answer or reference would be appreciated. Let us assume that $G$ is a simply ...
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33 views

set of positive roots made negative by a Weyl group element

If $w$ is a Weyl group element of a simple lie algebra and the reduced expression for $w$ is $s_{i_1}s_{i_2}...s_{i_k}$ what are the positive roots (in terms of reflections from reduced expressions of ...
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44 views

Lifts of embeddings of Lie algebras to their universal enveloping algebras

Let $k$ be an algebraically closed field, and let $(\mathfrak{h},[\;,\;])$ be a finite dimensional abelian Lie algebra $k$. Let $(\mathfrak{g},[\;,\;])$ be a finite dimensional Lie algebra over $k$ ...
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44 views

An $\mathrm{Ad}$-invariant inner product that agrees with the trace

Let $\mathfrak{g}$ be a real semisimple Lie algebra. Then, we have an obvious $\mathrm{Ad}$-invariant inner product (I don't care about positive definiteness) on $\mathfrak{g}$, namely the Killing ...
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34 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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33 views

Root space of a Semi simple group an LVS?

A semi-simple Lie group has a Cartan Subalgebra ($H$) (CSA) -an LVS, Dual to this CSA LVS is root space($H^*$), which is set funtionals that map elements of CSA to real numbers and hence useful in ...
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62 views

Quaternions, Lie Groups and Lie Algebras. Steps to realize a paper. [closed]

I have to realize a paper about quaternions and Lie Groups and Lie Algebras. How can I realize the links between quaternions and Lie Groups & Algebras. Which books do you recommend me? First, I ...
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Representations of Nilpotent Lie Algebras

Let $\mathfrak{g}$ be a rational, nilpotent Lie algebra. Then its adjoint representation will consist of elements which are nilpotent matrices over rationals. But this representation generally is not ...
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Dual basis to $e_{i+1}-e_{i} \in \ker ((1,1,…1)^\vee\in(\Bbb E^{n+1})^\vee)$

Studying the root system $A_n$ given by the simple roots $v_i:=e_{i+1}-e_i \in \Bbb E^{n+1}/\Bbb R(1,1,...,1)$ for $i = 1,...,n$, I came across the following dual basis: $v_i^\vee:= ...
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43 views

Is every element of a complex semisimple Lie algebra a commutator?

Let $L$ be a (finite-dimensional) complex semisimple Lie algebra. Then we know that $L = [L,L]$. Is it true that every element of $L$ must be a commutator? Since a complex semisimple Lie algebra is ...
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38 views

Why the square of ideal in Lie algebra is also ideal?

Let $L$ be a Lie algebra over field $F$, $I$ - ideal in this algebra. It's stated that $I^2$ (and so any item of central series) is also ideal in $L$. 1) For any $a, b \in I^2: [a,b] \in I^2$. True, ...
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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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Recovering a restricted Lie algebra from its restricted enveloping algebra

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $\mathfrak{g}$ be a restricted Lie algebra, with restricted enveloping algebra $u(\mathfrak{g})$. We can place a Hopf ...
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Are there always nontrivial primitive elements in a Hopf algebra?

Let $k$ be an algebraically closed field of arbitrary characteristic. Let $H$ be a Hopf algebra over $k$. We say $x\in H$ is a primitive element if $\Delta(x)=1\otimes x+x\otimes 1$, where $\Delta$ ...
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Can the equality $e^{-tY}Me^{tY} = e^{tX}M $ be shown by showing it only to 1st order? (Lie representations)

We have that A and B belong to different representations of the same Lie group. The representations have the same dimension. X and Y are elements of the respective Lie algebra representations. $$A = ...
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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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Ranks of matrix Lie groups and Lie algebra of SU(1,1), SO(2,1)

I was trying to find out by Googling, but had no luck. Am I right in thinking that for the Lie GROUPS: rank SL(n,R) = n, rank SO(n,R) = n (not sure about this one), rank SU(n,C) = n-1 and ...
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23 views

What is a $\mathbb{Z}$-form of an algebra?

A homework problem I have is to describe the Lie algebra associated to a Kac-Moody root datum $\mathcal{K} =(I,A,\Lambda ,(c_i)_{i\in I},(h_i)_{i\in I})$ as well as to describe the universal ...
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Obtaining representations of $G$ from $\mathrm{Lie}(G)$.

Suppose $\mathfrak{g}$ is a semisimple Lie algebra over $\mathbb{C}$, and $\tilde{G}$ is the unique connected, simply connected Lie group whose Lie algebra is $\mathfrak{g}$. Let $C$ be any discrete ...
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A construction of $\mathfrak{e}_8$ in Fulton and Harris

In section $22.4$ of "Representation Theory: A First Course" by Fulton and Harris, the exceptional Lie algebra $\mathfrak{e}_8$ is constructed using a method of Freudenthal. For background, I will ...
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29 views

Category , lie algebras …

I want a reference book about Lie algebras that have the definition of universal enveloping algebra by the categorical point of view. All references that i found use the construction by the quotient ...
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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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29 views

Root spaces of Lie Algebras — semisimple vs. general

(I am mainly following the notation of Roger Carter's Lie Algebras of Finite and Affine Type). Letting $L$ denote a (finite-dimensional) Lie algebra with roots $\Phi$ and Cartan subalgebra $H$, we ...