Lie algebra Definition and 171 Threads

How can we deform a given Lie algebra? In particular, in the attachment file how can we arrive at the commutation relations (20) by starting from the commutation relation (19)?

While studying Yang-Mills theory, I've come across the statement that there exists a positive-definite inner product on the lie algebra ##\mathfrak g## iff the group ##G## is compact and simple. Why is this true, and how it is proved?

Definition/Summary A simple Lie algebra ("Lee") is a nonabelian Lie algebra with no nontrivial ideals. Every finite simple Lie algebra is known, and every semisimple Lie algebra is a combination of simple ones. Every one has a maximal though non-unique abelian subalgebra, its Cartan...

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Homework Statement We have 16 Bosonic Generators E_{ij} = - E_{ji}; i,j = 1 \dots 4 E_{\bar{i} \bar{j}} = E_{\bar{i} \bar{j}}; \bar{i}, \bar{j} = 5 \dots 8 and 16 fermionic generators E_{j \bar{j}}; j = 1 \dots 4, \bar{j} = 5 \dots 8 that span the Lie superalgebra OSp(4|4). By the...

If \alpha and \beta are simple roots, then \alpha-\beta is not. This means that E_{-\vec{\alpha}}|E_{\vec{\beta}}\rangle = 0 Now, according to the text I read, this means that q in the formula \frac{2\vec{\alpha}\cdot \vec{\mu}}{\vec{\alpha}^2}=-(p-q) is zero, where \vec{\mu} is...

Hi, I want to know how can we arrive at the generators of Lie algebra if we have killing equation ? on the other words, In this attached image I want to know how can I arrive at communication relation (16) by starting from killing vector (14) and its constraint...

I am currently working with Lie algebras and my research requires me to have matrix representations for any given Lie algebra and highest weight. I solved this problem with a program for cases where all weights in a representation have multiplicity 1 by finding how E_\alpha acts on each node of...

Why generator matrices of a compact Lie algebra are Hermitian?I know that generators of adjoint representation are Hermitian,but how about the general representaion of Lie groups?

I am studying QFT and got stuck on one question which may be simple. Why is structure constants (in Lie algebra) real and independent on representations? Can you give me a detailed proof? Thanks!

Suppose we have $$[Q^a,Q^b]=if^c_{ab}Q^c$$ where Q's are generators of a Lie algebra associated a SU(N) group. So Q's are traceless. Also we have $$[P^a,P^b]=0$$ where P's are generators of a Lie algebra associated to an Abelian group. We have the following relation between these generators...

So, I just went through the derivation of the Lie algebra for SO(n). in order to do so, we considered ##b^{-1}ab##, and related it to ##U\left(b^{-1}ab\right)##, and since we have a group homomorphism, ##U^{-1}\left(b\right)U\left(a\right)U\left(b\right)##, all of which correspond to the...

Homework Statement Use the Jacobi identity in the form $$ \left[e_i, \left[e_j,e_k\right]\right] + \left[e_j, \left[e_k,e_i\right]\right] + \left[e_k, \left[e_i,e_j\right]\right] $$ and ## \left[e_i,e_j\right] = c^k_{ij}e_k ## to show that the structure constants ## c^k_{ij} ## satisfy the...

Homework Statement The problem statement is to prove the following identity (the following is the solution provided on the worksheet): Homework Equations The definitions of L_{\mu \nu} and P_{\rho} are apparent from the first line of the solution. The Attempt at a Solution I get to the...

Hello! I'm currently reading Ryder - Quantum Field Theory and am a bit confused about his discussion on the correpsondence between Lorentz transformations and SL(2,C) transformations on 2-spinor. He writes that the Lie algebra of Lorentz transformations can be satisfied by setting \vec{K}...

I must apologize if this question sounds dump but if an isomorphism is established between two groups, is it true that their lie algebra is an isomorphism too? My idea is that since the tangent space is sent to tangent space also in the matrix space, both groups' lie algebra will be isomorphism...

Now suppose the derived algebra has dimension 1. Then there exits some non-zero X_{1} \in g such that L' = span{X_{1}}. Extend this to a basis {X_{1};X_{2};X_{3}} for g. Then there exist scalars$\alpha, \beta , \gamma \in R (not all zero) such that [X_{1},X_{2}] = \alpha X_{1} [X_{1},X_{3}] =...

I read in mark wildon book "introduction to lie algebras" "Let F be any field. Up to isomorphism there is a unique two-dimensional nonabelian Lie algebra over F. This Lie algebra has a basis {x, y} such that its Lie bracket is described by [x, y] = x" and I'm curious, How can i proof...

My friend was telling me that charge arises as a coefficient of the Lie Algebra. Can someone give me a demonstration of this please, I was most fascinated by it because I had never heard of this before. Regards!

Homework Statement I have the Casimir second order operator: C= Ʃ gij aiaj and the Lie Algebra for the bases a: [as,al]= fpsl ap where f are the structure factors. I need to show that C commutes with all a, so that: [C,ar]=0 Homework Equations gij = Ʃ fkilfljk (Jacobi identity...

I am currently trying to up my understanding of Lie algebras as the brief introductions I have had from various QFT textbooks feels insufficient, but have been stuck on one small point for a couple days now. I am reading through the lecture notes / book by Robert Cahn found here...

If V is a 3-dimensional Lie algebra with basis vectors E,F,G with Lie bracket relations [E,F]=G, [E,G]=0, [F,G]=0 and V' is the Lie algebra consisting of all 3x3 strictly upper triangular matrices with complex entries then would you say the following 2 mappings (isomorphisms) are different? I...

I'm trying to derive the SO(2,1) algebra from the SO(3) algebra. The generators for SO(3) are given by: M^{ij}_{ab}=i*(\delta_{a}^{i}\delta_{b}^{j}-\delta_{b}^{i}\delta_{a}^{j}) where "a" represents the row, and "b" represents the column, and "ij" represents the generator (12=spin in...

Hi all, I've been wondering about this for some time. While I am only familiar with the basics of differential geometry, I have come across the Lie bracket commutator in a few places. Firstly, what is the intuitive explanation of the Lie bracket [X,Y] of two vectors, if there is one? In...

Hi everyone, I was just wondering if anyone had any suggestions of more-mathematically-rigorous textbooks on Lie groups and Lie algebras for (high-energy) physicists than, say, Howard Georgi's book. I have been eying books such as "Symmetries, Lie Algebras And Representations: A Graduate...

Please teach me this: Why is Casimir operator T^{a}T^{a} be an invariant of the coresponding Lie algebra? I know that Casimir operator commutes with all the group generators T^{a}. Thank you very much for your kind helping.

Sorry for such a simple question, usually I'd go to my physics teacher for help on this question, but it's break and I really don't know the answer. I'm currently studying from an online book (http://www.math.harvard.edu/~shlomo/docs/lie_algebras.pdf) and on the bottom of page 8 he states...

Is this statement correct as it stands? The Lie algebra of a Lie group G consists of extending the group operation of G to the elements of the tangent space of the identity element of G. or does it need to be qualified in some way?

I've been trying for many hours to wrap my head around this problem. Schutz, in his Geometrical Methods of Mathematical Physics book goes through great lengths defining a left-translation map on a Lie Group G, and then defining left-invariant vector fields on G, and then he goes on to say that...

Hi, I want to get to know the structure of Lie algebras better. I just got a really great book, I've found it really clear and well paced: "Symmetries, Lie Algebras, and Representations" from Cambridge monographs on mathematical physics. But I would like some others, and I'm hoping for some...

Given [M^{\mu \nu},M^{\rho\sigma}] = -i(\eta^{\mu\rho}M^{\nu\sigma}+\eta^{\nu\sigma}M^{\mu\rho}-\eta^{\mu\sigma}M^{\nu\rho}-\eta^{\nu\rho}M^{\mu\sigma}) and [ P^{\mu},P^{\nu}]=0 I need to show that [M^{\mu\nu},P^{\mu}] = i\eta^{\mu\rho}P^{\nu} - i\eta^{\nu\rho}P^{\mu}We've been given the...

Homework Statement Let \mathfrak{g} be the vector subspace in the general linear lie algebra \mathfrak{gl}_4 \mathbb{C} consisting of all block matrices A=\begin{bmatrix} X & Z\\ 0 & Y \end{bmatrix} where X,Y are any 2x2 matrices of trace 0 and Z is any 2x2 matrix. You are given that...

Homework Statement Find the derived lie algebra of \mathfrak{so}_3 \mathbb{C}, the 3x3 antisymmetric matrices with entries in \mathbb{C} with Lie bracket the matrix commutator [X,Y]=XY-YX for any X,Y\in \mathfrak{so}_3 \mathbb{C}. The Attempt at a Solution Since \mathfrak{so}_3...

Homework Statement Let \mathfrak{g} be any lie algebra and \mathfrak{h} be any ideal of \mathfrak{g}. The canonical homomorphism \pi : \mathfrak{g} \to \mathfrak{g/h} is defined \pi (x) = x + \mathfrak{h} for all x\in\mathfrak{g}. For any ideal \mathfrak{f} of the quotient lie algebra...

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 & 0...

hi friends ! it is well known that a Lie algebra over K is a K-vector space g equipped of a K-bilinear, called Lie bracket. I ask how can we determines the Lie algebra of any vector space then? For example we try the Lie algebra of horizontal space.

Homework Statement Write the generators of the product of representation t_{a}^{R_1 \otimes R_2} in terms of t_{a}^{R_1}, t_{a}^{R_2} Homework Equations [t_{a}^{R_i} , t_{b}^{R_i}] = i f_{abc} t_{c}^{R_i} I don't believe that the BCH formula is relevant here since that relates...

The following question may be trivial, but I just can't get it figuered out: Consider the real numbers R with the addition operation + as a Lie Group (R,+). What is the Lie Algebra of this Lie Group? Is it again (R,+), this time considered as a vector space? If so, what is the exponential map...

I've tried to understand this concept, but even the so-called "simplest" resources are too complicated to understand. Can anyone please explain this to me? Thanks in advance!

How come that a Lie algebra is defined as being non-associative and at same time it describes a Lie group locally? I wonder because groups are, again by definition, asscioative. thanks

Lie algebra \mathfrak{sl}(2,\mathbb{C}) consists of all 2x2 complex traceless matricies. The space of these matricies is 6-dimensional vector space over real numbers field but is 3-dimensional space over complex numbers field. Number of different representations of this algebra depend on how...

What I am trying to do is start with a Dynkin diagram for a semi-simple Lie algebra, and construct the generators of the algebra in matrix form. To do this with su(3) I found the root vectors and wrote out the commutation relations in the Cartan-Weyl basis. This gave me the structure constants...

I'm a physics undergrad and doing some undergrad study on QFT, and I found that Lie algebra is often invoked in texts, so I decide to take a Lie algebra this sem but I've not taken any abstract algebra course before.The first day's class really beats me because the lecturer used many concepts...

So the generators of the Lorentz Lie algebra relations obey [M^{\rho \sigma}, M^{\tau \mu}] = g^{\sigma \tau} M^{\rho \mu} - g^{\sigma \mu} M^{\rho \tau} + g^{\rho \mu} M^{\sigma \tau} - g^{\rho \tau} M^{\sigma \mu} where (M^{\rho \sigma})^\mu{}_\nu = g^{\rho \mu} \delta^\sigma{}_\nu -...

Okey, I have problem with the foundation of lie algebra. This is my understanding: We have a lie group which is a differentiable manifold. This lie group can for example be SO(2), etc. Then we have the Lie algebra which is a vectorspace with the lie bracket defined on it: [. , .]. This...

Hi, If anyone has a copy of this book, I'm just struggling to understand a point in the chapter about Lie Groups. In particular the section on qunatization of the roots, on p177-178, he generates the diagrams in fig 9.2 from those of 9.1. I thought I understand this, but I don't understand...

Hello there! Above is a problem that has to do with Lie Theory. Here it is: The operators P_{i},J,T (i,j=1,2) satisfy the following permutation relations: [J,P_{i}]= \epsilon_{ij}P_{ij},[P_{i},P_{j}]= \epsilon_{ij}T, [J,T]=[P_{i},T]=0 Show that these operators generate a Lie algebra. Is that...

I have seen it mentioned in various places that the Lie algebra of the diffeomorphism group of a manifold M is identifiable with the Lie algebra of all vector fields on M, but I have not found a demonstration of this. I can show that the map \rho: Lie(Diff(M)) \to Vect(M), ~~~ \rho(X)_p =...

how do i show the lie algebra of the group SO(p,q) is \mathfrak{so}(p,q) = \Biggl\{ \left( \begin{array}{cc} X & Z \\ Z^t & Y \end{array} \right) \vline X \in M_p ( \mathbb{R} ), Y \in M_q ( \mathbb{R} ), Z \in M_{p \times q} ( \mathbb{R} ), X^t=-X, Y^t=-Y \Biggr\} i'm not really getting...

What can we tell about Lie group if we know its Lie algebra. Let's consider the following example: we have three elements of Lie algebra which fulfill condition [L_i,L_j]=i \epsilon_{ijk}L_k . The corresponding Lie group is SU(2) or SO(3) (are there any other?). Does anyone know what...