En matemáticas , la representación adjunta (o acción adjunta ) de un grupo de Lie G es una forma de representar los elementos del grupo como transformaciones lineales del álgebra de Lie del grupo , considerada como un espacio vectorial . Por ejemplo, si G es, El grupo Lie de bienes n -by- n matrices invertibles , entonces la representación adjunta es el homomorfismo grupo que envía una invertible n -by- n matriz a un endomorfismo del espacio vectorial de todas las transformaciones lineales de definido por: .
Para cualquier grupo de Lie, esta representación natural se obtiene linealizando (es decir, tomando el diferencial de) la acción de G sobre sí mismo por conjugación . La representación adjunta se puede definir para grupos algebraicos lineales sobre campos arbitrarios .
Definición
Sea G un grupo de Lie , y sea
ser el mapeo g ↦ Ψ g , con Aut ( G ) el grupo de automorfismo de G y Ψ g : G → G dado por el automorfismo interno (conjugación)
Este Ψ es un homomorfismo de grupo de Lie .
Para cada g en G , defina Ad g como la derivada de Ψ g en el origen:
donde d es el diferencial yes el espacio tangente en el origen e ( siendo e el elemento de identidad del grupo G ). Desdees un automorfismo de grupo de Lie, Ad g es un automorfismo de álgebra de Lie; es decir, una transformación lineal invertible dea sí mismo que conserva el corchete de Lie . Además, dado que es un homomorfismo de grupo, también es un homomorfismo de grupo. [1] Por lo tanto, el mapa
es una representación de grupo llamado la representación adjunta de G .
Si G es un subgrupo de Lie inmerso del grupo lineal general (llamado grupo de Lie inmersivamente lineal), luego el álgebra de Lie consta de matrices y el mapa exponencial es la matriz exponencialpara matrices X con normas de operador pequeño. Por tanto, para g en G y una pequeña X en, tomando la derivada de en t = 0, se obtiene:
donde a la derecha tenemos los productos de matrices. Sies un subgrupo cerrado (es decir, G es un grupo de Lie de matriz), entonces esta fórmula es válida para todo g en G y todo X en.
Sucintamente, una representación adjunta es una representación isotropía asociada a la acción de la conjugación de G alrededor del elemento identidad de G .
Derivado de anuncio
Siempre se puede pasar de una representación de un grupo de Lie G a una representación de su álgebra de Lie tomando la derivada en la identidad.
Tomando la derivada del mapa adjunto
en el elemento de identidad da la representación adjunta del álgebra de Liede G :
where is the Lie algebra of which may be identified with the derivation algebra of . One can show that
for all , where the right hand side is given (induced) by the Lie bracket of vector fields. Indeed,[2] recall that, viewing as the Lie algebra of left-invariant vector fields on G, the bracket on is given as:[3] for left-invariant vector fields X, Y,
where denotes the flow generated by X. As it turns out, , roughly because both sides satisfy the same ODE defining the flow. That is, where denotes the right multiplication by . On the other hand, since , by chain rule,
as Y is left-invariant. Hence,
- ,
which is what was needed to show.
Thus, coincides with the same one defined in § Adjoint representation of a Lie algebra below. Ad and ad are related through the exponential map: Specifically, Adexp(x) = exp(adx) for all x in the Lie algebra.[4] It is a consequence of the general result relating Lie group and Lie algebra homomorphisms via the exponential map.[5]
If G is an immersely linear Lie group, then the above computation simplifies: indeed, as noted early, and thus with ,
- .
Taking the derivative of this at , we have:
- .
The general case can also be deduced from the linear case: indeed, let 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 x in the algebra generates a vector field X in the group G. Similarly, the adjoint map adxy = [x,y] of vectors in is homomorphic[clarification needed] to the Lie derivative LXY = [X,Y] of vector fields on the group G considered as a manifold.
Further see the derivative of the exponential map.
Representación adjunta de un álgebra de Lie
Let be a Lie algebra over some field. Given an element x of a Lie algebra , one defines the adjoint action of x on as the map
for all y in . It is called the adjoint endomorphism or adjoint action. ( is also often denoted as .) Since a bracket is bilinear, this determines the linear mapping
given by x ↦ adx. Within End, the bracket is, by definition, given by the commutator of the two operators:
where denotes composition of linear maps. Using the above definition of the bracket, the Jacobi identity
takes the form
where x, y, and z are arbitrary elements of .
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 and is called the adjoint representation of the algebra .
If is finite-dimensional, then End is isomorphic to , the Lie algebra of the general linear group of the vector space and if a basis for it is chosen, the composition corresponds to matrix multiplication.
In a more module-theoretic language, the construction says that is a module over itself.
The kernel of ad is the center of (that's just rephrasing the definition). On the other hand, for each element z in , the linear mapping obeys the Leibniz' law:
for all x and y in the algebra (the restatement of the Jacobi identity). That is to say, adz is a derivation and the image of under ad is a subalgebra of Der, the space of all derivations of .
When is the Lie algebra of a Lie group G, ad is the differential of Ad at the identity element of G (see #Derivative of Ad above).
There is the following formula similar to the Leibniz formula: for scalars and Lie algebra elements ,
- .
Constantes de estructura
The explicit matrix elements of the adjoint representation are given by the structure constants of the algebra. That is, let {ei} be a set of basis vectors for the algebra, with
Then the matrix elements for adei are given by
Thus, for example, the adjoint representation of su(2) is the defining rep of so(3).
Ejemplos de
- 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 (i.e. a closed subgroup of GL(n,ℂ)), then its Lie algebra is an algebra of n×n matrices with the commutator for a Lie bracket (i.e. a subalgebra of ). In this case, the adjoint map is given by Adg(x) = gxg−1.
- If G is SL(2, R) (real 2×2 matrices with determinant 1), the Lie algebra of G consists of real 2×2 matrices with 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 forms.
Propiedades
The following table summarizes the properties of the various maps mentioned in the definition
Lie group homomorphism: | Lie group automorphism: |
Lie group homomorphism: | Lie algebra automorphism:
|
Lie algebra homomorphism:
| Lie algebra derivation:
|
The image of G under the adjoint representation is denoted by Ad(G). If G is connected, the kernel of the adjoint representation coincides with the kernel of Ψ which is just the center 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 of the identity component G0 of G. By the first isomorphism theorem we have
Given a finite-dimensional real Lie algebra , by Lie's third theorem, there is a connected Lie group whose Lie algebra is the image of the adjoint representation of (i.e., .) It is called the adjoint group of .
Now, if is the Lie algebra of a connected Lie group G, then is the image of the adjoint representation of G: .
Raíces de un grupo de Lie semisimple
If G is semisimple, the non-zero weights of the adjoint representation form a root system.[6] (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(t1, ..., tn) as our maximal torus T. Conjugation by an element of T sends
Thus, T acts trivially on the diagonal part of the Lie algebra of G and with eigenvectors titj−1 on the various off-diagonal entries. The roots of G are the weights diag(t1, ..., tn) → titj−1. This accounts for the standard description of the root system of G = SLn(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:
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
with . The Lie algebra of the maximal torus is the Cartan subalgebra consisting of the matrices
If we conjugate an element of SL(2, R) by an element of the maximal torus we obtain
The matrices
are then 'eigenvectors' of the conjugation operation with eigenvalues . The function Λ which gives 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).
Variantes y análogos
The adjoint representation can also be defined for algebraic groups over any field.[clarification needed]
The co-adjoint representation is the contragredient representation of the adjoint representation. Alexandre Kirillov observed that the orbit of any vector in a co-adjoint representation is a symplectic manifold. According to the philosophy in representation theory 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 groups.
Notas
- ^ Indeed, by chain rule,
- ^ Kobayashi & Nomizu 1996, page 41
- ^ Kobayashi & Nomizu 1996, Proposition 1.9.
- ^ Hall 2015 Proposition 3.35
- ^ Hall 2015 Theorem 3.28
- ^ Hall 2015 Section 7.3
Referencias
- Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
- Kobayashi, Shoshichi; Nomizu, Katsumi (1996). Foundations of Differential Geometry, Vol. 1 (New ed.). Wiley-Interscience. ISBN 978-0-471-15733-5.
- Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 978-3319134666.