What is Adjoint representation: Definition and 30 Discussions
In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is
G
L
(
n
,
R
)
{\displaystyle GL(n,\mathbb {R} )}
, the Lie group of real n-by-n invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible n-by-n matrix
g
{\displaystyle g}
to an endomorphism of the vector space of all linear transformations of
R
n
{\displaystyle \mathbb {R} ^{n}}
defined by:
x
↦
g
x
g
−
1
{\displaystyle x\mapsto gxg^{-1}}
.
For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of G on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields.
For transformations, A and B are similar if A = S-1BS where S is the change of basis matrix.
For Lie groups, the adjoint representation Adg(b) = gbg-1, describes a group action on itself.
The expressions have similar form except for the order of the inverses. Is there there any connection...
I'm following the lecture notes by https://www.thphys.uni-heidelberg.de/~weigand/QFT2-14/SkriptQFT2.pdf.
On page 169, section 6.2 he is briefly touching on the non-abelian gauge symmetry in the SM.
The fundamental representation makes sense to me. For example, for ##SU(3)##, we define the...
Hi!
I'm doing my master thesis in AdS/CFT and I've read several times that "Fields transforms in the adjoint representation" or "Fields transforms in the fundamental representation". I've had courses in Advanced mathematics (where I studied Group theory) and QFTs, but I don't understand (or...
I'm having a bit of an issue wrapping my head around the adjoint representation in group theory. I thought I understood the principle but I've got a practice problem which I can't even really begin to attempt. The question is this:
My understanding of this question is that, given a...
Suppose I have a hermitian ##N \times N## matrix ##M##. Let ##U \in SU(N)## be the matrix that diagonalizes ##M##: ##M = U\Lambda U^\dagger##, where ##\Lambda## is the matrix of eigenvalues of ##M##. This transformation can be considered as the adjoint action ##Ad## of ##SU(N)## over its...
I am trying to work out the weights of the adjoint representation of SU(3) by calculating the 2 Cartan
generators as follows:
I obtain the structure constants from λa and λ8 using:
[λa,λb] = ifabcλc
I get:
f312 = 1
f321 = -1
f345 = 1/2
f354 = -1/2
f367 = -1/2
f376 = 1/2
f845 = √3/2
f854 =...
Homework Statement
[/B]
I am looking at this document. http://www.math.columbia.edu/~woit/notes3.pdf
Homework Equations
[/B]
ad(x)y = [x,y]
Ad(X) = gXg-1The Attempt at a Solution
[/B]
I understand how ad(S1) and X is found but I don't understand what g and g-1 to use to find Ad(X). Also...
Let U ∈ SU(N) and {ta} be the set of generators of su(N), a = 1, ..., N2 - 1. The action of the adjoint representation of U on some generator ta can be written as
Ad(U)ta = Λ(U)abtb
I want to characterize the matrix Λ(U), i. e., I want to see which of its elements are independent. It's known...
So, we know that if g is a Lie algebra, we can take the cartan subalgebra h ⊂ g and diagonalize the adjoint representation of h, ad(h). This generates the Cartan-Weyl basis for g. Now, let G be the Lie group with Lie algebra g. Is there a way to diagonalize the adjoint representation Ad(T) of...
In some Yang-Mills theory with gauge group ##G##, the gauge fields ##A_{\mu}^{a}## transform as
$$A_{\mu}^{a}
\to A_{\mu}^{a} \pm \partial_{\mu}\theta^{a} \pm f^{abc}A_{\mu}^{b}\theta^{c}$$
$$A_{\mu}^{a}
\to A_{\mu}^{a} \pm...
Not sure if this is the correct forum but here goes.
I am trying to prove [Ta,Tb] = ifabcTc
Where (Ta)bc = -ifabc and fabcare the structure constants for SU(3).
I picked f123 and generated the three 8 x 8 matrices .. T1, T2 and T3.
The matrices components are all 0 except for,
(T1)23 = -i...
Given that ##g T_a g^{-1} = D^b_a T_b## one can show that the generators in the adjoint representation of a group ##G## are the structure constants of the lie algebra satisfied by the ##T_a##.
Write ##g## infinitesimal, so that ##g = 1 + \mathrm {i} \alpha^a T_a## and ##D^c_a = \delta^c_a + i...
Homework Statement
[/B]
1)Show that the kinetic term for a Dirac spinor is invariant under the symmetry group ##U(N) \otimes U(N)##
2) Show that if ##T_a## are the generators of ##O(N)##, the bilinears ##\phi^T T^a \phi## transform according to the adjoint representation.
Homework Equations...
Hey,
There are some posts about the reps of SO, but I'm confused about some physical understanding of this.
We define types of fields depending on how they transform under a Lorentz transformation, i.e. which representation of SO(3,1) they carry.
The scalar carries the trivial rep, and lives...
hi!
in the first page of the attached pdf, after the title " 't hooft double line notation", he says that we have to consider the gluon as NxN traceless hermitian matrices to convince ourselves about the double line notation.
there is my question: if you want the indices a,b to run from 1 to...
Homework Statement
My textbooks takes for granted that, given a Lie group ##g## and its algebra ##\mathfrak{g}##, we have that ##AXA^{-1} \in \mathfrak{g}##.
Homework Equations
For ##Y## to be in ##\mathfrak{g}## means that ##e^{tY} \in G## for each ##t \in \mathbf{R}##
The Attempt at a...
Hi,
I am trying to work through a proof/argument to show that the adjoint representation of a semisimple Lie algebra is completely reducible.
Suppose S denotes an invariant subspace of the Lie algebra, and we pick Y_i in the invariant subspace S. The rest of the generators X_r are such that...
In my limited study of abstract Lie groups, I have come across the adjoint representation ##Ad: G \to GL(\mathfrak{g})## on the lie algebra ##\matfrak{g}##. It is defined through the conjugation map ##C_g(h) = ghg^{-1}## as the pushforward ##C_{g*}|_{g=e}: \mathfrak{g} \to \mathfrak{g}##...
Hello, people.
I'm studying (as an exercise) the breaking of an SU(3) gauge group to SU(2) x U(1) via a Higgs mechanism. The scalar responsible for the breaking is \Phi, who transforms under the adjoint representation of SU(3) (an octet). First of all I want to construct the most general...
This is a short question. I don't know why, but somehow I have the impression that scalar in adjoint representation should be real. Now I highly doubt this statement, but I have no idea how to disprove it. Can anyone give me a clear no?
Thanks,
Hi,
I'm getting a bit confused about the adjoint representation. I learned about Lie algrebras using the book by Howard Georgi (i.e. it is very "physics-like" and we did not distinguish between the abstract approach to group theory and the matrix approach to group theory). He defines the...
Please teach me this:
What is the adjoint representation in Lie group? Where is the vector space that the ''elements of the group'' act on in this representation(adjoint representation)?
Thank you very much for your kind helping.
Basic question, but nevertheless.
In a non-Abelian gauge theory, the fermions transform in the fundamental representation, i.e. doublets for SU(2), triplets for SU(3), while the gauge fields transform in the adjoint representation, which can be taken straight from the structure constants of...
Homework Statement
given that
N\otimes\bar{N} = 1 \oplus A
consinder the SU(2) subgroup of SU(N), that acts on the two first components of the fundamental representation N of SU(N). Under this SU(2) subgroup, the repsentation N of SU(N) transforms as 2 \oplus (N-2)
with info...
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...
Dear All,
I'm reading Georgi's text about Lie algebra, 2nd edition.
In chap 6, he introduced "Roots and Weights". What I didn't understand is the discussion of section 6.2 about the adjoint representation. He said: "The adjoint representation, is particularly important. Because the rows...
I read this in a paper: Suppose there is a theory describing fermions transforming nontrivially under SU(3) gauge symmetry.
L = \Psi^{\bar}(\gamma^A D_A+Y(\Phi))\Psi. The covariant derivative is: D_A\Psi=(\partial_A-i E_A^{\alpha}T_{\alpha})\Psi. Where E_A^{\alpha} are SU(3) gauge fields...
Why has the adjoint representation a higher dimension than the basis matrices it acts on? for example here
Why is e_1 two dim and ad(e_1) four dim?
Isn't ad(X) Y a simple matrix multiplication here? But then multiplying 4x4 with 2x2 matrices, what does it mean?
thanks
what is the sufficient condition for the kernel of an adjoint representation to be the center of the Lie group?
Does the Lie group have to be non-compact and connected, etc?