Lie algebra representation

Lie groups

Classical groups General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n)
Simple Lie groups List of simple Lie groups Infinite simple Lie groups: A_n, B_n, C_n, D_n, Exceptional simple Lie groups: G₂ F₄ E₆ E₇ E₈
Other Lie groups Circle group Lorentz group Poincaré group Conformal group Diffeomorphism group Loop group
Lie algebras Exponential map Adjoint representation of a Lie group Adjoint representation of a Lie algebra Killing form Lie point symmetry
Structure of semi-simple Lie groups Dynkin diagrams Cartan subalgebra Root system Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Representation of a Lie group Representation of a Lie algebra
Lie groups in Physics Particle physics and representation theory Representation theory of the Lorentz group Representation theory of the Poincaré group Representation theory of the Galilean group

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 given by the commutator.

The notion is closely related to that of a representation of a Lie group. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the universal cover of a Lie group are the integrated form of the representations of its Lie algebra.

1 Formal definition
2 Infinitesimal Lie group representations
3 Properties
4 Classification
- 4.1 Finite-dimensional representations of semisimple Lie algebras
5 Representation on an algebra
6 See also
7 References

Formal definition

A representation of a Lie algebra $\mathfrak g$ is a Lie algebra homomorphism

$\rho\colon \mathfrak g \to \mathfrak{gl}(V)$

from $\mathfrak g$ to the Lie algebra of endomorphisms on a vector space $V$ (with the commutator as the Lie bracket), sending an element x of $\mathfrak g$ to an element ρ_x of $\mathfrak{gl}(V)$ .

Explicitly, this means that

$\rho_{[x,y]} = [\rho_x,\rho_y] = \rho_x\rho_y - \rho_y\rho_x\,$

for all $x,y$ in $\mathfrak g$ . The vector space $V$ , together with the representation $\rho$ , is called a $\mathfrak g$ -module. (Many authors abuse terminology and refer to $V$ itself as the representation).

One can equivalently define a $\mathfrak g$ -module as a vector space $V$ together with a bilinear map $\mathfrak g \times V\to V$ such that

$[x,y]\cdot v = x\cdot(y\cdot v) - y\cdot(x\cdot v)$

for all $x,y$ in $\mathfrak g$ and $v$ in $V$ . This is related to the previous definition by setting $x\cdot v = \rho_x(v).$

Infinitesimal Lie group representations

If $\phi:G\to H$ is a homomorphism of Lie groups, and $\mathfrak g$ and $\mathfrak h$ are the Lie algebras of $G$ and $H$ respectively, then the induced map $\phi_*: \mathfrak g \to \mathfrak h$ on tangent spaces is a Lie algebra homomorphism. In particular, a representation of Lie groups

$\phi: G\to \mathrm{GL}(V)\,$

determines a Lie algebra homomorphism

$\phi_*: \mathfrak g \to \mathfrak{gl}(V)$

from $\mathfrak g$ to the Lie algebra of the general linear group $\mathrm{GL}(V)$ , i.e. the endomorphism algebra of $V$ .

A partial converse to this statement says that every representation of a finite-dimensional (real or complex) Lie algebra lifts to a unique representation of the associated simply connected Lie group, so that representations of simply-connected Lie groups are in one-to-one correspondence with representations of their Lie algebras.

Properties

Representations of a Lie algebra are in one-to-one correspondence with algebra representations of the associated universal enveloping algebra. This follows from the universal property of that construction.

If the Lie algebra is semisimple, then all reducible representations are decomposable. Otherwise, that's not true in general.

If we have two representations, with V₁ and V₂ as their underlying vector spaces and ·[·]₁ and ·[·]₂ as the representations, then the product of both representations would have $V_1\otimes V_2$ as the underlying vector space and

$x[v_1\otimes v_2]=x[v_1]\otimes v_2%2Bv_1\otimes x[v_2] .$

If L is a real Lie algebra and $\rho:L\times V\rightarrow V$ is a complex representation of it, we can construct another representation of L called its dual representation as follows.

Let V^∗ be the dual vector space of V. In other words, V^∗ is the set of all linear maps from V to C with addition defined over it in the usual linear way, but scalar multiplication defined over it such that $(z\omega)[X]=\bar{z}\omega[X]$ for any z in C, ω in V^∗ and X in V. This is usually rewritten as a contraction with a sesquilinear form 〈·,·〉. i.e. 〈ω,X〉 is defined to be ω[X].

We define $\bar{\rho}$ as follows:

〈 $\bar{\rho}$ (A)[ω],X〉 + 〈ω, $\rho$ A[X]〉 = 0,

for any A in L, ω in V^∗ and X in V. This defines $\bar{\rho}$ uniquely.

Classification

Finite-dimensional representations of semisimple Lie algebras

For more details on this topic, see Weight (representation theory).

Similarly to how semisimple Lie algebras can be classified, the finite-dimensional representations of semisimple Lie algebras can be classified. This is a classical theory, widely regarded as beautiful, and a standard reference is (Fulton & Harris 1992).

Briefly, finite-dimensional representations of a semisimple Lie algebra are completely reducible, so it suffices to classify irreducible (simple) representations. Semisimple Lie algebras are classified in terms of the weights of the adjoint representation, the so-called root system; in a similar manner all finite-dimensional irreducible representations can be understood in terms of weights; see weight (representation theory) for details.

Representation on an algebra

If we have a Lie superalgebra L, then a representation of L on an algebra is a (not necessarily associative) Z₂ graded algebra A which is a representation of L as a Z₂ graded vector space and in addition, the elements of L acts as derivations/antiderivations.

More specifically, if H is a pure element of L and x and y are pure elements of A,

H[xy] = (H[x])y + (−1)^xHx(H[y])

Also, if A is unital, then

H[1] = 0

Now, for the case of a representation of a Lie algebra, we simply drop all the gradings and the (−1) to the some power factors.

A Lie (super)algebra is an algebra and it has an adjoint representation of itself. This is a representation on an algebra: the (anti)derivation property is the super Jacobi identity.

If a vector space is both an associative algebra and a Lie algebra and the adjoint representation of the Lie algebra on itself is a representation on an algebra (i.e., acts by derivations on the associative algebra structure), then it is a Poisson algebra. The analogous observation for Lie superalgebras gives the notion of a Poisson superalgebra.

References

Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics, 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN 978-0-387-97527-6
J.Humphreys, Introduction to Lie algebras and representation theory, Birkhäuser, 2000