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Mathematical term
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
Adjoint_representation
Group of unitary complex matrices with determinant of 1
Tr(T_{a}T_{b})={\frac {1}{2}}\delta _{ab}.} In the (n2 − 1)-dimensional adjoint representation, the generators are represented by (n2 − 1) × (n2 − 1) matrices
Special_unitary_group
Mathematical operation on vector spaces
d ( V ) {\displaystyle \mathrm {End} (V)} . In fact it is the adjoint representation ad(u) of E n d ( V ) {\displaystyle \mathrm {End} (V)} . Given
Tensor_product
Algebraic structure used in analysis
Lie algebra g {\displaystyle {\mathfrak {g}}} , the adjoint representation is the representation ad : g → g l ( g ) {\displaystyle \operatorname {ad}
Lie_algebra
Branch of mathematics that studies abstract algebraic structures
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of
Representation_theory
Quotient of special unitary group by its center
representations. PU(n) has an adjoint action on SU(n), thus it has an ( n 2 − 1 ) {\displaystyle (n^{2}-1)} -dimensional representation. When n = 2 this corresponds
Projective_unitary_group
Writing Lie algebra sets as matrices
v = ρ(X)(v). The most basic example of a Lie algebra representation is the adjoint representation of a Lie algebra g {\displaystyle {\mathfrak {g}}} on
Lie_algebra_representation
Correspondence between topics in Lie theory
every representation of the Lie algebra of SO(3) does give rise to a representation of SU(2). An example of a Lie group representation is the adjoint representation
Lie group–Lie algebra correspondence
Lie_group–Lie_algebra_correspondence
Direct sum of simple Lie algebras
lemma.) As the adjoint representation is injective, a semisimple Lie algebra is a linear Lie algebra under the adjoint representation. This may lead to
Semisimple_Lie_algebra
Index of articles associated with the same name
Hermitian adjoint (adjoint of a linear operator) in functional analysis Adjoint endomorphism of a Lie algebra Adjoint representation of a Lie group Adjoint functors
Adjoint
Group of real 2×2 matrices with unit determinant
real Lie group of dimension 3 with applications in geometry, topology, representation theory, and physics. SL(2, R) acts on the complex upper half-plane by
SL2(R)
248-dimensional exceptional simple Lie group
compact Lie groups in that its non-trivial representation of smallest dimension is the adjoint representation (of dimension 248) acting on the Lie algebra
E8_(mathematics)
{g}})\subset \mathrm {GL} ({\mathfrak {g}})} be the (left) adjoint representation of G. The adjoint bundle of P is the associated bundle a d P = P × A d g
Adjoint_bundle
mathematics, the coadjoint representation K {\displaystyle K} of a Lie group G {\displaystyle G} is the dual of the adjoint representation. If g {\displaystyle
Coadjoint_representation
Group representation
transpose in the definition of the dual representation may be identified with the ordinary matrix transpose. Since the adjoint of a matrix is the complex conjugate
Dual_representation
Representation of the symmetry group of spacetime in special relativity
the two above representations are the adjoint representation of the Lie algebra and the adjoint representation of the group respectively. The corresponding
Representation theory of the Lorentz group
Representation_theory_of_the_Lorentz_group
Symmetric bilinear form in mathematics
their fundamental matrix representation):[citation needed] The table shows that the Dynkin index for the adjoint representation is equal to twice the dual
Killing_form
Elementary particle that mediates the strong force
the fundamental representation (triplet, denoted 3) of the color gauge group, SU(3). The gluons are vectors in the adjoint representation (octets, denoted
Gluon
First case of a Lie group that is both compact and non-abelian
{\displaystyle l=1} ) is the 3 representation, the adjoint representation. It describes 3-d rotations, the standard representation of SO(3), so real numbers
Representation theory of SU(2)
Representation_theory_of_SU(2)
above, with ρ = ad {\displaystyle \rho =\operatorname {ad} } , the adjoint representation. (Note the relation between f {\displaystyle f} and ρ {\displaystyle
Lie algebra–valued differential form
Lie_algebra–valued_differential_form
Conjugate transpose of an operator in infinite dimensions
{\displaystyle A} on an inner product space defines a Hermitian adjoint (or adjoint) operator A ∗ {\displaystyle A^{*}} on that space according to the
Hermitian_adjoint
52-dimensional exceptional simple Lie group
379848, 412776, 420147, 627912... The 52-dimensional representation is the adjoint representation, and the 26-dimensional one is the trace-free part of
F4_(mathematics)
Theory of the strong nuclear interactions
are the gluon fields, dynamical functions of spacetime, in the adjoint representation of the SU(3) gauge group, indexed by a, b and c running from 1 {\displaystyle
Quantum_chromodynamics
Subatomic particle; lightest meson
triplet representation or the adjoint representation 3 of SU(2). By contrast, the up and down quarks transform according to the fundamental representation 2
Pion
Aspect of mathematical representation theory
in the space of tensors. The adjoint representation of the simple Lie group of type E8 is a fundamental representation. The irreducible representations
Fundamental_representation
Operation measuring the failure of two entities to commute
repeated derivatives of a product, can be written abstractly using the adjoint representation: x n y = ∑ k = 0 n ( n k ) ad x k ( y ) x n − k . {\displaystyle
Commutator
Natural number
SO(8). The special unitary group SO(3) has an eight-dimensional adjoint representation whose colors are ascribed gauge symmetries that represent the vectors
8
highest long root, making the quasi-minuscule representation be the adjoint representation.) The minuscule representations are indexed by the weight lattice
Minuscule_representation
Concept in mathematics
supersymmetry: its fundamental representation is the so-called spinor representation, while its adjoint representation generates the Lorentz group SO(3
Special_linear_Lie_algebra
Topological quantum field theory
exterior derivative operator d and the connection A, transforms in the adjoint representation of the gauge group G. The square of the covariant derivative with
Chern–Simons_theory
Property of some binary operations
equivalent to the following identity between the operators of the adjoint representation: ad [ x , y ] = [ ad x , ad y ] . {\displaystyle \operatorname {ad}
Jacobi_identity
133-dimensional exceptional simple Lie group
133-dimensional adjoint representation, there is a 56-dimensional "vector" representation, to be found in the E8 adjoint representation. The characters
E7_(mathematics)
Theorem about the dual of a Hilbert space
The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes
Riesz_representation_theorem
Species of elementary particle
respectively. The pions are assigned to the triplet (the spin-1, 3, or adjoint representation) of SU(2). Though there is a difference from the theory of spin:
Flavour_(particle_physics)
Nilpotent subalgebra of a Lie algebra
addition g {\displaystyle {\mathfrak {g}}} is semisimple, then the adjoint representation presents g {\displaystyle {\mathfrak {g}}} as a linear Lie algebra
Cartan_subalgebra
Relativistic quantum mechanical wave equation
connection on the principal bundle, which necessarily transforms in the adjoint representation of the gauge group. The covariant derivative then takes the form
Dirac_equation
Subgroup of a root system's isometry group
T ) {\displaystyle (K,T)} ; the roots are the nonzero weights of the adjoint action of T {\displaystyle T} on the Lie algebra of K {\displaystyle K}
Weyl_group
In mathematics, a type of algebra
(ii) a d ( g ) {\displaystyle {\rm {ad}}({\mathfrak {g}})} , the adjoint representation of g {\displaystyle {\mathfrak {g}}} , is solvable. (iii) There
Solvable_Lie_algebra
Concept in mathematics
direct sum of the two modules g(A) and ν(A) and should be called adjoint representation. Note however that in the more general case where ρ does not have
Representation_up_to_homotopy
{\displaystyle {\mathfrak {g}}} are locally nilpotent. For example, the adjoint representation of a Kac–Moody algebra is integrable. Kac 1990, § 3.6. Kac 1990
Integrable_module
Concept in mathematics
needed for computing that kernel. What we have is the action of the adjoint representation on g ; {\displaystyle {\mathfrak {g}};} we need it on U ( g ) .
Universal_enveloping_algebra
Coefficients of an algebra over a field
representations, and in fact, give exactly the matrix elements of the adjoint representation. The Killing form and the Casimir invariant also have a particularly
Structure_constants
Relationship between two functors abstracting many common constructions
this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics
Adjoint_functors
Simple Lie group; the automorphism group of the octonions
10556, 11571, 11648, 12096, 13090.... The 14-dimensional representation is the adjoint representation, and the 7-dimensional one is action of G2 on the imaginary
G2_(mathematics)
Linear operator equal to its own adjoint
In mathematics, a self-adjoint operator on a complex vector space V {\displaystyle V} with inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot
Self-adjoint_operator
bilinear form. Compatibility means that it is invariant under the adjoint representation. Examples of such are semisimple Lie algebras, such as su(n) and
Quadratic_Lie_algebra
Universal construction of a complex Lie group from a real Lie group
π) ⊂ Z(GC), which follows from the fact that the kernel of the adjoint representation of GC equals its centre, combined with the equality ( C Φ ( k )
Complexification_(Lie_group)
Group of flat spacetime symmetries
Poincaré group is a group extension of the Lorentz group by a vector representation of it; it is sometimes dubbed, informally, as the inhomogeneous Lorentz
Poincaré_group
Group of gauge symmetries in Yang–Mills theory
fiber is a group G {\displaystyle G} which acts on itself by the adjoint representation. The unit element of G ( X ) {\displaystyle G(X)} is a constant
Gauge_group_(mathematics)
Geometric arrangements of points, foundational to Lie theory
modulo centers. In each case, the roots are non-zero weights of the adjoint representation. We now give a brief indication of how irreducible root systems
Root_system
Group representation
Representation theory of the Lorentz group Representation theory of Hopf algebras Adjoint representation of a Lie group List of Lie group topics Symmetry
Representation_of_a_Lie_group
Representation theory of an important group in physics
In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group
Representation theory of the Poincaré group
Representation_theory_of_the_Poincaré_group
Concept in Lie algebra representation theory
reflecting that not every representation of the Lie algebra comes from a representation of the group.) For the adjoint representation a d : g → End ( g )
Weight (representation theory)
Weight_(representation_theory)
Group that is also a differentiable manifold with group operations that are smooth
theory becomes an 11-dimensional theory when N becomes infinite. Adjoint representation of a Lie group Haar measure Homogeneous space Lie point symmetry
Lie_group
Algebraic generalization of the derivative
derivation on the tensor algebra of a manifold. It follows that the adjoint representation of a Lie algebra is a derivation on that algebra. The Pincherle
Derivation (differential algebra)
Derivation_(differential_algebra)
Partial differential equations whose solutions are instantons
compact Lie algebra admits an invariant inner product under the adjoint representation. Since X {\displaystyle X} is Riemannian, there is an inner product
Yang–Mills_equations
operations. adjoint The adjoint representation of a Lie group G is the representation given by the adjoint action of G on the Lie algebra of G (an adjoint action
Glossary of representation theory
Glossary_of_representation_theory
In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, hence the name. More concretely, a Lie algebra is reductive
Reductive_Lie_algebra
Group of matrices with determinant 1
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Special_linear_group
y]=0} for every x , y {\displaystyle x,y} in the algebra. adjoint 1. An adjoint representation of a Lie group: Ad : G → GL ( g ) {\displaystyle \operatorname
Glossary of Lie groups and Lie algebras
Glossary_of_Lie_groups_and_Lie_algebras
Distinguished element of a Lie algebra's center
symmetric homogeneous polynomials in the symmetric algebra of the adjoint representation ad g . {\displaystyle \operatorname {ad} _{\mathfrak {g}}.} : C
Casimir_element
Feature of a system that is preserved under some transformation
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Symmetry_(physics)
Complex matrix A* obtained from a matrix A by transposing it and conjugating each entry
2022). "1.6: Matrix Representation of Complex Numbers". Applied Linear Algebra and Differential Equations. LibreTexts. "Adjoint matrix", Encyclopedia
Conjugate_transpose
Mathematical game
spin-1/2 representation belongs to the fundamental representation, and the spin-1 is the adjoint representation. The notion of double-covering used here is a
Tangloids
via the adjoint representation, and the property of being nilpotent is invariant under this action. A nilpotent orbit is an orbit of the adjoint action
Nilpotent_orbit
Smooth manifold with an inner product on each tangent space
Riemannian metric can be computed explicitly in terms of ge, the adjoint representation of G, and the Lie algebra associated to G. These formulas simplify
Riemannian_manifold
Yang–Mills coupled to a Higgs field
{g}}} under some representation; in particular here we are concerned with the adjoint representation, and the trace on this representation is the Killing
Yang–Mills–Higgs_equations
homomorphism Fundamental representation Antifundamental representation Bifundamental representation Adjoint representation Weight (representation theory) Cartan's
List of representation theory topics
List_of_representation_theory_topics
Manifold with inversion symmetry
product on h {\displaystyle {\mathfrak {h}}} , invariant under the adjoint representation and σ, induces a Riemannian structure on H / K, with H acting by
Hermitian_symmetric_space
78-dimensional exceptional simple Lie group
through breaking to SO(10) × U(1). The adjoint 78 representation breaks, as explained above, into an adjoint 45, spinor 16 and 16 as well as a singlet
E6_(mathematics)
Topological field
which has as "dynamical" fields a 2-form B taking values in the adjoint representation of G, and a connection form A for G. The action is given by S =
BF_model
For a square matrix, the transpose of the cofactor matrix
classical adjoint adj(A) of a square matrix A is the transpose of its cofactor matrix. It is occasionally known as adjunct matrix, or "adjoint", though
Adjugate_matrix
Invariance of operations under geometric translation
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Translational_symmetry
Map from a Lie algebra to its Lie group
Then the following diagram commutes: In particular, when applied to the adjoint action of a Lie group G {\displaystyle G} , since Ad ∗ = ad {\displaystyle
Exponential_map_(Lie_theory)
Concept in mathematics
isomorphic to the product of n copies of the integers, Zn. The adjoint representation is the action of G by conjugation on its Lie algebra g {\displaystyle
Reductive_group
Root system associated to a symmetric space
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Restricted_root_system
Physics property associated with symmetries
representations of the Lie group. Their product then forms the adjoint representation of the group. Thus, a common example is that the product of two
Charge_(physics)
Construction in group theory
projective linear group is called a projective representation of the group G, by analogy with a linear representation (a homomorphism G → GL(V)). These were studied
Projective_linear_group
Quantum number related to the strong force
contains an octet of fields (see gluon field), and belongs to the adjoint representation (8), and can be written using the Gell-Mann matrices as A μ = A
Color_charge
Physics-mathematics connection
There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner. It links the properties
Particle physics and representation theory
Particle_physics_and_representation_theory
one has to use a linear representation of the group to take traces of elements. The most natural one is the adjoint representation. It turns out that for
Trace field of a representation
Trace_field_of_a_representation
Representation theory of the symmetries of non-relativistic quantum space
Wigner's classification of relativistic mechanics) in terms of the representation theory of the Galilean group, which is the spacetime symmetry group
Representation theory of the Galilean group
Representation_theory_of_the_Galilean_group
algebra was termed regular if it is semisimple and the kernel of its adjoint representation is a Cartan subalgebra). An element g ∈ G {\displaystyle g\in G}
Regular element of a Lie algebra
Regular_element_of_a_Lie_algebra
character formula Representation of a Lie group Representation of a Lie algebra Adjoint representation of a Lie group Adjoint representation of a Lie algebra
List_of_Lie_groups_topics
Isometry group of Euclidean space
size n + 1, as explained for the affine group. Details for the first representation are given in the next section. In the terms of Felix Klein's Erlangen
Euclidean_group
Mathematical transformation in physics
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Time-translation_symmetry
Lie algebra, usually infinite-dimensional
{End} ({\mathfrak {g}}),\operatorname {ad} (x)(y)=[x,y],} is the adjoint representation of g {\displaystyle {\mathfrak {g}}} . Under a "symmetrizability"
Kac–Moody_algebra
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Real_form_(Lie_theory)
Type of mathematical group
group of 2 × 2 matrices, but it has a faithful representation as 3 × 3 matrices (the adjoint representation), which can be used in the general case. Many
Linear_group
the highest root, so that V λ {\displaystyle V_{\lambda }} is the adjoint representation, the Dynkin index I ( λ ) {\displaystyle I(\lambda )} is equal to
Dynkin_index
Dual to the Dirac spinor
In quantum field theory, the Dirac adjoint defines the dual operation of a Dirac spinor. The Dirac adjoint is motivated by the need to form well-behaved
Dirac_adjoint
Mathematical expression for linear operators
(x)_{n}=\operatorname {ad} (x_{n})} . The adjoint representation is a very natural and general representation of any Lie algebra. The argument above illustrates
Jordan–Chevalley decomposition
Jordan–Chevalley_decomposition
Special low-energy state in quantum mechanics
be decomposed into the sum of an adjoint representation (the triplet or spin 1 state) and a trivial representation (the singlet or spin 0 state). While
Singlet_state
Mathematical group
play a central role in symplectic geometry, Hamiltonian mechanics, and representation theory. A related but different family is the compact symplectic group
Symplectic_group
Topics referred to by the same term
derivatives AD16, the hexadecimal number equal to decimal number 173 Adjoint representation of a Lie group, abbreviated "Ad" in mathematics Axiom of determinacy
AD_(disambiguation)
Matrices named after Élie Cartan
denote the weight lattice and root lattice, respectively. In modular representation theory, and more generally in the theory of representations of finite-dimensional
Cartan_matrix
Subgroup of the group of invertible n×n matrices
automorphism of g {\displaystyle {\mathfrak {g}}} , giving the adjoint representation: Ad : G → Aut ( g ) . {\displaystyle \operatorname {Ad} \colon
Linear_algebraic_group
Concept in topology
inner product on g {\displaystyle {\mathfrak {g}}} . Under the adjoint representation, K is the subgroup of G that preserves this inner product. If H
Maximal_compact_subgroup
Matrix equal to its conjugate-transpose
mathematics, more precisely in linear algebra, a Hermitian matrix (or self-adjoint matrix) is a square matrix that is equal to its own conjugate transpose—that
Hermitian_matrix
Group of 𝑛 × 𝑛 invertible matrices
contractible – see Kuiper's theorem. List of finite simple groups SL2(R) Representation theory of SL2(R) Representations of classical Lie groups Here rings
General_linear_group
ADJOINT REPRESENTATION
ADJOINT REPRESENTATION
Girl/Female
Biblical
To call, to anoint.
Female
Irish
Variant spelling of Irish Éadan, ÉADAOIN means "face" or perhaps "against" or "opposite."
Girl/Female
Latin
Adroit; skillful.
Girl/Female
Indian, Telugu
Love; To Joint
Boy/Male
Muslim/Islamic
Skillful Adroit
Surname or Lastname
English
English : presumably from Old French joint ‘united’, ‘joined’. The application as a surname is unclear.
Boy/Male
Muslim
Skillful, Adroit (1)
Male
Japanese
(1-å·§, 2-åŒ , 3-å·¥) Japanese name TAKUMI means 1) "adroit," 2) "artisan," or 3) "skilful."
Girl/Female
Hindu, Indian
Representation of Love
Male
English
Anglicized form of Irish Gaelic Comhghall, COWAL means "joint pledge."
Male
Irish
Contracted form of Irish Gaelic Comhghall, COMGAL means "joint pledge."
Girl/Female
Arabic, Muslim
Connection; Joint
Boy/Male
Arabic, Australian, Muslim, Sindhi
Skillful; Adroit
Biblical
to call; to anoint
Surname or Lastname
English
English : from a pet form of Fulcher.German (also Füge) : nickname for a skillful, adroit person, from Middle High German vüege ‘skillful’, ‘fitting’ (see Fiegel).
Girl/Female
British, English, Latin
Dyer; Skillful; Dexterous; Adroit; Right-handed
ADJOINT REPRESENTATION
ADJOINT REPRESENTATION
Girl/Female
Hindu, Indian
Star
Female
Egyptian
, Peace of Nebt.
Boy/Male
Tamil
Shadananan | ஷாதநாநந
Lord Subramanyan
Boy/Male
Hindu
Very pure
Girl/Female
Greek American
Sweetly speaking, sweet-spoken. Famous bearer; 4th century Spanish martyr St Eulalia.
Male
Greek
(Σάββας) Variant spelling of Greek Sabbas, SAVVAS means "Saturday, the Sabbath."
Girl/Female
Arabic, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Muslim, Oriya, Parsi, Punjabi, Sanskrit, Sikh, Tamil, Telugu
Friend
Female
English
English short form of Latin Diana, DIDI means "divine, heavenly."Â Compare with masculine Didi.
Boy/Male
Muslim
Happy
Boy/Male
Hindu
ADJOINT REPRESENTATION
ADJOINT REPRESENTATION
ADJOINT REPRESENTATION
ADJOINT REPRESENTATION
ADJOINT REPRESENTATION
a.
Shared by, or affecting two or more; held in common; as, joint property; a joint bond.
n.
The place or part where two things or parts are joined or united; the union of two or more smooth or even surfaces admitting of a close-fitting or junction; junction as, a joint between two pieces of timber; a joint in a pipe.
n.
A joining of two things or parts so as to admit of motion; an articulation, whether movable or not; a hinge; as, the knee joint; a node or joint of a stem; a ball and socket joint. See Articulation.
v. t.
To reunite the joints of; to joint anew.
v. t.
To provide with a joint or joints; to articulate.
v. i.
To fit as if by joints; to coalesce as joints do; as, the stones joint, neatly.
n.
An adjunct; a helper.
v. t.
To unite by a joint or joints; to fit together; to prepare so as to fit together; as, to joint boards.
v. i.
To join one's self.
p. pr. & vb. n.
of Adjoin
a.
Dexterous in the use of the hands or in the exercise of the mental faculties; exhibiting skill and readiness in avoiding danger or escaping difficulty; ready in invention or execution; -- applied to persons and to acts; as, an adroit mechanic, an adroit reply.
a.
Joined; united; combined; concerted; as joint action.
v. i.
To lie or be next, or in contact; to be contiguous; as, the houses adjoin.
v. t.
To separate the joints; of; to divide at the joint or joints; to disjoint; to cut up into joints, as meat.
imp. & p. p.
of Adjoin
a.
United, joined, or sharing with another or with others; not solitary in interest or action; holding in common with an associate, or with associates; acting together; as, joint heir; joint creditor; joint debtor, etc.
n.
The space between the adjacent surfaces of two bodies joined and held together, as by means of cement, mortar, etc.; as, a thin joint.
v. t.
To join or unite to; to lie contiguous to; to be in contact with; to attach; to append.
n.
The part or space included between two joints, knots, nodes, or articulations; as, a joint of cane or of a grass stem; a joint of the leg.