Lie algebras Definition and 83 Threads

I need some help with understanding the basics of Lie algebra theory. I suspect my problems are due to a fundamental misunderstanding somewhere so apologies in advance for the naivety of the questions and for the restatement of elementary mathematics. As I understand it elements of Lie groups...

If a mapping between Lie algebras \varphi : \mathfrak{g} \to \mathfrak{h} takes a basis in \mathfrak{g} to a basis in \mathfrak{h} is it an isomorphism of vector spaces?

Homework Statement • \mathfrak{g} is the Lie algebra with basis vectors E,F,G such that the following relations for Lie brackets are satisfied: [E,F]=G,\;\;[E,G]=0,\;\;[F,G]=0. • \mathfrak{h} is the Lie algebra consisting of 3x3 matrices of the form \begin{bmatrix} 0 & a & c \\ 0...

Homework Statement Show the map \varphi : \mathfrak{g} \to \mathfrak{h} defined by \varphi (aE + bF + cG) = \begin{bmatrix} 0 & a & c \\ 0 & 0 & b \\ 0 & 0 & 0 \end{bmatrix} is bijective. \mathfrak{g} is the Lie algebra with basis vectors E,F,G such that the following relations for...

Homework Statement Let \mathfrak{g} , \mathfrak{h} be Lie algebras over \mathbb{C}. (i) When is a mapping \varphi : \mathfrak{g} \to \mathfrak{h} a homomorphism? (ii) When are the Lie algebras \mathfrak{g} and \mathfrak{h} isomorphic? (iii) Let \mathfrak{g} be the Lie algebra with...

Howard Georgi wrote a book called "Lie Algebras In Particle Physics: from Isospin To Unified Theories (Frontiers in Physics)" Are there any other books like this one - that covers the same stuff - except that uses a more mathematically formal tone? I know representation theory and some the...

Hi folks, If I have a Lie algebra \mathfrak{g} with an invariant (under the adjoint action ad of the Lie algebra) scalar product, what are the conditions that this scalar product is also invariant under the adjoint action Ad of the group? For instance, the Killing form is invariant under...

I need to solve two assignments in Lie algebras. These assignments are not very difficult, but my knowledges in Lie algebras aren't good. 1. Let \delta be a derivation of the Lie algebra \Im. Show that if \delta commutes with every inner derivation, then \delta(\Im)\subseteqC(\Im), where...

I'd like to open a discussion thread for version 2 of the draft of my book ''Classical and Quantum Mechanics via Lie algebras'', available online at http://lanl.arxiv.org/abs/0810.1019 , and for the associated thermal interpretation of quantum mechanics, espoused in the book. The goal of the...

Very often we've identified sets of particles with the weights of a semi-simple Lie algebra - for example, the 8 particles of the baryon octet with the weights of the 8 representation of SU(3) global flavour symmetry (in the old days of the Eightfold Way), or the 3 weak bosons with the weights...

I have a very superficial understanding of this subject so apologies in advance for what's probably a stupid question. Can someone please explain to me why if we have a Lie Group, G with elements g, the adjoint representation of something, eg g^{-1} A_\mu g takes values in the Lie Algebra of G...

Alright, I understand that there are redundant degrees of freedom in the Lagrangian, and because transformations between these possible "gauges" can be parametrized by a continuous variable, we can form a Lie Group. What I am not so firm upon is how Lie Algebras, specifically, the Lie Algebra...

I have some questions regarding the exceptional Lie algebras e(n), n=6,7,8. Can anybody explain to me what prevents us from constructing e(9) from e(8)? What goes wrong? One can use the e(8) lattice vectors and try to construct an e(9) vector; one could go even further and try e(10) etc. I...

I know we need metric, curvature, what is Lie algebras used for? What is the physics meaning?

I am studying Lie Algebras at home by my own, without access to any teacher. Being autodidact, it would much help me to have a book with solved exercises of Lie Algebras or Lie Groups. Does anyone know if is there any book that could help me solving problems, execises? Thank you, Luis

Could anyone recommend a good book on Lie Algebras for self-study? I need to deeply understand Dynkin diagrams and classification of algebras. Other question: does anybody know a book on problems of Lie Groups or Lie Algebras? Thank you.

I'm reviewing my Group theory notes at the moment. I have a few questions. 1)what is the "connected component of the identity"? (How would you go about working this out?) 2)Why is Lie(SO(n))=Lie(O(n))? 3)Why is the SU(2) Lie group isomorphic to a sphere in 3D?

Let: g_{j}(t) be a curve in a group G, which goes through the identity element; g_j(t=0) = identity. and: \xi_{j}=\frac{d}{dt}g_{j}(t)\right|_{t=0} We know that: \xi_{j}{\in}Lie(G) Why can we say: 1) hg(t)h^{-1} (h is an element of the Group) is also a curve in the group, which goes...

Hey All, I want to start learning about Lie Algebras and I was wondering if anyone can recommend any good textbooks. I am an engineer, so I don't really care for a pure maths 'prove this' 'prove that' approach. Also - does anyone have any two cents on 'Manifolds, Tensor Analysis, And...

Hello, Can anyone explain in extremely simple words what Lie Algebras deal with, and are useful for? Could you also point out a very simple, toy example, in which the use of Lie Algebras is vital? Thanks!

Hey there! I have a simple question concerning infinitesimal generators: In order to get properties of the group, one always linearizes the group element (g = \exp(\mathrm{i}\theta^a T^a) = 1 + \theta^a T^a + \hdots); in this way, one can show, say, that the antisymmetric matrices (where both...

We are asked to show that L(SO(4)) = L(SU(2)) (+) L(SU(2)) where L is the Lie algebra and (+) is the direct product. We are given the hint to consider the antisymmetric 4 by 4 matrices where each row and column has a single 1/2 or -1/2 in it. By doing this we generate four matrices...

I know that the highest weight of an irreducible representation is unique (up to scaling). Is the converse true, however? I have seen some proofs of the irreducibility of certain representations that proceed by showing the uniqueness of the highest weight. How do we know that this is a valid...

Homework Statement Derive the commutation relations for the generators of the Galilean group directly from group multiplication law (without using our results for the Lorentz group). Include the most general set of central charges that cannot be eliminated by redefinition of the group...

I know that all Lie algebras comprised of strictly upper triangular matrices are nilpotent. It would seem that there are also nilpotent Lie algebras that are not comprised of strictly upper triangular matrices, but I can't think of any. Does anybody have any examples?

Does anybody can propose a good book on classification of the complex simple Lie algebras. I know that they fall into several simple (An, Bn, Cn, Dn) and exceptional groups (E, F, G), but I find only a pieces of information about these algebras. Also, they could be represented by...

Georgi "Lie Algebras In..." It is a old book, I took it from the library two days ago. And I am ashamed that my instructor did not suggest it during our undergraduate group theory. The selection of exersices is very good. And CarlB will enjoy the one at the end of first chapter: to show that...

This speculative effort may be only wishful thinking, but if mathematical [geometrical] objects can be represented by Lie [and other] Algebras and Groups as well as by Mathematical Games, then this may aid in the search for a GUT / TOE. This effort is not rigorous, but a tenuous association...

I've been studying some things about Lie algebras and I've got some questions about it. We know that for a given Lie group G, the tangent space at the identity has the structure of a Lie algebra, where the Lie bracket is given by [X,Y]=ad(X)(Y). This map ad is the differential at e of the...

I understand the notation [\mathcal{L},\mathcal{L}]=0 to be "the whole Lie algebra commutes with itself", but the precise meaning of [\mathcal{L},\mathcal{L}]=\mathcal{L} I'm unsure of. Does it mean that every element of \mathcal{L} can be generated by taking the commutator of two elements...

Hi guys - long time reader first time poster! I'm currently getting to grips with the topic of Lie Algebras, and I've come across something that's baffled me somewhat. I've been asked to show: so(4) = su(2) \oplus su(2) Where the lower so(n) denotes the Lie Algebra of SO(n) etc. Now, in a...

Hi I'm looking for a guide to introduce muyself in the study of compact and non compact Lie algebras. Please take a minute to signal me some bibliography al the respect. Thank very much Guillom

C will stand for the complexnumbers Let V be a vectorspace over the reals. V x V is the cartesian product, the set of ordered pairs (u,v) We can turn that into a vectorspace over the complexnumbers in what I guess is an obvious or at least very natural way which I'll spell out just for...