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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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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76 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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34 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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48 views

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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30 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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43 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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42 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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33 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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56 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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39 views

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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15 views

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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37 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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15 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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35 views

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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21 views

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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59 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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50 views

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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33 views

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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41 views

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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24 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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27 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 ...
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50 views

Lie algabra of R^n

Until now the only example of lie groups I have seen are subgroups of $GL_n$. Today I had the idea, that also $G=(\mathbb R^n,+)$ must be a lie group ($(\mathbb R^n,+)$ is a group with the ...
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36 views

In which course one learns Lie Group&Algebra and to which category of mathematics this subject belongs?

I'm a junior and i have never leanred this subject. I think "Lie Group&Algebra" is really deep and massive theory since the wikipedia page for it is quite long. Nevertheless, i'm not sure ...
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19 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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Computing a Lie Bracket: General Questions

I'm asked to compute the following Lie Bracket: $\left [ -y \dfrac{\partial}{\partial x} + x\dfrac{\partial}{\partial y} , \dfrac{\partial}{\partial x} \right] $ on $\mathbb{R}^2$. Just writing it ...
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99 views

What is the kernel of a Maurer-Cartan form?

The Maurer-Cartan form on the Lie group $Gl(n,\mathbb{R})$ is a one-form taking values in $\mathfrak{gl}(n,\mathbb{R})$ as defined in the link. It has a rather concrete "extrinsic definition" as ...
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18 views

For a matrix group $G$ of dimension $d$, I am trying to prove that the function $Ad : G \rightarrow GL_d(\mathbb{R})$ is smooth.

For a matrix group $G$ of dimension $d$, I am trying to prove that the function $Ad : G \rightarrow GL_d(\mathbb{R})$ is smooth. So where I am starting is by extending $Ad : G \rightarrow ...
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18 views

Double cover $Sp(1) \times SO(3) \rightarrow SO(4)$

I am working on understanding double covers at the moment. And I have come across a few double covers such as $Sp(1) \times Sp(1) \rightarrow SO(4)$ and $Sp(1) \rightarrow SO(3)$. And it seems to me ...
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36 views

Inner automorphisms of Lie groups

I have a few questions about $Aut(G)$, when $G$ is a Lie group. It was proven by Hochschild that if $G/G_0$ is finitely generated, then $Aut(G)$ is a Lie group with at most countably many components. ...
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2answers
99 views

Why does exponentiating the derivative yield the shift operator?

If we formally exponentiate the derivative operator $\frac{d}{dx}$ on $\mathbb{R}$, we get $$e^\frac{d}{dx} = I+\frac{d}{dx}+\frac{1}{2!}\frac{d^2}{dx^2}+\frac{1}{3!}\frac{d^3}{dx^3}+ \cdots$$ ...
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69 views

If the Lie algebra is a direct sum, then the Lie group is a direct product?

I am reading the corollary 21.6 in the book "Morse Theory" by John Milnor, but I've encountered a statement for which I have no ideas. Let $G$ be a simply connected Lie group with a bi-invariant ...
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79 views

Representation theory and particle physics

Are there good books which explain clearly explain the connections between modern particle physics and representation theory of groups and lie algebras?
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23 views

Uniqueness of decomposition of $\mathfrak{sl}(2,\mathbb{C})$-modules

By Weyl's Theorem, I know that every $\mathfrak{sl}(2,\mathbb{C})$-module is completely reducible. I'm under the impression that, up to isomorphism, this decomposition is unique, and I would go about ...
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23 views

Cartan subalgebra of product

i have a simple question what is the Cartan subalgebra of Lie algebra associated to the Lie group ?
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What's wrong with my proof that reductive Lie algebras are semisimple?

If $L$ is a Lie algebra, $\text{Rad}(L)$ denotes its largest solvable ideal. Then $L$ is reductive if $\text{Rad}(L) = Z(L)$ (the center of $L$). An exercise in Humphreys asks: $L$ is reductive if ...
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38 views

Advice on proving a tricky inequality

Im a little out of my depth here and am not well versed in combinatorics. Im not sure if this problem is too hard to solve or if there exists well known results to prove it. Here is part 1 which might ...
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68 views

$\mathfrak{so}(n)$ has trivial center when $n\geq 3$

Is there a nice way to show that $$\mathfrak{so}(n)=\{A \in M(n,\mathbb{R}): A+A^t=0\} $$ has zero center for $n \geq 3$?
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24 views

Kac-Moody root datum introductory text?

I have been given a project to describe the construction of the Lie algebra associated to a Kac-Moody root datum $D=(I,A,\Lambda, (c_i)_{i\in I}, (h_i)_{i\in I})$. I only know basic definitions: that ...
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45 views

The trace as an integral over a sphere [duplicate]

Let $V$ be a real vector space of dimension $n$ and let $\langle \, \cdot\, , \,\cdot\, \rangle$ be an inner product on $V$. We can define a linear functional on the space of endomorphisms of $V$ by ...
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Is there a good list of accidental Lie algebra isomorphisms?

The Wikipedia page Exceptional isomorphisms contains some lie algebra isomorphisms. Is there a list more complete than that, especially including real algebras in low dimensions?
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A1 Lie algebra statement from Jürgen Fuchs' book “Affine Lie algebras and quantum groups”

On page 11, there is a statement saying that applying twice $ad_{E_\pm}$ to an arbitrary $$x = \xi_+E_+ + \xi_-E_- + \zeta H,$$ renders (obviously) $-2\xi_\pm E_\pm,$ the conclusion being that any ...
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31 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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24 views

Is the restricted enveloping algebra local?

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

Infinite series of nested commutators

I'm trying to show the following: If $S_i$ are a set of three matrices such that $$ [S_i, S_j] = \epsilon_{ijk} S_k $$ then $$\exp\big( \alpha_i [S_i, \cdot]\big) S_j = (\exp (M) \vec{S})_j$$ ...
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37 views

Exponents of a semisimple Lie algebra

I'd like to compute the exponents of a semisimple complex Lie algebra $\mathfrak{g}$. According to http://math.berkeley.edu/~theojf/LieQuantumGroups.pdf proposition 8.1.2.18, this amounts to ...