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 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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77 views

Computing the fundamental groups of simple algebraic groups of type $A$

I'm interested in seeing the computation for the fundamental groups of the simple algebraic groups of type $A$. Below is the definition of the fundamental group for a simple algebraic group $G$. Let ...
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314 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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42 views

explicit realization of irreducible representations of simple lie algebras

I know explicit realization of irreducible representations of simple lie algebra $sl_n$ when the highest weight of that representation is a fundamental weight.Is there any explicit realization of any ...
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85 views

Does the Lie Bracket automatically exist?

Let $g$ be a Matrix Lie Group. The Lie Algebra of $g := Lie(g)$ is defined as $ Lie(g) = \{ \dot{\gamma}(0) | \gamma:(-\epsilon, \epsilon) \rightarrow g, \gamma \in C^1, \gamma(0) = \mathbb{I} \} $ ...
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85 views

A Isomorphism between the extension group and cohomology group of Lie algebras

Within the book An introduction to homological algebra by Weibel, I am trying to prove the following isomorphism, but I am not sure this is true. But I really want to know how to prove or disprove ...
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58 views

PBW theorem for restricted Lie algebras

I'm looking at the proof of the PBW theorem for restricted Lie algebras to be found in Ponto and May's "More Concise Algebraic Topology", page 361 (367 in linked file). I either see an error in their ...
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310 views

Representation of Complexification of Lie Algebra

Is the following obvious? I think it is, but wanted to make sure before an exam tomorrow! "There is a bijection between the complex representations of a real Lie algebra and the complex ...
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77 views

Weights set spans

Definition Let $T$ be a torus and $R: G \to GL(V)$ a representation. $R(T)$ is a collection of commuting matrices and therefore can be simultaniously diagonalized. For a character $\lambda \in ...
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120 views

adjoint representation completely reducible

Let $\mathcal{A}$ be a Lie algebra. Suppose that adjoint representation of $\mathcal{A}$ is completely reducible (or semisimple). Show that $\mathcal{A}$ can be written as a direct sum of semisimple ...
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111 views

Usage and determination of “rank” and “dimension” of groups & representations

Physicist here. I seem to see conflicting statements about the rank of some groups I've come across lately. A paper I'm reading states that $SO(6)$ is rank 3 and therefore its Cartan subalgebra ...
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119 views

Regarding the definition of vector field flow

To make the connection to the Lie derivative, let $t \mapsto \Phi^X_t$ be the 1-parameter group of diffeomorphisms (or flow) generated by the vector field $ X $. The differential $ d\Phi^X_t ...
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202 views

The enveloping algebra of a finite dimensional Lie algebra has no zero divisor

Let $L$ be a complex, finite dimensional Lie algebra. It is well-known that the graded associative algebra of the enveloping algebra $U(L)$ is isomorphic to the symmetric algebra $S(L)$. Therefore ...
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346 views

Standard parabolic Lie subalgebras and conjugacy

Let $\mathfrak g$ be a given semisimple Lie algebra with corresponding adjoint Lie group $G$. A parabolic subalgebra is any subalgebra containing a Borel subalgebra. We can pick a Borel ...
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132 views

Integral forms of loop algebras.

The question following is about integral forms for semisimple Lie algebras and loop algebras constructed from them. Let $\frak g$ a finite-dimensional Lie algebra over $\mathbb C$ and $L(\frak ...
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115 views

Question concerning semisimple Lie algebras

I'm currently solving a problem in Fulton's Representation Theory A first course and I'm not sure why a particular result is true. One part of the problem (exercise 14.15 if anyone is interested) ...
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124 views

The Weyl Group of $F_4$

The Weyl Group of $F_4$ is of order $1152=2^{7} \cdot 3^{2}$. By Burnside's theorem the group is solvable. Is there a way to see solvability from the root system? Is it possible to see the order of ...
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351 views

Invariant inner product $\langle\,,\rangle$ on a Lie algebra

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. We can use the Killing form to identify $\mathfrak{h}$ and $\mathfrak{h}^*$ ...
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31 views

Lagrangian densities, Lie Groups and Lie Algebras

I'm quite new to Physics and I was having a look for the first time to the Standard model. I'm not sure if the mechanism that I'm describing is directly from Weyl or from others but what I found quite ...
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60 views

The cohomology of $\mathrm{GL}_n$ over an algebraically closed field

How does one go about computing the cohomology groups $H^*(\mathrm{GL}_{\kern{0.1em}{m}}(\overline{\mathbb{F}}_p),M)$? I am particularly interested in the case when $M$ is an algebraic representation. ...
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29 views

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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34 views

Properties of the complexification functor for Lie algebras

The complexification of Lie algebras determines a functor from real Lie algebras to complex Lie algebras, whose right adjoint is the restriction of scalars functor. Thus, we know that complexification ...
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67 views

Weyl group of complex Lie group

Let $G$ be a compact connected Lie group with maximal torus $T$. The Weyl group is defined by $$W:=N_G(T)/T.$$ Now, $G$ has a complexification $G_{\Bbb C}$ with maximal torus $T_{\Bbb C}$ which is the ...
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56 views

Differential of a representation of a linear algebraic group

I asked a question if a representation of the lie algebra of a simply connected algebraic group G induces a representation of the group itself here: \link {Representation of the lie algebra of a ...
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12 views

Find basis $\beta$ of $M$ such that $\phi(H)$ sits inside diagonal matrices, $\phi(S)$ sits inside upper triangular matrices w.r.t. basis $\beta$

Following question appeared in my Lie Algebra exam,but unfortunately i could not solve this question. Let $L$ be a semi simple complex Lie Algebra and $ \phi:L \to gl(M)$ be a finite dimensional ...
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28 views

Taylor series identity for polynomial using Lie group

The following question is from Kirillov's Introduction to Lie Groups and Lie Algebras, and my attempt is the following: $$\sum_{n\geq ...
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39 views

Why are $\mathfrak{pgl}_n\simeq\mathfrak{sl}_n$ when characteristic does not divide $n$?

Suppose $k$ is some algebraically closed field whose characteristic does not divide $n$. Why can we identify the lie algebras $\mathfrak{pgl}_n\simeq\mathfrak{sl_n}$ of the projective linear group and ...
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51 views

The simplest way to present a Lie algebra to a wide audience?

I would like to get suggestions from you as to the best way to present the idea and contents of Lie algebras to a wide public of people with no detailed background in maths. What wlould you explain to ...
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40 views

Can the parameter $t$ in the exponential map $e^{tX}$ be complex?

From a Lie algebra to a Lie group, can the parameter $t$ in the exponential map $t\rightarrow e^{tX}$ be complex? If the Lie algebra is a complex one, this is legal, right?
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34 views

Action of $sl(2,\mathbb{C})$ on Dual of Polynomials does not Exponentiate

Let $V$ be the space of holomorphic polynomial functions in two complex variables $\xi,\eta$ and let $V^\ast$ be its dual space with subspace $W$ of linear functionals of the form $Df(1,0)$ where $D$ ...
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48 views

Is the exponential map of complex spin group surjective?

The complex spin group $Spin(n,C)$ is defined as the double cover of $SO(n,C)$. If the the exponential map is surjective, it will give a parametrization of this Lie group. Is it true for this ...
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15 views

Quotient of the abelian Lie group $(\mathbb{C}, +)$ by a full rank lattice

How can I show that a quotient of the abelian Lie group $(\mathbb{C}, +)$ by a full rank lattice has no faithful finite-dimensional linear representation as a complex Lie group? I was thinking of ...
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27 views

Prove that a subgroup in a Lie group is homogeneous

Let $\mathbb E:=(\mathbb R^4, \cdot)$ be a Carnot group whose Lie algebra is given by $\mathfrak g=V_1\oplus V_2 \oplus V_3$, where $V_1=span\{X_1,X_2\},$ $V_2=span\{X_3\},$ $V_3=span\{X_4\}$, the ...
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51 views

Nonstandard analysis, Lie groups and universal enveloping algebras

The idea of nonstandard analysis is to combine finite quantities with infinitesimals. And, back in the day, Lie algebras were roughly considered the "infinitesimal elements" of Lie groups. Say we want ...
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54 views

Root system of an abelian lie subalgebra.

Let $L$ be a lie algebra and $H$ an abelian subalgebra of $L$ such that each element of $h \in H$ is diagonalizable under the adjoint representation. So there exists a basis of common eigenvectors for ...
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31 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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81 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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32 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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33 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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25 views

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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21 views

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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46 views

Outer automorphism group of $\mathfrak{su}(n)$

I'm reading about the special unitary Lie algebras, and seen it said that complex conjugation is not an inner automorphism of $\mathfrak{su}(n)$ for $n>2$. If there an easy way to see this? I ...
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24 views

$24\otimes 24$ notation for representations of $SO(24)$

I was working through a set of string theory notes, and I came across the notation $24 \otimes 24$ to denote a reducible representation of $SO(24)$, but I am not familiar with the notation. I have ...
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20 views

Two different definitions for Lie Algebras for closed subgroup of $GL_n(\mathbb R)$

Let $G$ be a closed subgroup of $GL_n(\mathbb R)$. There are two definitions for $\mathrm{Lie}(G)$ $\mathrm{Lie}(G) = \{ \gamma'(0) : \gamma : (-\epsilon, \epsilon) \rightarrow G \text{ is ...
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35 views

How does a Lie algebra act on a tensor product of L-modules?

What is the $L$-module structure on $V\otimes W$ where $L$ is a Lie-algebra and $V,W$ are $L$-modules? The following question is related but I can't find the definition in there: tensor product of ...
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57 views

Lie Bracket on Lie Algebra vs. Lie derivative on corresponding group orbit tangent vector field

Suppose I have a matrix group, for example the 2-d affine group and two elements of Lie algebra, A and B, expressed as 3x3 matrices. The Lie bracket is [A,B]=AB-BA. Suppose on the other hand I ...
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25 views

Local Lie derivative on $G$-space at the zero of a vector field

Let a Lie group $G$ act on a manifold $M$ and let $X\in Lie(G)$. For now suppose $G=T$ is a torus (but the answer to this question should hold for $G$ abelian). $L_X$ is a vector field on $M$, at a ...
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57 views

Derivatives of solution to Schrödinger equation

Consider the differential equation (Schrödinger, but rewritten to be pleasing to Lie algebraic eyes): $\frac{d U(t)}{dt} = c(t)U(t)$ where $c(t)=a+w(t)b(t)$, $a,b \in \mathfrak{su}(n)$ and $w$ is a ...
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45 views

Relationship between solutions of two matrix differential equations

Given a ($4\times4$ in the important case) matrix differential equation: $\frac{d U_t}{dt}= A_t U_t$ where $U_t \in SU(n)$ and $A_t \in \mathfrak{su}(n)$. What is the relationship between the ...
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61 views

Why doesn't the “naive” scalar product for $SO(n)$ yield something invariant?

By definition, for $SO(n)$ we have $g^T g=1$ for $g \in SO(n)$. Given some vector $v \in V$ and some representation $R: SO(N) \rightarrow \mathrm{Lin}(V)$, the defining condition above tells us ...