HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the adjoint representation (or adjoint action) of a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
''G'' is a way of representing the elements of the group as
linear transformations In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
of the group's
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
, considered as a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
. For example, if ''G'' is GL(n, \mathbb), 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 to an
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
of the vector space of all linear transformations of \mathbb^n defined by: x \mapsto g x g^ . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
of ''G'' on itself by
conjugation Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the chang ...
. The adjoint representation can be defined for
linear algebraic group In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n wh ...
s over arbitrary
fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
.


Definition

Let ''G'' be a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
, and let :\Psi: G \to \operatorname(G) be the mapping , with Aut(''G'') the
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of ''G'' and given by the
inner automorphism In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group itse ...
(conjugation) :\Psi_g(h)= ghg^~. This Ψ is a
Lie group homomorphism In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the add ...
. For each ''g'' in ''G'', define to be the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of at the origin: :\operatorname_g = (d\Psi_g)_e : T_eG \rightarrow T_eG where is the differential and \mathfrak = T_e G is the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
at the origin ( being the identity element of the group ). Since \Psi_g is a Lie group automorphism, Ad''g'' is a
Lie algebra automorphism In abstract algebra, an automorphism of a Lie algebra \mathfrak g is an isomorphism from \mathfrak g to itself, that is, a linear map preserving the Lie bracket. The set of automorphisms of \mathfrak are denoted \text(\mathfrak), the automorphism g ...
; i.e., an invertible
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
of \mathfrak g to itself that preserves the
Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...
. Moreover, since g \mapsto \Psi_g is a group homomorphism, g \mapsto \operatorname_g too is a group homomorphism. Hence, the map :\mathrm\colon G \to \mathrm(\mathfrak g), \, g \mapsto \mathrm_g is a
group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to re ...
called the adjoint representation of ''G''. If ''G'' is an immersed Lie subgroup of the general linear group \mathrm_n(\mathbb) (called immersely linear Lie group), then the Lie algebra \mathfrak consists of matrices and the exponential map is the matrix exponential \operatorname(X) = e^X for matrices ''X'' with small operator norms. Thus, for ''g'' in ''G'' and small ''X'' in \mathfrak, taking the derivative of \Psi_g(\operatorname(tX)) = ge^g^ at ''t'' = 0, one gets: :\operatorname_g(X) = gX g^ where on the right we have the products of matrices. If G \subset \mathrm_n(\mathbb) is a closed subgroup (that is, ''G'' is a matrix Lie group), then this formula is valid for all ''g'' in ''G'' and all ''X'' in \mathfrak g. Succinctly, an adjoint representation is an
isotropy representation In differential geometry, the isotropy representation is a natural linear representation of a Lie group, that is acting on a manifold, on the tangent space to a fixed point. Construction Given a Lie group action (G, \sigma) on a manifold ''M'', ...
associated to the conjugation action of ''G'' around the identity element of ''G''.


Derivative of Ad

One may always pass from a representation of a Lie group ''G'' to a representation of its Lie algebra by taking the derivative at the identity. Taking the derivative of the adjoint map :\mathrm : G \to \mathrm(\mathfrak g) at the identity element gives the adjoint representation of the Lie algebra \mathfrak g = \operatorname(G) of ''G'': :\begin \mathrm : & \, \mathfrak g \to \mathrm(\mathfrak g) \\ & \,x \mapsto \operatorname_x = d(\operatorname)_e(x) \end where \mathrm(\mathfrak g) = \operatorname(\operatorname(\mathfrak)) is the Lie algebra of \mathrm(\mathfrak g) which may be identified with the
derivation algebra In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A natu ...
of \mathfrak g. One can show that :\mathrm_x(y) = ,y, for all x,y \in \mathfrak g, where the right hand side is given (induced) by the
Lie bracket of vector fields In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields ''X'' and ''Y'' on a smooth m ...
. Indeed, recall that, viewing \mathfrak as the Lie algebra of left-invariant vector fields on ''G'', the bracket on \mathfrak g is given as: for left-invariant vector fields ''X'', ''Y'', :
, Y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
= \lim_ (d \varphi_(Y) - Y) where \varphi_t: G \to G denotes the flow generated by ''X''. As it turns out, \varphi_t(g) = g\varphi_t(e), roughly because both sides satisfy the same ODE defining the flow. That is, \varphi_t = R_ where R_h denotes the right multiplication by h \in G. On the other hand, since \Psi_g = R_ \circ L_g, by
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
, :\operatorname_g(Y) = d (R_ \circ L_g)(Y) = d R_ (d L_g(Y)) = d R_(Y) as ''Y'' is left-invariant. Hence, :
, Y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
= \lim_ (\operatorname_(Y) - Y), which is what was needed to show. Thus, \mathrm_x coincides with the same one defined in below. Ad and ad are related through the exponential map: Specifically, Adexp(''x'') = exp(ad''x'') for all ''x'' in the Lie algebra. It is a consequence of the general result relating Lie group and Lie algebra homomorphisms via the exponential map. If ''G'' is an immersely linear Lie group, then the above computation simplifies: indeed, as noted early, \operatorname_g(Y) = gYg^ and thus with g = e^, :\operatorname_(Y) = e^ Y e^. Taking the derivative of this at t = 0, we have: :\operatorname_X Y = XY - YX. The general case can also be deduced from the linear case: indeed, let G' be an immersely linear Lie group having the same Lie algebra as that of ''G''. Then the derivative of Ad at the identity element for ''G'' and that for ''G'' coincide; hence, without loss of generality, ''G'' can be assumed to be ''G''. The upper-case/lower-case notation is used extensively in the literature. Thus, for example, a vector in the algebra \mathfrak generates a vector field in the group . Similarly, the adjoint map of vectors in \mathfrak is homomorphic to the
Lie derivative In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
of vector fields on the group considered as a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. Further see the
derivative of the exponential map In the theory of Lie groups, the exponential map is a map from the Lie algebra of a Lie group into . In case is a matrix Lie group, the exponential map reduces to the matrix exponential. The exponential map, denoted , is analytic and has as su ...
.


Adjoint representation of a Lie algebra

Let \mathfrak be a Lie algebra over some field. Given an element of a Lie algebra \mathfrak, one defines the adjoint action of on \mathfrak as the map :\operatorname_x : \mathfrak \to \mathfrak \qquad\text\qquad \operatorname_x (y) =
, y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
/math> for all in \mathfrak. It is called the adjoint endomorphism or adjoint action. (\operatorname_x is also often denoted as \operatorname(x).) Since a bracket is bilinear, this determines the
linear mapping In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
:\operatorname:\mathfrak \to \mathfrak(\mathfrak) = (\operatorname(\mathfrak), given by . Within End(\mathfrak), the bracket is, by definition, given by the commutator of the two operators: : , S= T \circ S - S \circ T where \circ denotes composition of linear maps. Using the above definition of the bracket, the
Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the asso ...
: ,_[y,_z_+_[y,_[z,_x.html"_;"title=",_z.html"_;"title=",_[y,_z">,_[y,_z_+_[y,_[z,_x">,_z.html"_;"title=",_[y,_z">,_[y,_z_+_[y,_[z,_x_+_[z,_[x,_y.html" ;"title=",_z">,_[y,_z_+_[y,_[z,_x.html" ;"title=",_z.html" ;"title=", [y, z">, [y, z + [y, [z, x">,_z.html" ;"title=", [y, z">, [y, z + [y, [z, x + [z, [x, y">,_z">,_[y,_z_+_[y,_[z,_x.html" ;"title=",_z.html" ;"title=", [y, z">, [y, z + [y, [z, x">,_z.html" ;"title=", [y, z">, [y, z + [y, [z, x + [z, [x, y = 0 takes the form :\left([\operatorname_x, \operatorname_y]\right)(z) = \left(\operatorname_\right)(z) where , , and are arbitrary elements of \mathfrak. This last identity says that ad is a Lie algebra homomorphism; i.e., a linear mapping that takes brackets to brackets. Hence, ad is a
representation of a Lie algebra In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is g ...
and is called the adjoint representation of the algebra \mathfrak. If \mathfrak is finite-dimensional and a basis for it is chosen, then \mathfrak(\mathfrak) is the Lie algebra of square matrices and the composition corresponds to
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
. In a more module-theoretic language, the construction says that \mathfrak is a module over itself. The kernel of ad is the
center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
of \mathfrak (that's just rephrasing the definition). On the other hand, for each element in \mathfrak, the linear mapping \delta = \operatorname_z obeys the Leibniz' law: :\delta (
, y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
= delta(x),y+ , \delta(y)/math> for all and in the algebra (the restatement of the Jacobi identity). That is to say, ad''z'' is a
derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...
and the image of \mathfrak under ad is a subalgebra of Der(\mathfrak), the space of all derivations of \mathfrak. When \mathfrak = \operatorname(G) is the Lie algebra of a Lie group ''G'', ad is the differential of Ad at the identity element of ''G''. There is the following formula similar to the Leibniz formula: for scalars \alpha, \beta and Lie algebra elements x, y, z, :(\operatorname_x - \alpha - \beta)^n , z= \sum_^n \binom \left \operatorname_x - \alpha)^i y, (\operatorname_x - \beta)^ z\right


Structure constants

The explicit matrix elements of the adjoint representation are given by the
structure constants In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting prod ...
of the algebra. That is, let be a set of
basis vectors In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as componen ...
for the algebra, with : ^i,e^j\sum_k_k e^k. Then the matrix elements for adei are given by :^j = _k ~. Thus, for example, the adjoint representation of su(2) is the defining representation of so(3).


Examples

*If ''G'' is abelian of dimension ''n'', the adjoint representation of ''G'' is the trivial ''n''-dimensional representation. *If ''G'' is a
matrix Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the add ...
(i.e. a closed subgroup of \mathrm(n, \Complex)), then its Lie algebra is an algebra of ''n''×''n'' matrices with the commutator for a Lie bracket (i.e. a subalgebra of \mathfrak_n(\Complex)). In this case, the adjoint map is given by Ad''g''(''x'') = ''gxg''−1. *If ''G'' is SL(2, R) (real 2×2 matrices with
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
1), the Lie algebra of ''G'' consists of real 2×2 matrices with
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' ...
0. The representation is equivalent to that given by the action of ''G'' by linear substitution on the space of binary (i.e., 2 variable)
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...
s.


Properties

The following table summarizes the properties of the various maps mentioned in the definition The
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of ''G'' under the adjoint representation is denoted by Ad(''G''). If ''G'' is
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
, the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of the adjoint representation coincides with the kernel of Ψ which is just the
center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
of ''G''. Therefore, the adjoint representation of a connected Lie group ''G'' is faithful if and only if ''G'' is centerless. More generally, if ''G'' is not connected, then the kernel of the adjoint map is the
centralizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
of the
identity component In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element. In point set topology, the identity componen ...
''G''0 of ''G''. By the
first isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist f ...
we have :\mathrm(G) \cong G/Z_G(G_0). Given a finite-dimensional real Lie algebra \mathfrak, by
Lie's third theorem In the mathematics of Lie theory, Lie's third theorem states that every finite-dimensional Lie algebra \mathfrak over the real numbers is associated to a Lie group ''G''. The theorem is part of the Lie group–Lie algebra correspondence. Histori ...
, there is a connected Lie group \operatorname(\mathfrak) whose Lie algebra is the image of the adjoint representation of \mathfrak (i.e., \operatorname(\operatorname(\mathfrak)) = \operatorname(\mathfrak).) It is called the adjoint group of \mathfrak. Now, if \mathfrak is the Lie algebra of a connected Lie group ''G'', then \operatorname(\mathfrak) is the image of the adjoint representation of ''G'': \operatorname(\mathfrak) = \operatorname(G).


Roots of a semisimple Lie group

If ''G'' is
semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
, the non-zero weights of the adjoint representation form a
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representati ...
. Section 7.3 (In general, one needs to pass to the complexification of the Lie algebra before proceeding.) To see how this works, consider the case ''G'' = SL(''n'', R). We can take the group of diagonal matrices diag(''t''1, ..., ''t''''n'') as our
maximal torus In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefore ...
''T''. Conjugation by an element of ''T'' sends :\begin a_&a_&\cdots&a_\\ a_&a_&\cdots&a_\\ \vdots&\vdots&\ddots&\vdots\\ a_&a_&\cdots&a_\\ \end \mapsto \begin a_&t_1t_2^a_&\cdots&t_1t_n^a_\\ t_2t_1^a_&a_&\cdots&t_2t_n^a_\\ \vdots&\vdots&\ddots&\vdots\\ t_nt_1^a_&t_nt_2^a_&\cdots&a_\\ \end. Thus, ''T'' acts trivially on the diagonal part of the Lie algebra of ''G'' and with eigenvectors ''t''''i''''t''''j''−1 on the various off-diagonal entries. The roots of ''G'' are the weights diag(''t''1, ..., ''t''''n'') → ''t''''i''''t''''j''−1. This accounts for the standard description of the root system of ''G'' = SL''n''(R) as the set of vectors of the form ''ei''−''ej''.


Example SL(2, R)

When computing the root system for one of the simplest cases of Lie Groups, the group SL(2, R) of two dimensional matrices with determinant 1 consists of the set of matrices of the form: : \begin a & b\\ c & d\\ \end with ''a'', ''b'', ''c'', ''d'' real and ''ad'' − ''bc'' = 1. A maximal compact connected abelian Lie subgroup, or maximal torus ''T'', is given by the subset of all matrices of the form : \begin t_1 & 0\\ 0 & t_2\\ \end = \begin t_1 & 0\\ 0 & 1/t_1\\ \end = \begin \exp(\theta) & 0 \\ 0 & \exp(-\theta) \\ \end with t_1 t_2 = 1 . The Lie algebra of the maximal torus is the Cartan subalgebra consisting of the matrices : \begin \theta & 0\\ 0 & -\theta \\ \end = \theta\begin 1 & 0\\ 0 & 0 \\ \end-\theta\begin 0 & 0\\ 0 & 1 \\ \end = \theta(e_1-e_2). If we conjugate an element of SL(2, ''R'') by an element of the maximal torus we obtain : \begin t_1 & 0\\ 0 & 1/t_1\\ \end \begin a & b\\ c & d\\ \end \begin 1/t_1 & 0\\ 0 & t_1\\ \end = \begin a t_1 & b t_1 \\ c / t_1 & d / t_1\\ \end \begin 1 / t_1 & 0\\ 0 & t_1\\ \end = \begin a & b t_1^2\\ c t_1^ & d\\ \end The matrices : \begin 1 & 0\\ 0 & 0\\ \end \begin 0 & 0\\ 0 & 1\\ \end \begin 0 & 1\\ 0 & 0\\ \end \begin 0 & 0\\ 1 & 0\\ \end are then 'eigenvectors' of the conjugation operation with eigenvalues 1,1,t_1^2, t_1^. The function Λ which gives t_1^2 is a multiplicative character, or homomorphism from the group's torus to the underlying field R. The function λ giving θ is a weight of the Lie Algebra with weight space given by the span of the matrices. It is satisfying to show the multiplicativity of the character and the linearity of the weight. It can further be proved that the differential of Λ can be used to create a weight. It is also educational to consider the case of SL(3, R).


Variants and analogues

The adjoint representation can also be defined for
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
s over any field. The co-adjoint representation is the
contragredient representation In mathematics, if is a group and is a linear representation of it on the vector space , then the dual representation is defined over the dual vector space as follows: : is the transpose of , that is, = for all . The dual representation ...
of the adjoint representation.
Alexandre Kirillov Alexandre Aleksandrovich Kirillov (russian: Алекса́ндр Алекса́ндрович Кири́ллов, born 1936) is a Soviet and Russian mathematician, known for his works in the fields of representation theory, topological groups a ...
observed that the
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
of any vector in a co-adjoint representation is a
symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...
. According to the philosophy in
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
known as the orbit method (see also the Kirillov character formula), the irreducible representations of a Lie group ''G'' should be indexed in some way by its co-adjoint orbits. This relationship is closest in the case of
nilpotent Lie group In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series terminates with . Intuit ...
s.


See also

*


Notes


References

* * * . {{DEFAULTSORT:Adjoint Representation Of A Lie Group Representation theory of Lie groups Lie groups