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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Is there a name for this Lie algebra?

Consider the three dimensional, complex Lie algebra with basis $\{a,a^\dagger, I\}$ and the following structure relations: \begin{align} [a,a^\dagger] = I, \qquad [a,I] = 0, \qquad [a^\dagger, I] = ...
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1answer
78 views

Definition of (co)homology of groups and Lie algebras: actions and augmentations

In the Chevalley-Eilenberg chain complex, what is $ux_i$? What does "trivial $\frak{g}$-module $k$" mean? Below I denote $R=k$ (any commutative unital ring). How is the augmentation (last map in the ...
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1answer
74 views

What is $\mathfrak{gl}(\infty)$

As title says, I know what is $\mathfrak{gl}(n,\mathbb{C})$, but what is $\mathfrak{gl}(\infty)$? Where can I find good reference for this?
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52 views

Commutativity and Maximal Tori in Connected, Compact Lie Groups

Let $G$ be a path-connected, compact Lie Group. Let $x \in G$ and let $T_x \subset G$ denote the union of all the maximal tori in $G$ that contain $x$. Question: Is it true that if $y \notin T_x$, ...
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52 views

$[[x,y],z]=[x,[y,z]] \Rightarrow [x,y]=0$?

I got the next problem: Let $A$ be a Lie algebra, prove that if the bracket associates $([[x,y],z]=[x,[y,z]]$) then the bracket is zero $([x,y]=0)$. Can't get the result using the properties ...
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73 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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78 views

Real representations of SL(2,C)

Is there a classification of real-linear (rather than complex-linear) finite-dimensional representations of SL(2,C)?
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132 views

Zero-product property for enveloping algebras

Let $L$ be a finite-dimensional Lie algebra $L$ over a field $k$. Let $(U(L), i)$ be a universal enveloping algebra of $L$. If $x,y \in U(L) - \{0\}$ is there something contradictory about the ...
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49 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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1k views

How should I show that the Lie algebra so(6) of SO(6) is isomorphic to the Lie algebra su(4) of SU(4)?

As far as I can see, an isomorphism of Lie algebras is a bijective map which preserves the Lie bracket. I need to show that $\mathfrak{so}(6)$ (the Lie algebra of SO(6)) is isomorphic to the ...
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602 views

What is the correspondence between structure constants and a Lie group?

Let $T^a$ (with $a = 1,2,\ldots,n$) be a set of generators of a Lie group that satisfy the commutation relations: \begin{equation} [T^a,T^b] = i \sum_{c=1}^n f^{abc} T^c \,, \end{equation} where ...
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137 views

Vector fields on smooth manifolds and Lie algebras

I'm currently studying differential geometry on smooth manifolds using differential forms and I'm trying to apply it to what I have learned earlier about Lie groups, but something doesn't seem to ...
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37 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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111 views

Open questions in theory of Lie groups and Lie algebras

I am just about to finish an introductory book 'Lie groups, Lie algberas & Representations' by Brian Hall and am curious to know what are the current directions of research in this area. I learn ...
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1answer
134 views

Question regarding isomorphisms in low rank Lie algebras

I am reading Brian Hall's book 'Lie Groups, Lie Algebras, & Representations' and on p.271 I find that in low rank Lie algberas there are some isomorphisms. For example, ...
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62 views

Irreducibility of Lie algebra representations

Let $\mathfrak{g}$ be a finite dimensional real Lie algebra and $\pi: \mathfrak{g} \to \mathfrak{gl}(V)$ be a homomorphism of real Lie algebras where $V$ is a finite dimensional real vector space. But ...
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68 views

definition of derived algebra $[L,L]$ of a Lie algebra $L$

Definition of derived algebra of a Lie algebra $L$ is given by linear span of commutators $[x,y]$ for $x,y \in L$. but here why do we take linear span and why cant we just consider collection of all ...
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37 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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104 views

semisimplicity of Lie algebra

Let $L$ be a lie algebra. Then if $L$ is semisimple, we have $L = L_1 \oplus \cdots\oplus L_n$ for some simple ideals $L_i$. But we can also consider the adjoint representation. In this ...
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680 views

video lectures on Lie algebra

Is there any video lecture on first course on Lie algebra available online? , by the first course I mean, The complete book of Introduction of Lie algebra and its representation theory by James ...
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2answers
53 views

gl(n,C) as a direct sum?

I've been trying to prove this: $gl_n(\mathbb{C})=Sl_n(\mathbb{C})\oplus \mathbb{C}I_n $ where $gl_n(\mathbb{C})$ is the General Linear Lie Algebra and $Sl_n(\mathbb{C})$ is the Special Linear ...
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228 views

Definition of Verma modules

I have a question regarding different (but equivalent!?) definitions of Verma modules of semisimple Lie algebras: Let F be a field and denote the following: $ \mathfrak{g}$ , a semisimple Lie ...
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2answers
39 views

Image of a nilpotent Lie algebra

Suppose $L_1$ and $L_2$ are two Lie algebras, and that $f: L_1\to L_2$ is a Lie algebra homomorphism. If $L_1$ is nilpotent, does it follow that $f(L_1)$ is nilpotent? Remark 1. The corresponding ...
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121 views

Equation On Root Systems (Humphreys Exercise 9.10)

I am stuck in the following problem from Humphreys. Let $\alpha, \beta$ be roots in a root system $\Phi$. Let the $\alpha$-string through $\beta$ be $\beta - r\alpha, \ldots, \beta + q\alpha$ and let ...
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205 views

Properties of Lie derivative

Let's have Lie derivative: $$ L_{V}\varphi = V^{\mu}\partial_{\nu}\varphi , \quad L_{V}A_{\mu} = V^{\nu}\partial_{\nu}A_{\mu} + (\partial_{\mu}V^{\nu})A_{\nu}. $$ How to show that for scalar and ...
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49 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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1answer
77 views

What is the smallest Lie subalgebra of $ {{\frak{gl}}_{n}}(\mathbb{R}) $ whose center is the set of $ (n \times n) $-scalar matrices?

We know that the center of the Lie algebra $ {{\frak{gl}}_{n}}(\mathbb{R}) $ of all $ (n \times n) $-matrices is the Lie subalgebra of all $ (n \times n) $-scalar matrices. The Lie algebra $ ...
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149 views

When is the connected component of the identity of a matrix Lie group the union of the one parameter subgroups?

When is it the case that all elements in the connected component of the identity of a matrix Lie group can be written as a single exponential of some element of the corresponding Lie algebra?
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How can I show that $\mathfrak{sl}_n(\mathbb{C})$ is a simple Lie algebra?

The question is in the title: how can I show $\mathfrak{sl}_n(\mathbb{C})$ is simple? In every book I scoured, they say $\mathfrak{sl}_n(\mathbb{C})$ is simple but they do not provide a proof! Is ...
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211 views

Jordan–Chevalley decomposition

I'm trying to understand the proof of it in Humphreys(Humphreys 1972, Prop. 4.2, p. 17). And I've not got over which field we are working. The characteristic polynomial may not have roots in the ...
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94 views

Problem 4.4 - Lie Algebras - Humphreys

I read the exercise 4.4 in the book Introduction to Lie algebras and representation theory of J. Humphreys, and I do not quite understand the sentence : We start with $L\leq\mathfrak{gl}(p,F)$ as ...
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51 views

Lie algebra homomorphism

I'm sure I'm missing something really obvious here. This seems too stupid. On page 47 of Erdmann & Wildon's Introduction to Lie Algebras, we have the following set up. Let $L$ be a Lie subalgebra ...
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414 views

Does every Lie algebra come from commutator of some associative product operation?

Suppose $\mathfrak{g}$ is an Lie algebra. Is it possible to define an associative product operation $\star$ on $\mathfrak{g}$ such that $[A,B]=A\star B - B \star A$ ? If it is not possible to do so ...
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198 views

Show that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is isomorphic to V + iV

Let V be a real n-dimensional vector space. Show that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is isomorphic to V + iV. Note that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is a real vector space and is ...
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155 views

Faithfulness of adjoint representation of Lie algberas

Are there any simple or useful conditions (necessary & sufficient) under which the adjoint representation lie algebra is faithful ? One sufficient condition is semisimplicity, but perhaps this is ...
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90 views

Perturbation in characteristic p, or Why, really, does Lie's theorem fail?

While recalling some basics of Lie theory, I found a funny proof of the main lemma in Lie's theorem on triangularity of representations of solvable Lie algebras. It turns out that this proof has a ...
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84 views

Lie algebra associated to a linear form

Let $F$ be a finite dimensional vector space over a field $k$, if $f : F \to k$ is any linear form, I can define on $F$ a Lie algebra bracket by the following rule $$ [x,y]=f(x)y-f(y)x, $$ or in terms ...
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1answer
635 views

Dimension of the Lie algebra of $SO(n,k)$

Let $SO(n,k)$ denote the special generalized orthogonal group. Of course, $SO(n,k)$ is a Lie group. I know that the Lie algebra $so(n,k)$ of $SO(n,k)$ coincides with the Lie algebra of $O(n,k)$. I'd ...
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70 views

Complete reducibility of tensor product

Let $L$ be a Lie algebra (over a algebraically closed field, not sure if it is relevant). If $V$ and $W$ are two completely reducible $L$-modules, can anyone give a hint on how to show that $V\otimes ...
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189 views

Existence of an irreducible $L$-submodule

Suppose $L$ is a finite dimensional Lie algebra. Let $V$ be an $L$-module (i.e. $V$ is a vector space which $L$ acts upon). We are assuming that $V$ has a finite dimension. My question is the ...
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118 views

O(p,q; C) isomorphic to the usual orthogonal group O(p + q; C) for complex field

I'm trying to make sense of this statement that appears on wiki: "The group O(p,q) is defined for vector spaces over the reals. For complex spaces, all groups O(p,q; C) are isomorphic to the usual ...
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167 views

How to prove a property of Lie derivatives

I know that there are five properties for Lie derivative. But one of them I don't know how to prove. It is $ L_x[\omega(Y)]=(L_x \omega)(Y)+\omega(L_x Y)$ Note : Here $\omega$ is a covariant vector ...
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234 views

Derived algebra of a lie algebra contained in an ideal

Let $\mathfrak{g}$ be a Lie algebra over $\mathbb{R}$ or $\mathbb{C}$. Assume $\mathfrak{i}$ is an ideal with $\mathfrak{g/i}$ abelian. Then the derived algebra $[\mathfrak{g},\mathfrak{g}]\subseteq ...
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132 views

The Poincare group: rank and casimirs.

It is often stated without proof in the particle physics literature that the Poincare group has rank $2$ and that as a result, the corresponding Lie algebra has exactly $2$ casimirs. How can one show ...
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223 views

Schur's lemma and Invariant subspaces of direct sums of irreducible representations

There is a corollary to Schur's lemma which says that : If $V$ is a finite dimensional irreducible complex representation of a group G or Lie algebra and $\phi :V \rightarrow V$ is an intertwining ...
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381 views

When is the Cayley transform of a matrix $J$-orthogonal?

The (real) general linear group is defined $GL(n)=\{A \in \mathbb{R}^{n\times n} \mid \operatorname{det}(A) \neq 0\}$. It is a matrix Lie group. Let $J$ be a constant $n$-by-$n$ real matrix. The ...
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2answers
301 views

Reference request for studying Lie group & Lie algebra representations

I am learning representation theory of Lie groups & Lie algebras from the book by Brian Hall. Unfortunately, this does not discuss infinite dimensional representations. Which books should I study ...
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286 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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69 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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155 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 ...