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




{\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


{\displaystyle g}
to an endomorphism of the vector space of all linear transformations of



{\displaystyle \mathbb {R} ^{n}}
defined by:





{\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.

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  1. K

    I Is There a Connection Between Conjugation and Change of Basis?

    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...
  2. V

    A Adjoint representation and spinor field valued in the Lie algebra

    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...
  3. J

    A Fields transforming in the adjoint representation?

    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...
  4. A

    I Adjoint Representation Confusion

    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...
  5. Luck0

    A Diagonalizing Hermitian matrices with adjoint representation

    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...
  6. N

    SU(3) Cartan Generators in Adjoint Representation

    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 =...
  7. N

    What Is the Adjoint Representation of SU(2)?

    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...
  8. Luck0

    A Characterizing the adjoint representation

    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...
  9. Luck0

    A Diagonalization of adjoint representation of a Lie Group

    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...
  10. S

    A Gauge transformation of gauge fields in the adjoint representation

    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...
  11. N

    I Bases for SU(3) Adjoint representation

    What are the bases for the adjoint representation for SU(3)?
  12. N

    I Adjoint representation of SU(3)

    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...
  13. C

    I Adjoint representation and the generators

    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...
  14. C

    Bilinears in adjoint representation

    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...
  15. G

    Adjoint representation of Lorentz group

    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...
  16. N

    The double line notation and the adjoint representation

    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...
  17. A

    Proof that the adjoint representation is an endomorphism

    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...
  18. maverick280857

    The adjoint representation of a semisimple Lie algebra is completely reducible

    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...
  19. C

    Adjoint representation vs the adjoint of a matrix

    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}##...
  20. M

    Scalar in adjoint representation

    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...
  21. K

    Is scalar in adjoint representation always real

    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,
  22. A

    Confusion about the definition of adjoint representation and roots.

    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...
  23. N

    What is adjoint representation in Lie group?

    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.
  24. S

    Fundamental and Adjoint Representation of Gauge Groups

    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...
  25. A

    Transformation of the adjoint representation

    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...
  26. Y

    Answer: Lie Algebras: Adjoint Representations of Same Dimension as Basis

    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...
  27. I

    Adjoint representation correspondence?

    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...
  28. A

    How does a scalar transform under adjoint representation of SU(3)?

    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...
  29. R

    Why Adjoint Representation Has a Higher Dimension Than Basis Matrices?

    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
  30. kakarukeys

    Sufficient Condition for Adjoint Representation Kernel to be Lie Group Center

    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?