Lie algebra

A Lie algebra is an easy example of an algebraic structure that is not associative. Lie algebras describe infinitesimal symmetries or transformations. In short a Lie algebra is a vector space together with a skew-symmetric bilinear operation denoted as bracket that is subject to the Jacobi identity [X,[Y,Z]] +[Y,[Z,X]] +[Z,[X,Y]] = 0 where X, Y, and Z run over all elements of the Lie algebra.

In particular to every Lie group there is associated a Lie algebra that covers the infinitesimal structure of that group.

Examples

The simplest example is the three dimensional space R³ together with the vector product. In the standard base i,j,k this is defined as i×j=k, j×k=i, k×i=j, and extended skew-symmetric and linear. This Lie algebra is also denoted so(3) or su(2) as it is the Lie algebra associated to either of the Lie groups SO(3) or SU(2).

Other examples are:

• kⁿ where k is a field with 0-bracket [X,Y]=0 called abelian Lie algebras.
• Mat_n(k) the n×n matrices over a field k together with the commutator of matrix multiplication, i.e. $[A,B]=A\circ B-B\circ A$. A straightforward computation shows that the Jacobi identity holds.
• subalgebras, such as: so(n), also denoted o(n), the real n×n matrices that are skew-symmetric,
• sl_n(k) the n×n matrices that are traceless, i.e. the sum of the diagonal elements is 0.
• u(n) the complex n×n matrices that are skew-hermitean. Note that these form only a real Lie algebra, as multiplication with a general complex number does not preserve skew-hermiticity.
• su(n) the skew-hermitean matrices with trace 0.
• sp_n(k) the 2n×2n matrices that preserve the standard symplectic form ω – that is the 2n×2n matrix J that has 2×2 block structure and Id (the identity matrix) in the lower left block, -Id in the upper right block, and 0 in the rest. A 2n×2n matrix A is an infinitesimal symmetry of J if

$A^{T}\circ J+J\circ A=0.$

• an analog construction can be done with arbitrary linear geometric maps, e.g. in the case so(n) the structure is the Euclidian metric g with matrix δ_ab the Kronecker delta, i.e. 1 for a=b and 0 otherwise.

The theorem of Ado states that every finite dimensional Lie algebra over an algebraically closed field has a faithful finite dimensional representation. Therefore every (finite dimensional) Lie algebra is isomorphic to a Lie subalgebra of a matrix Lie algebra (the End_n(k) in the above list; after algebraic closure of the field).

Lie algebra associated to a Lie group

To a Lie group G over the real or complex numbers we can associate a Lie algebra in the following way. The vector space is the tangent space at the identity of G. This vector space is also canonically isomorphic to the left-invariant vector fields on G. The commutator of two left-invariant vector fields is again left-invariant and moreover k-linear and skew-symmetric. This endows the vector space with a bracket.

The above examples are all Lie algebras associated to Lie groups, namely o(n) associated to O(n) the Lie groups of orthogonal transformations. so(n) to SO(n) the orientation preserving orthogonal maps. u(n) to U(n) the unitary transformations, su(n) to SU(n) the volume preserving unitary transformations. sl_n(k) to SL_n(k) the volume preserving linear transformations. gl_n(k) to GL_n(k) the linear transformations of kⁿ.

The three theorems of Lie have the following content: Lie's first theorem states that (finite dimensional) groups of symmetries are Lie groups.

Lie's second theorem states that given a connected Lie group and its associated Lie algebra, then Lie subgroups correspond bijectively with Lie subalgebras.

Lie's third theorem states that every complex Lie algebra integrates uniquely to a connected simply connected Lie group. In the light of Ado's theorem, this can be proven starting from a faithful matrix representation, using the matrix exponential map to construct an associated matrix Lie group, and then constructing its universal cover which is the simply connected Lie integrating the Lie algebra.

Homomorphisms, subalgebras, and ideals

Analogous to associative algebras, we define a Lie algebra homomorphism between Lie algebras (g,[.,.]) and (h,[.,.]') as linear map φ:g→h such that for all elements X,Y ∈ g

φ[X,Y] = [φ(X),φ(Y)]'.

Given a Lie algebra it is easier to find further Lie algebras associated to it. Namely, a vector subspace h ⊆ g is a Lie subalgebra iff the bracket operation is closed on h. Every Lie algebra g has two trivial Lie subalgebras, 0 and g. The image of a homomorphism im φ = {φ(X): X ∈ g} is a Lie subalgebra.

A more special subalgebra is an ideal. Let g be a Lie algebra and I a vector subspace. I is an ideal if

[I,g] ⊆ I.

0 and g are also trivial ideals of g.

Analogous to rings, an ideal is good for the quotient construction. Let g/I be the quotient vector space of a Lie algebra g by an ideal I. It is endowed with a Lie algebra structure via representatives, i.e.

[X+I,Y+I] = [X,Y]+I

which is representation independent since I is an ideal. In particular the quotient map π:g→g/I is a surjective Lie algebra homomorphism.

Conversely the kernel kerφ = {X∈g : φ(X)=0} of every Lie algebra homomorphism φ:g→h is an ideal.

Classification

Lie algebras can be classified by the following properties:

Abelian, nilpotent, and solvable

Abelian means that the Lie bracket vanishes, i.e. [X,Y]=0. The only examples are the kⁿ.

Consider the definition of the lower central series of a Lie algebra g

g > [g,g] > [[g,g],g] > [[[g,g],g],g] > …

A Lie algebra is called nilpotent if the lower central series finally becomes 0. A series of length n means that arbitrary iterated commutators of length greater than n always vanish.

Next, consider the definition of the derived series of a Lie algebra g

D⁰g=g, Dⁿ⁺¹g=[Dⁿg,Dⁿg].

A Lie algebra is called solvable if its derived series finally becomes 0.

The complete classification of solvable Lie algebras is still an open problem and beyond the means of finitely many invariants. It is thus a problem that is not Turing computable.

Simple and Semisimple

The opposite of a solvable Lie algebra is a simple Lie algebra which means that it has no proper ideals. The classification of simple complex Lie algebras (see also the page about Lie algebra/root systems) states that these fall into four infinite series and five exceptional Lie algebras: A_n=sl_n+1(C), B_n=so(2n+1)⊗C, C_n=sp_n(C), D_n=so(2n)⊗C, and the exeptions e₈, e₇, e₆, f₄, g₂.

A Lie algebra is semi-simple if every element can be written as the commutator of two elements, i.e. for the Lie algebra g, [g,g]=g. Simple Lie algebras are semi-simple. Moreover every semi-simple Lie algebra decomposes into the direct sum of its minimal ideals which are simple Lie algebras.

Killing form and quadratic Lie algebras

Given an arbitrary Lie algebra g we obtain a symmetric bilinear form g on it via

$g(X,Y)=\mathrm {tr} (\mathrm {ad} _{X}\circ \mathrm {ad} _{Y})$

where X, Y ∈ g and ad is the adjoint reprsentation. This is called the Carthan–Killing form of the Lie algebra. The two authors proved that it is non-degenerate iff the Lie algebra is semi-simple. The Carthan–Killing form has another property, namely for X, Y, Z ∈ g

$g([X,Y],Z)+g(Y,[X,Z])=0$

which is called ad-invariance of the metric g.

Conversely, a Lie algebra g is called quadratic if it permits an ad-invariant symmetric non-degenerate bilinear form.

Beside semi-simple Lie algebras whose ad-invariant metric is unique up to a non-zero factor, also abelian Lie algebras permit a quadratic structure.

While Lie algebroids are the vector bundle generalization of Lie algebras, they generally do permit only trivial quadratic structures, i.e. either the anchor map is 0 or the bilinearform 0, there is a notion of a quadratic algebroid called Courant algebroid.

Remarks about infinite dimensional Lie algebras

In the above definition we did not restrict to finite dimensional vector spaces even though this is usually implied when talking about Lie algebras. In infinite dimensional Lie algebras there is the obvious question of continuity of the Lie bracket and one requires thus at least a topological vector space and often demands continuity of the bracket (and operations of the vector space).

The easiest version of infinite dimensional Lie algebras are Lie algebroids, and a particular example of that is the tangent bundle of a smooth manifold.

Lie superalgebras

In the physics of quantum field theory it was first observed that beside Lie algebras also odd operators play an interesting role. The simplest example of a graded object is a Lie superalgebra. This is a supervector space g⁰⊕g¹ together with a bracket [.,.] of degree d, i.e. linear maps [.,.]:gⁱ×g^j→g^i+j+d (where the addition is modulo 2) that are graded skew symmetric, i.e.

$[Y,X]=-(-1)^{(|X|+d)(|Y|+d)}[X,Y]$

where X∈g^|X| and Y∈g^|Y| and fulfill the graded Jacobi identity, i.e.

$[X,[Y,Z]]=[[X,Y],Z]+(-1)^{(|X|+d)(|Y|+d)}[Y,[X,Z]]$

where X and Y as before and Z∈g^•.

Examples

Examples are beside ordinary Lie algebras where g¹=0, d=0 and also the following:

Let (A,·) be an associative Z/(2) graded algebra over a field k and use the same vector space g=A together with the graded commutator

$[X,Y]=X\cdot Y-(-1)^{|X|\,|Y|}Y\cdot X$

where again X∈g^|X| and Y∈g^|Y|.

As a more particular example of a Z/(2) graded algebra consider the endomorphisms (linear maps) of V⁰⊕V¹ that come in 2×2 block form. Declare the 00-block and 11-block to be even, and the 01- and 10-block to be odd, this is a consistent Z/(2) grading and permits thus the above construction.

References

1. V.S. Varadarajan: Lie groups, Lie algebras, and their Representations, Springer (1984), ISBN 0-387-90969-9.