Lie algebra extension explained

In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension is an enlargement of a given Lie algebra by another Lie algebra . Extensions arise in several ways. There is the trivial extension obtained by taking a direct sum of two Lie algebras. Other types are the split extension and the central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations. Such a Lie algebra will contain central charges.

Starting with a polynomial loop algebra over finite-dimensional simple Lie algebra and performing two extensions, a central extension and an extension by a derivation, one obtains a Lie algebra which is isomorphic with an untwisted affine Kac–Moody algebra. Using the centrally extended loop algebra one may construct a current algebra in two spacetime dimensions. The Virasoro algebra is the universal central extension of the Witt algebra.

Central extensions are needed in physics, because the symmetry group of a quantized system usually is a central extension of the classical symmetry group, and in the same way the corresponding symmetry Lie algebra of the quantum system is, in general, a central extension of the classical symmetry algebra. Kac–Moody algebras have been conjectured to be symmetry groups of a unified superstring theory.^[1] The centrally extended Lie algebras play a dominant role in quantum field theory, particularly in conformal field theory, string theory and in M-theory.

A large portion towards the end is devoted to background material for applications of Lie algebra extensions, both in mathematics and in physics, in areas where they are actually useful. A parenthetical link, (background material), is provided where it might be beneficial.

History

Due to the Lie correspondence, the theory, and consequently the history of Lie algebra extensions, is tightly linked to the theory and history of group extensions. A systematic study of group extensions was performed by the Austrian mathematician Otto Schreier in 1923 in his PhD thesis and later published.^[2] The problem posed for his thesis by Otto Hölder was "given two groups and, find all groups having a normal subgroup isomorphic to such that the factor group is isomorphic to ".

Lie algebra extensions are most interesting and useful for infinite-dimensional Lie algebras. In 1967, Victor Kac and Robert Moody independently generalized the notion of classical Lie algebras, resulting in a new theory of infinite-dimensional Lie algebras, now called Kac–Moody algebras. They generalize the finite-dimensional simple Lie algebras and can often concretely be constructed as extensions.

Notation and proofs

Notational abuse to be found below includes for the exponential map given an argument, writing for the element in a direct product (is the identity in), and analogously for Lie algebra direct sums (where also and are used interchangeably). Likewise for semidirect products and semidirect sums. Canonical injections (both for groups and Lie algebras) are used for implicit identifications. Furthermore, if,, ..., are groups, then the default names for elements of,, ..., are,, ..., and their Lie algebras are,, ... . The default names for elements of,, ..., are,, ... (just like for the groups!), partly to save scarce alphabetical resources but mostly to have a uniform notation.

Lie algebras that are ingredients in an extension will, without comment, be taken to be over the same field.

The summation convention applies, including sometimes when the indices involved are both upstairs or both downstairs.

Caveat: Not all proofs and proof outlines below have universal validity. The main reason is that the Lie algebras are often infinite-dimensional, and then there may or may not be a Lie group corresponding to the Lie algebra. Moreover, even if such a group exists, it may not have the "usual" properties, e.g. the exponential map might not exist, and if it does, it might not have all the "usual" properties. In such cases, it is questionable whether the group should be endowed with the "Lie" qualifier. The literature is not uniform. For the explicit examples, the relevant structures are supposedly in place.

Definition

Lie algebra extensions are formalized in terms of short exact sequences. A short exact sequence is an exact sequence of length three,such that is a monomorphism, is an epimorphism, and . From these properties of exact sequences, it follows that (the image of)

ak h is an ideal in ak e . Moreover, ak g \cong ak e/\operatorname{Im} i = ak e/ \operatorname{Ker} s, but it is not necessarily the case that ak g is isomorphic to a subalgebra of ak e . This construction mirrors the analogous constructions in the closely related concept of group extensions.

If the situation in prevails, non-trivially and for Lie algebras over the same field, then one says that

ak e is an extension of ak g by ak h .

Properties

The defining property may be reformulated. The Lie algebra

ak e is an extension of ak g by ak h ifis exact. Here the zeros on the ends represent the zero Lie algebra (containing only the zero vector) and the maps are the obvious ones; \iota maps to and \sigma maps all elements of ak g to . With this definition, it follows automatically that is a monomorphism and is an epimorphism.

An extension of

ak g by ak h is not necessarily unique. Let ak e, ak e' denote two extensions and let the primes below have the obvious interpretation. Then, if there exists a Lie algebra isomorphism f \colon ak e → ak e' such thatf \circ i = i', s' \circ f = s, then the extensions ak e and ak e' are said to be equivalent extensions. Equivalence of extensions is an equivalence relation.

Extension types

Trivial

A Lie algebra extension

ak h \overset i \hookrightarrow ak t \overset s \twoheadrightarrow ak g, is trivial if there is a subspace such that and is an ideal in .

Split

A Lie algebra extension

ak h \overset i \hookrightarrow ak s \overset s \twoheadrightarrow ak g, is split if there is a subspace such that as a vector space and is a subalgebra in .

An ideal is a subalgebra, but a subalgebra is not necessarily an ideal. A trivial extension is thus a split extension.

Central

Central extensions of a Lie algebra by an abelian Lie algebra can be obtained with the help of a so-called (nontrivial) 2-cocycle (background) on . Non-trivial 2-cocycles occur in the context of projective representations (background) of Lie groups. This is alluded to further down.

A Lie algebra extension

ak h \overset i \hookrightarrow ak e \overset s \twoheadrightarrow ak g, is a central extension if is contained in the center of .

Properties

• Since the center commutes with everything, in this case is abelian.
• Given a central extension of, one may construct a 2-cocycle on . Suppose is a central extension of by . Let be a linear map from to with the property that, i.e. is a section of . Use this section to define by

\epsilon(G _{1,} G _{2)} = l([G _{1,} G _{2])} - [l(G _{1),} l(G _{2)],} G _{1,} G _{2} \in ak g. The map satisfies\epsilon(G _{1,} [G _{2,} G _{3])} + \epsilon(G _{2,} [G _{3,} G _{1])} + \epsilon(G _{3,} [G _{1,} G _{2])} = 0 \in ak e. To see this, use the definition of on the left hand side, then use the linearity of . Use Jacobi identity on to get rid of half of the six terms. Use the definition of again on terms sitting inside three Lie brackets, bilinearity of Lie brackets, and the Jacobi identity on, and then finally use on the three remaining terms that and that so that brackets to zero with everything.It then follows that satisfies the corresponding relation, and if in addition is one-dimensional, then is a 2-cocycle on (via a trivial correspondence of with the underlying field).

A central extension

0 \overset \iota \hookrightarrow ak h \overset i \hookrightarrow ak e \overset s \twoheadrightarrow ak g \overset \sigma \twoheadrightarrow 0 is universal if for every other central extension0 \overset \iota \hookrightarrow ak h' \overset {i'} \hookrightarrow ak e' \overset {s'} \twoheadrightarrow ak g \overset \sigma \twoheadrightarrow 0 there exist unique homomorphisms \Phi : ak e \to ak e' and \Psi : ak h \to ak h' such that the diagramcommutes, i.e. . By universality, it is easy to conclude that such universal central extensions are unique up to isomorphism.

Construction

By direct sum

Let

ak g , ak h be Lie algebras over the same field F . Defineak e = ak h x ak g, and define addition pointwise on ak e . Scalar multiplication is defined by\alpha(H, G) = (\alpha H, \alpha G), \alpha \in F, H \in ak h, G \in ak g. With these definitions, ak h x ak g \equiv ak h ⊕ ak g is a vector space over F . With the Lie bracket:ak e is a Lie algebra. Define furtheri: ak h \hookrightarrow ak e; H \mapsto (H, 0 ), s: ak e \twoheadrightarrow ak g; (H, G) \mapsto G. It is clear that holds as an exact sequence. This extension of ak g by ak h is called a trivial extension. It is, of course, nothing else than the Lie algebra direct sum. By symmetry of definitions, ak e is an extension of ak h by ak g as well, but ak h ⊕ ak g ≠ ak g ⊕ ak h . It is clear from that the subalgebra 0 ⊕ ak g is an ideal (Lie algebra). This property of the direct sum of Lie algebras is promoted to the definition of a trivial extension.

By semidirect sum

Inspired by the construction of a semidirect product (background) of groups using a homomorphism, one can make the corresponding construct for Lie algebras.

If is a Lie algebra homomorphism, then define a Lie bracket on

ak e = ak h ⊕ ak g byWith this Lie bracket, the Lie algebra so obtained is denoted and is called the semidirect sum of and .

By inspection of one sees that is a subalgebra of and is an ideal in . Define by and by . It is clear that . Thus is a Lie algebra extension of by .

As with the trivial extension, this property generalizes to the definition of a split extension.

(a _{2,} Λ _{2)(a} _{1,} Λ _{1)} = (a _{2} + Λ _{2a} _{1,} Λ _{2Λ} _{1),} a _{1,} a _{2} \in T \subset P, Λ _{1,} Λ _{2} \in O(3,1) \subset P, (where and are identified with their images in). From it follows immediately that, in the Poincaré group, . Thus every Lorentz transformation corresponds to an automorphism of with inverse and is clearly a homomorphism. Now define\overline P = T ⊗ _{S} O(3, 1 ), endowed with multiplication given by . Unwinding the definitions one finds that the multiplication is the same as the multiplication one started with and it follows that . From follows that and then from it follows that .

By derivation

Let be a derivation (background) of and denote by the one-dimensional Lie algebra spanned by . Define the Lie bracket on by^[3]

[G _{1} + H _{1,} G _{2} + H _{2]} = [ λ \delta + H _{1,} \mu\delta + H _{2]} = [H _{1,} H _{2]} + λ \delta(H _{2)} - \mu \delta(H _{1).}

It is obvious from the definition of the bracket that is and ideal in in and that is a subalgebra of . Furthermore, is complementary to in . Let be given by and by . It is clear that . Thus is a split extension of by . Such an extension is called extension by a derivation.

If is defined by, then is a Lie algebra homomorphism into . Hence this construction is a special case of a semidirect sum, for when starting from and using the construction in the preceding section, the same Lie brackets result.

By 2-cocycle

If is a 2-cocycle (background) on a Lie algebra and is any one-dimensional vector space, let (vector space direct sum) and define a Lie bracket on by

[\mu H + G _{1,} \nu H + G _{2]} = [G _{1,} G _{2]} + \varepsilon(G _{1,} G _{2)H,} \mu, \nu \in F.

Here is an arbitrary but fixed element of . Antisymmetry follows from antisymmetry of the Lie bracket on and antisymmetry of the 2-cocycle. The Jacobi identity follows from the corresponding properties of and of . Thus is a Lie algebra. Put and it follows that . Also, it follows with and that

Notes and References

1. The Beacon of Kac–Moody Symmetry for Physics. (free access)
2. [Otto Schreier]
3. To show that the Jacobi identity holds, one writes everything out, uses the fact that the underlying Lie algebras have a Lie product satisfying the Jacobi identity, and that .