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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2
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1answer
114 views

$\mathfrak{h}_1,\mathfrak{h}_2$ Cartan subalgebras with $\mathfrak{h}_1\cap\mathfrak{h}_2=0$

Let $\mathfrak{g}$ be a finite dimensional simple Lie Algebra over an algebraically closed field $K$. I'm having trouble to show that always exists Cartan subalgebras $\mathfrak{h}_1,\mathfrak{h}_2$ ...
2
votes
1answer
72 views

Weights of a locally finite-dimensional module

Let $\frak g$ be a complex finite-dimensional simple Lie algebra and $V$ be a $\frak g$-module with weights bounded by above by some fixed weight and suppose that $V$ is locally finite-dimensional. ...
8
votes
1answer
193 views

Schur -Weyl duality for $sl_2$ and $S_n$

$V$ is an $m$ dimensional vector space having a structure of $sl_2(\mathbb{C})$-module, where $sl_2(\mathbb{C})$ is the Lie algebra of the Lie group $SL_2(\mathbb{C})$. The symmetric group $S_n$ acts ...
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0answers
40 views

Inverse boson operator realization of $\mathfrak{so}(3)$

This is actually a homework problem. The inverse boson operators $a^{-1}$ and $\left(a^\dagger\right)^{-1}$ are defined as $$a^{-1} |n\rangle = \frac{1}{\sqrt{n+1}} |n+1\rangle$$ ...
1
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1answer
57 views

#(cardinality of irreps Lie algebras) > #(irreps of ssociative algebras). Proof?

I know that irreducible representations of associative $*$-algebras are fairly restricted: any $*$-algebra $A$ is isomorphic to a finite sum of simple algebras ...
3
votes
2answers
179 views

Lie algebras from differentiation

I noticed that the characterizations of the Lie algebras of matrix Lie groups can be obtained by differentiation. For example: $$O(n) = XX^t = \mathbb{1} \implies \mathfrak{o}(n) = X + X^t = ...
5
votes
0answers
103 views

Pullback of a 3-form to SU2

I have a left invariant 3-form, $\sigma$ on an simply connected Lie group, $G$ whose value at the identity is $\sigma=\langle[x,y],z\rangle$, where $\langle\cdot,\cdot\rangle$ denotes an invariant ...
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0answers
30 views

set of roots satisfying a minimal condition related to the induced Killing form

Let $\mathfrak{g}$ a finite-dimensional complex simple Lie algebra with Cartan subalgebra $\frak h$. Let denote $(\cdot,\cdot)$ the non-degenerate bilinear form on $\frak h^*$ induced by the Killing ...
0
votes
1answer
432 views

Vector space generated by the tensor products of pauli matrices

Let $\sigma_0,\sigma_x,\sigma_y,\sigma_z$ stand for the $2\times 2$ identity matrix and the well known pauli matrices: \begin{equation} ...
1
vote
1answer
306 views

Eigenvalues of ad (Adjoint action)

The Question: Let $A$ be an $n \times n$ matrix with distinct eigenvalues $\lambda_{1},...,\lambda_{n}$. Show that ad$_{A}$ acting on the space of all $n \times n$ complex matrices has $n^{2}$ ...
4
votes
1answer
260 views

Why is the Lie algebra corresponding to group SO(3) called o(3)?

How is Lie algebra named? Why is it $\mathfrak{su}(2)$ for group $SU(2)$, but $\mathfrak{o}(3)$ for group $SO(3)$? What does the "$\mathfrak{s}$" in algebra $\mathfrak{su}(2)$ mean?
6
votes
0answers
172 views

An exercise in Serre's Lie algebra book

Let $k$ be a commutative ring. Prove that a Lie $k$-algebra $\mathfrak{g} = 0$ iff $U\mathfrak{g} = k$. Use the adjoint representaion. Here is my attempt at it: The only non-trivial statement is ...
6
votes
3answers
207 views

A general pattern to find the roots of the classical lie algebras

Is there any general pattern for the roots of each of the classical lie algebras? So, can I tell all the roots of each of the $nth$ rank classical lie algebras $A_n, B_n, C_n, D_n$, as a linear ...
3
votes
1answer
271 views

Is a basis for the Lie algebra of a Lie group also a set of infinitesimal generators for the Lie group?

Let $G$ be a (EDIT: connected) Lie group of dimension $n$, and let $\mathfrak{g}$ be the associated Lie algebra. If $x_1,\ldots,x_n$ is a basis for $\mathfrak{g},$ is it necessarily true that the ...
3
votes
2answers
93 views

“How many” matrices generate 2d Lie algebra (i.e. $[c_1,c_2]=k_1 c_1 +k_2 c_2$)?

Consider a pair of matrices $(c_1, c_2)$. The words "it generates the 2-dimensional Lie algebra", means that there exists a pair of scalars $k_1$, $k_2$, such that $$[c_1, c_2] = k_1 c_1 + k_2 c_2,$$ ...
6
votes
0answers
243 views

Root Systems of Lie Groups.

Let $G$ be a compact Lie group assumed to be a subgroup of $U(n)$. Also, let $T$ be a maximal torus of G. Then there exists a basis $\{v_1, \ldots ,v_d\}$ of the Lie algebra of $G$, $\mathfrak{g}$, ...
1
vote
2answers
158 views

Is there any good example about Lie algebra homomorphisms?

My textbook gave an example of the trace, but I think to get a better comprehension, more examples are still needed. Any example will be helpful ~
5
votes
1answer
118 views

What is good about simple Lie algebras?

Recently I've been reading Naive Lie Theory by John Stillwell. In the book our aim usually concerns finding whether Lie algebras or Lie groups are simple. I wonder what beautiful properties does a ...
8
votes
1answer
203 views

Ties between Lie algebras and ring theory

I would like to get a general understanding of the relationship between (noncommutative) ring theory and Lie algebra theory. All Lie algebras are finite dimensional and over a field $k$ of ...
12
votes
1answer
220 views

How to prove these Lie algebra relations

This is a bit of a basic computational question concerning Lie algebras, but I'm getting kind of bamboozled so I thought I'd post it. I'm confused about how to perform some computations in Serre's ...
2
votes
1answer
403 views

What is the Lie algebra of the ``indefinite orthogonal group''?

For $p,q \geq 0$ and $n=p+q\geq 1$, give $\mathbb{R}^n$ the indefinite inner product (written as a matrix) $$ \begin{pmatrix} I_p & \\ & -I_q \end{pmatrix}, $$ where $I_m$ is the $m \times m$ ...
2
votes
0answers
64 views

Basis of the Engel algebra

If I have a connected, simply connected nilpotent lie group given by the commutators between the elements of a basis of its Lie algebra how can I recover the left invariant vector fields? For ...
9
votes
1answer
364 views

Inscrutable proof in Humphrey's book on Lie algebras and representations

This is a question pertaining to Humphrey's Introduction to Lie Algebras and Representation Theory Is there an explanation of the lemma in §4.3-Cartan's Criterion? I understand the proof given there ...
1
vote
1answer
87 views

Question on Lie's theorem

I am looking a Lie's theorem in Lie algebra liturature but I do not fully understand one part of the proof. The following proof is given in these notes on page 12. Thm. Let $\mathfrak{g}\subset ...
0
votes
0answers
122 views

Derivative/Chain Rule (for MANLYfolds) Computation

Embarrasingly, I can't compute the following derivative. $dh(X)=\left.\frac{d}{dt}h(e^{XT}]\right|_{t=0}$, where $X$ resides in the lie algebra of $\rm SL(3,\Bbb C)$ [ie $\mathfrak{sl}(3,\Bbb C)$] ...
3
votes
2answers
418 views

Lie Algebra Homomorphism Question

So this is a bit of a follow-up to my recent question. I don't mean to inundate the feed with my quandaries, but as I move through the theory I keep hitting stumbling blocks (which y'all so kindly ...
0
votes
1answer
57 views

Question about Lie algebra, where am I going wrong?

Let $M:= \begin{pmatrix}1 & 2 & 3 & 4\\3 & 4 & 4 & 1\\1 & 9 & -1 & -3\\9 &5 & -2 & -4\end{pmatrix} = \begin{pmatrix} A & B\\C & ...
1
vote
1answer
120 views

Conjugate exponential integral formula for Lie algebra

Somewhere in my notes the following formula appears $\int_0^1 e^{s R} \frac{d R}{dt} e^{(1-s)R} ds = \frac{d}{dt} e^{R}$ where $R$ depends on $t$, and has values in a Lie algebra [$\mathfrak{so}(3)$ ...
2
votes
0answers
102 views

Finding the Killing form of $\mathfrak{sp}_{2n}(\mathbb{C})$

How can I find the Killing form of $\mathfrak{sp}_{2n}(\mathbb{C})$? I'm explicitly working with basis vectors in trying to compute $\operatorname{tr}(\operatorname{ad}(a)\operatorname{ad}(b))$ but ...
1
vote
0answers
63 views

Question about root space

Let $\mathfrak{g}$ be a Lie algebra and consider $\operatorname{Rad}(\mathfrak{g})$, the radical of $\mathfrak{g}$, that is, the sum of all solvable ideals in $\mathfrak{g}$. Suppose that we have the ...
1
vote
1answer
70 views

Understanding a statement about the existence of functionals

Thm. Let $L$ be a solvable subalgebra of $gl(V)$, $V$ a finite dimensional nonzero vector space. Then $V$ contains a common eigenvector for all the endomorphisms in $L$. The proof of this theorem is ...
3
votes
1answer
96 views

Lie brackets of $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$

Let $f$ be a smooth function, $f\colon\mathbb{R}^2\to \mathbb{R}$. What is $\left[\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right]$ ? I want to say it's $0$ since $\frac{\partial ...
4
votes
0answers
419 views

Computation of the Killing form of $\mathfrak{gl}_{m}$.

Consider the Killing form of the Lie algebra $\mathfrak{gl}_{m}$. Then $\{e_{ij}\}$ is a basis for this Lie algebra where $e_{ij}$ is a matrix with 1 in the $i$th row, $j$th column and 0 everywhere ...
3
votes
3answers
290 views

Is Cartan subalgebra of a Lie algebra unique?

Let $\mathfrak{g}$ be a Lie algebra. Is Cartan subalgebra of $\mathfrak{g}$ unique? I see in some places it is written "Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$".
4
votes
0answers
95 views

The Nambu bracket

Does anybody know how to show the Jacobi identity for the Nambu bracket in $\mathbb{R}^3$? The Nambu bracket with respect to $c \in \mathcal{F}(\mathbb{R}^3)$ is defined as $$\{F,G\}_c = \langle\nabla ...
0
votes
2answers
91 views

Can you cancel in characteristic p?

Lets say you have an ideal in some algebra of characteristic p. Yeah, so if you have a lie algebra with a field that is characteristic p. Can you cancel. So for example if you have a vector space ...
1
vote
1answer
169 views

Showing Witt Algebra is simple?

So I got the Witt Algebra over finite field http://en.wikipedia.org/wiki/Witt_algebra and need to show that it's simple. But, I don't know where to start. So let I be a nonzero ideal of witt algebra, ...
1
vote
1answer
72 views

algebraic group to the lie algebra and hom

Let $G$ be a linear algebraic group and let $\rho:G \rightarrow GL(V_{1})$ and $\psi:G \rightarrow GL(V_{2})$ be finite representations. Why is $Hom_{G}(V_{1},V_{2}) \subset Hom_{\mathfrak ...
2
votes
0answers
59 views

Representations of $U(d)$. Calculation of Gelfand-Zeitlin patterns for particular vectors.

Following structure is given $\left(\mathbb{C}^d\right)^{\otimes n}$. Consider irreducible representations of $U(d)$. And consider the fully symmetric subspace $T_{\alpha}$ in ...
0
votes
1answer
159 views

bilinear forms and linear algebraic groups

Let $G$ be a linear algebraic group and $\phi$ a finite dimensional regular representation of $G$ into $GL(V)$ I would like to know about bilinear forms on $V$ and when they are $G$-invariant. ...
5
votes
2answers
580 views

Universal Cover of $SL_{2}(\mathbb{R})$

Why does the universal cover of $SL_{2}(\mathbb{R})$ have no finite dimensional representations?
3
votes
1answer
134 views

What is the difference between $\ker( L \bigwedge L \overset{[-,-]}{\rightarrow} L )$ and $\langle a \wedge b \big| [a,b]=0\rangle$?

Let $L$ be a finite dimensional Lie algebra. We view the Lie bracket as a linear map on the exterior square: $$\pi:L \bigwedge L \rightarrow L$$ Define $$\bigwedge L := \langle a \wedge b \big| ...
2
votes
2answers
199 views

Is there some kind of character theory for representations of finite dimensional algebras?

We know that for a representation $V$ of a Lie algebra or a quantum group, we can define character of $V$ as $ch(V)=\sum_{\mu} dim(V_{\mu})e^{\mu}$, where $V_{\mu}$ is the weight space of $V$ with ...
1
vote
1answer
1k views

Lie derivative of a vector field equals the lie bracket

Let $X$ and $Y$ be vector fields on a smooth manifold $M$, and let $\phi_t$ be the flow of $X$, i.e. $\frac{d}{dt} \phi_t(p) = X_p$. I am trying to prove the following formula: $\frac{d}{dt} ...
0
votes
1answer
212 views

Lie algebra using skew-symmetry

Let g be a Lie algebra such that [[x,y],y]=0 for all $x,y \in g$. Show that 3[[x,y],z]=0 for all $x,y,z \in g$. [Hint: Observe that the mapping (x,y,z) to [[x,y],z] is skew-symmetric in x,y,z and make ...
2
votes
1answer
218 views

Lie algebra is associative

Prove that a Lie algebra g is associative iff the derived subalgebra of g is contained in the centre of g, that is $g^{(1)} \subset c(g)$. So we have the derived sub algebra is in the centre of g. We ...
1
vote
1answer
286 views

Derivation of a lie algebra

Let A be an algebra over K with multiplication $(x,y) \rightarrow x \cdot y$. A linear operator D on the vector space A is called a derivation of A if $D(x \cdot y)=(Dx) \cdot y + x \cdot (Dy)$ $( ...
3
votes
1answer
106 views

Adjoint endomorphism on $\mathfrak{sl}(2,k)$

Let $\{e,h,f\}$ be the standard basis of the Lie algebra $\mathfrak{sl}(2,k)$. Prove that $(\mbox{ad }e)^3=0$ http://en.wikipedia.org/wiki/Special_linear_Lie_algebra First I computed $(\mbox{ad ...
5
votes
4answers
548 views

What does the symbol $\operatorname{Tr}$ in the Yang-Mills action mean?

I find that many authors write the Yang-Mills action as follows: $$\mathcal{J}= \int \operatorname{Tr}(F \wedge \star F).$$ I have yet to find a formal description of the symbol $\operatorname{Tr}$ ...
2
votes
1answer
142 views

Computing the Lie algebra of $SU(n, \mathbb{C})$

The group $SU(n, \mathbb{C})$ is the set of $n \times n$ complex matrices $Q$ such that $\det Q = 1$ and $Q\overline{Q}^{T} = 1$. The Lie algebra is the set of traceless anti-Hermitian matrices. To ...