Lie algebra Definition and 171 Threads

I know I should be able to look this up but am having trouble this morning. I would like to know the Lie Algebra structure of the three sphere. In particular I'd like to express it in terms of the left invariant vector fields that are either tangent to the fibers of the Hopf fibration or...

Question is in the title. Seems a lot of people throw that statement around as if its obvious, but it isn't obvious to me. I can kind of see how it might be true. If you take a group element, differentiate it wrt the group parameters to pull down the generators, and then evaluate this...

Hello, I hope it's not the wrong forum for my question which is the following: Is there some list of Lie algebras, whose adjoint representations have the same dimension as their basic representation (like, e.g., this is the case for so(3))? How can one find such Lie algebras? Could you...

hi ,i see from a book su(2) has the form U\left(\hat{n},\omega\right)=1cos\frac{\omega}{2}-i\sigmasin\frac{\omega}{2} in getting the relation with so(3),why we choose \frac{\omega}{2},how about changing for \omega? thank you

I hope this is the right place to ask this question: Why does GUT Model has to have a rank of at least 4 (Such as Georgi-Glashaw Model of SU(5) )? In Georgi's Lie Algebras book it vaguely states that they correspond to the generators S, R, T3 and T8, where S and R are generators of U(1) and...

I have five generators of a lie algebra, g_1,g_2,g_3,g_4,g_5 which at first glance I believe are independent, although I could be wrong. I have calculated the structure constants, i.e. \left[g_i,g_j\right]=f_{ij}^k g_k And from that I have calculated a matrix rep using...

The problem statement Let \mathfrak{g} be a nilpotent Lie algebra. Prove that the Killing form of \mathfrak{g} vanishes identically. The attempt 1 \mathfrak{g} itself is a solvable ideal, so \textrm{rad}(\mathfrak{g})=\mathfrak{g} and \mathfrak{g} is not semisimple. By Cartan's criterion the...

I'm mainly hoping that somebody else might have done the same exercise earlier. In that case it could be possible to spot where I'm going wrong. Homework Statement I'm supposed to prove that Lie algebras \mathfrak{o}(3) and \mathfrak{sp}(2) are isomorphic. Homework Equations Let's...

Hi, Show that the Special linear Lie algebra is simple. I tried it with induction but without result.

[SOLVED] Difficult 3D Lie algebra Homework Statement Let \left(\begin{array}{cc} a & b \\ c & d \end{array}\right) \in GL_2(\mathbb{C}). Consider the Lie algebra \mathfrak{g}_{(a,b,c,d)} with basis {x,y,z} relations given by [x,y]= ay + cz [x,z] = by + dz [y,z] = 0 Show that...

I'm looking for a solid book on Lie groups and Lie algebras, there is too many choices out there. What is a classic text, if there is one?

I'm trying to prove that any representation of a semisimple Lie algebra can be uniquely decomposed into irreducible representations. I have seen some sketches of proofs that show that any representation \phi of a semisimple Lie algebra which acts on a finite-dimensional complex vector space...

I read in Knapp's book on Lie algebras that "a 2-dimensional nilpotent Lie algebra is abelian." Why is this the case? Can somebody who knows please tell me?

Suppose A\subset\mathfrak{g} and I\subset\mathfrak{g} are subalgebras of some Lie algebra, and I is an ideal. Is there something wrong with an isomorphism (A+I)/I \simeq A/I, a+i+I=a+I\mapsto a+I, for a\in A and i\in I? I cannot see what could be wrong, but all texts always give a theorem...

I'm reading about gauge theory and the text goes through some stuff about Lie groups and algebras rather quickly. I tried to prove one of the things they state without proof and got stuck. Suppose that M and N are manifolds and \phi:M\rightarrow N is a diffeomorphism. Then we can define a...

Homework Statement Take L = \left(\begin{array}{ccc}0 & -a & -b \\b & c & 0 \\a & 0 & -c\end{array}\right) where a,b,c are complex numbers. Homework Equations I find that a basis for the above Lie Algebra is e_1 = \left(\begin{array}{ccc}0 & -1 & 0 \\0 & 0 & 0 \\1 & 0 &...

The following matrices are written in Matlab codes form. The standard basis for so(3) is: L1 = [0 0 0; 0 0 -1; 0 1 0], L2 = [0 0 1; 0 0 0; -1 0 0], L3 = [0 -1 0; 1 0 0; 0 0 0]. Since [L1, L2] = L3, the structure constants of this Lie algebra are C(12, 1) = C(12, 2) = 0, C(12, 3) = 1...

The following matrices are written in Matlab codes form. The standard basis for so(3) is: L1 = [0 0 0; 0 0 -1; 0 1 0], L2 = [0 0 1; 0 0 0; -1 0 0], L3 = [0 -1 0; 1 0 0; 0 0 0]. Since [L1, L2] = L3, the structure constants of this Lie algebra are C(12, 1) = C(12, 2) = 0, C(12, 3) = 1...

Hi, I have spent all weekend reading Textbooks, where I concentrated on Cahn, trying to understand what is going on in Lie Algebra lecture notes. I am having a lot of trouble because I have no background in maths other than applied maths, and lie algebras is so different to applied maths and I...

OK, firstly I hope this is the rigth place for my question. I'm in a bit of a problem. I need to be able to calucalte the Killing for for a Lie algebra by next wek, but I'm stuck and won't be able to get any help in 'real life' until Friday, not leaving me enough time to sort out my problem. So...

OK, can someone please tell if 0 (zero) would belong to the center of a Lie algebra. By center I mean for a Lie algebra L center(L) = { z in L : [z,x]=0 for all x in L} I think it should, but I'm not too sure...I'm surely confusing myself somewhere along the line, as this shouldn't be...