In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension e is an enlargement of a given Lie algebra g by another Lie algebra h. 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. 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. The problem posed for his thesis by Otto Hölder was "given two groups G and H, find all groups E having a normal subgroup N isomorphic to G such that the factor group E/N is isomorphic to H". 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 eX for the exponential map exp given an argument, writing g for the element (g, eH) in a direct product G × H (eH is the identity in H), and analogously for Lie algebra direct sums (where also g + h and (g, h) 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 G, H, ..., are groups, then the default names for elements of G, H, ..., are g, h, ..., and their Lie algebras are g, h, ... . The default names for elements of g, h, ..., are G, H, ... (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 i is a monomorphism, s is an epimorphism, and ker s = im i. From these properties of exact sequences, it follows that (the image of) h {\displaystyle {\mathfrak {h}}} is an ideal in e {\displaystyle {\mathfrak {e}}} . Moreover,
g ≅ e / Im i = e / Ker s , {\displaystyle {\mathfrak {g}}\cong {\mathfrak {e}}/\operatorname {Im} i={\mathfrak {e}}/\operatorname {Ker} s,}
but it is not necessarily the case that g {\displaystyle {\mathfrak {g}}} is isomorphic to a subalgebra of e {\displaystyle {\mathfrak {e}}} . This construction mirrors the analogous constructions in the closely related concept of group extensions. If the situation in (1) prevails, non-trivially and for Lie algebras over the same field, then one says that e {\displaystyle {\mathfrak {e}}} is an extension of g {\displaystyle {\mathfrak {g}}} by h {\displaystyle {\mathfrak {h}}} .
Properties The defining property may be reformulated. The Lie algebra e {\displaystyle {\mathfrak {e}}} is an extension of g {\displaystyle {\mathfrak {g}}} by h {\displaystyle {\mathfrak {h}}} if
is exact. Here the zeros on the ends represent the zero Lie algebra (containing only the zero vector 0) and the maps are the obvious ones; ι {\displaystyle \iota } maps 0 to 0 and σ {\displaystyle \sigma } maps all elements of g {\displaystyle {\mathfrak {g}}} to 0. With this definition, it follows automatically that i is a monomorphism and s is an epimorphism. An extension of g {\displaystyle {\mathfrak {g}}} by h {\displaystyle {\mathfrak {h}}} is not necessarily unique. Let e , e ′ {\displaystyle {\mathfrak {e}},{\mathfrak {e}}'} denote two extensions and let the primes below have the obvious interpretation. Then, if there exists a Lie algebra isomorphism f : e → e ′ {\displaystyle f\colon {\mathfrak {e}}\rightarrow {\mathfrak {e}}'} such that
f ∘ i = i ′ , s ′ ∘ f = s , {\displaystyle f\circ i=i',\quad s'\circ f=s,}
then the extensions e {\displaystyle {\mathfrak {e}}} and e ′ {\displaystyle {\mathfrak {e}}'} are said to be equivalent extensions. Equivalence of extensions is an equivalence relation.
Extension types
Trivial A Lie algebra extension
h ↪ i t ↠ s g , {\displaystyle {\mathfrak {h}}\;{\overset {i}{\hookrightarrow }}\;{\mathfrak {t}}\;{\overset {s}{\twoheadrightarrow }}\;{\mathfrak {g}},}
is trivial if there is a subspace i such that t = i ⊕ ker s and i is an ideal in t.
Split A Lie algebra extension
h ↪ i s ↠ s g , {\displaystyle {\mathfrak {h}}\;{\overset {i}{\hookrightarrow }}\;{\mathfrak {s}}\;{\overset {s}{\twoheadrightarrow }}\;{\mathfrak {g}},}
is split if there is a subspace u such that s = u ⊕ ker s as a vector space and u is a subalgebra in s. 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 g by an abelian Lie algebra h can be obtained with the help of a so-called (nontrivial) 2-cocycle (background) on g. 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
h ↪ i e ↠ s g , {\displaystyle {\mathfrak {h}}\;{\overset {i}{\hookrightarrow }}\;{\mathfrak {e}}\;{\overset {s}{\twoheadrightarrow }}\;{\mathfrak {g}},}
is a central extension if ker s is contained in the center Z(e) of e. Properties
Since the center commutes with everything, h ≅ im i = ker s in this case is abelian. Given a central extension e of g, one may construct a 2-cocycle on g. Suppose e is a central extension of g by h. Let l be a linear map from g to e with the property that s ∘ l = Idg, i.e. l is a section of s. Use this section to define ε: g × g → e by
ϵ ( G 1 , G 2 ) = l ( [ G 1 , G 2 ] ) − [ l ( G 1 ) , l ( G 2 ) ] , G 1 , G 2 ∈ g . {\displaystyle \epsilon (G_{1},G_{2})=l([G_{1},G_{2}])-[l(G_{1}),l(G_{2})],\quad G_{1},G_{2}\in {\mathfrak {g}}.}
The map ε satisfies
ϵ ( G 1 , [ G 2 , G 3 ] ) + ϵ ( G 2 , [ G 3 , G 1 ] ) + ϵ ( G 3 , [ G 1 , G 2 ] ) = 0 ∈ e . {\displaystyle \epsilon (G_{1},[G_{2},G_{3}])+\epsilon (G_{2},[G_{3},G_{1}])+\epsilon (G_{3},[G_{1},G_{2}])=0\in {\mathfrak {e}}.}
To see this, use the definition of ε on the left hand side, then use the linearity of l. Use Jacobi identity on g to get rid of half of the six terms. Use the definition of ε again on terms l([Gi,Gj]) sitting inside three Lie brackets, bilinearity of Lie brackets, and the Jacobi identity on e, and then finally use on the three remaining terms that Im ε ⊂ ker s and that ker s ⊂ Z(e) so that ε(Gi, Gj) brackets to zero with everything. It then follows that φ = i−1 ∘ ε satisfies the corresponding relation, and if h in addition is one-dimensional, then φ is a 2-cocycle on g (via a trivial correspondence of h with the underlying field). A central extension
0 ↪ ι h ↪ i e ↠ s g ↠ σ 0 {\displaystyle 0\;{\overset {\iota }{\hookrightarrow }}{\mathfrak {h}}\;{\overset {i}{\hookrightarrow }}\;{\mathfrak {e}}\;{\overset {s}{\twoheadrightarrow }}\;{\mathfrak {g}}\;{\overset {\sigma }{\twoheadrightarrow }}\;0}
is universal if for every other central extension
0 ↪ ι h ′ ↪ i ′ e ′ ↠ s ′ g ↠ σ 0 {\displaystyle 0\;{\overset {\iota }{\hookrightarrow }}{\mathfrak {h}}'\;{\overset {i'}{\hookrightarrow }}\;{\mathfrak {e}}'\;{\overset {s'}{\twoheadrightarrow }}\;{\mathfrak {g}}\;{\overset {\sigma }{\twoheadrightarrow }}\;0}
there exist unique homomorphisms Φ : e → e ′ {\displaystyle \Phi :{\mathfrak {e}}\to {\mathfrak {e}}'} and Ψ : h → h ′ {\displaystyle \Psi :{\mathfrak {h}}\to {\mathfrak {h}}'} such that the diagram
commutes, i.e. i' ∘ Ψ = Φ ∘ i and s' ∘ Φ = s. By universality, it is easy to conclude that such universal central extensions are unique up to isomorphism.
Construction
By direct sum Let g {\displaystyle {\mathfrak {g}}} , h {\displaystyle {\mathfrak {h}}} be Lie algebras over the same field F {\displaystyle F} . Define
e = h × g , {\displaystyle {\mathfrak {e}}={\mathfrak {h}}\times {\mathfrak {g}},}
and define addition pointwise on e {\displaystyle {\mathfrak {e}}} . Scalar multiplication is defined by
α ( H , G ) = ( α H , α G ) , α ∈ F , H ∈ h , G ∈ g . {\displaystyle \alpha (H,G)=(\alpha H,\alpha G),\alpha \in F,H\in {\mathfrak {h}},G\in {\mathfrak {g}}.}
With these definitions, h × g ≡ h ⊕ g {\displaystyle {\mathfrak {h}}\times {\mathfrak {g}}\equiv {\mathfrak {h}}\oplus {\mathfrak {g}}} is a vector space over F {\displaystyle F} . With the Lie bracket:
e {\displaystyle {\mathfrak {e}}} is a Lie algebra. Define further
i : h ↪ e ; H ↦ ( H , 0 ) , s : e ↠ g ; ( H , G ) ↦ G . {\displaystyle i:{\mathfrak {h}}\hookrightarrow {\mathfrak {e}};H\mapsto (H,0),\quad s:{\mathfrak {e}}\twoheadrightarrow {\mathfrak {g}};(H,G)\mapsto G.}
It is clear that (1) holds as an exact sequence. This extension of g {\displaystyle {\mathfrak {g}}} by h {\displaystyle {\mathfrak {h}}} is called a trivial extension. It is, of course, nothing else than the Lie algebra direct sum. By symmetry of definitions, e {\displaystyle {\mathfrak {e}}} is an extension of h {\displaystyle {\mathfrak {h}}} by g {\displaystyle {\mathfrak {g}}} as well, but h ⊕ g ≠ g ⊕ h {\displaystyle {\mathfrak {h}}\oplus {\mathfrak {g}}\neq {\mathfrak {g}}\oplus {\mathfrak {h}}} . It is clear from (3) that the subalgebra 0 ⊕ g {\displaystyle 0\oplus {\mathfrak {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 G → Aut(H), one can make the corresponding construct for Lie algebras. If ψ:g → Der h is a Lie algebra homomorphism, then define a Lie bracket on e = h ⊕ g {\displaystyle {\mathfrak {e}}={\mathfrak {h}}\oplus {\mathfrak {g}}} by
With this Lie bracket, the Lie algebra so obtained is denoted e= h ⊕S g and is called the semidirect sum of h and g. By inspection of (7) one sees that 0 ⊕ g is a subalgebra of e and h ⊕ 0 is an ideal in e. Define i:h → e by H ↦ H ⊕ 0 and s:e → g by H ⊕ G ↦ G, H ∈ h, G ∈ g. It is clear that ker s = im i. Thus e is a Lie algebra extension of g by h. As with the trivial extension, this property generalizes to the definition of a split extension. ExampleLet G be the Lorentz group O(3, 1) and let T denote the translation group in 4 dimensions, isomorphic to ( R 4 , + ) {\displaystyle (\mathbb {R} ^{4},+)} , and consider the multiplication rule of the Poincaré group P
( a 2 , Λ 2 ) ( a 1 , Λ 1 ) = ( a 2 + Λ 2 a 1 , Λ 2 Λ 1 ) , a 1 , a 2 ∈ T ⊂ P , Λ 1 , Λ 2 ∈ O ( 3 , 1 ) ⊂ P , {\displaystyle (a_{2},\Lambda _{2})(a_{1},\Lambda _{1})=(a_{2}+\Lambda _{2}a_{1},\Lambda _{2}\Lambda _{1}),\quad a_{1},a_{2}\in \mathrm {T} \subset \mathrm {P} ,\Lambda _{1},\Lambda _{2}\in \mathrm {O} (3,1)\subset \mathrm {P} ,}
(where T and O(3, 1) are identified with their images in P). From it follows immediately that, in the Poincaré group, (0, Λ)(a, I)(0, Λ−1) = (Λ a, I) ∈ T ⊂ P. Thus every Lorentz transformation Λ corresponds to an automorphism ΦΛ of T with inverse ΦΛ−1 and Φ is clearly a homomorphism. Now define
P ¯ = T ⊗ S O ( 3 , 1 ) , {\displaystyle {\overline {\mathrm {P} }}=\mathrm {T} \otimes _{S}\mathrm {O} (3,1),}
endowed with multiplication given by (4). Unwinding the definitions one finds that the multiplication is the same as the multiplication one started with and it follows that P = P. From (5') follows that ΨΛ = AdΛ and then from (6') it follows that ψλ = adλ. λ ∈ o(3, 1).
By derivation Let δ be a derivation (background) of h and denote by g the one-dimensional Lie algebra spanned by δ. Define the Lie bracket on e = g ⊕ h by
[ G 1 + H 1 , G 2 + H 2 ] = [ λ δ + H 1 , μ δ + H 2 ] = [ H 1 , H 2 ] + λ δ ( H 2 ) − μ δ ( H 1 ) . {\displaystyle [G_{1}+H_{1},G_{2}+H_{2}]=[\lambda \delta +H_{1},\mu \delta +H_{2}]=[H_{1},H_{2}]+\lambda \delta (H_{2})-\mu \delta (H_{1}).}
It is obvious from the definition of the bracket that h is and ideal in e in and that g is a subalgebra of e. Furthermore, g is complementary to h in e. Let i:h → e be given by H ↦ (0, H) and s:e → g by (G, H) ↦ G. It is clear that im i = ker s. Thus e is a split extension of g by h. Such an extension is called extension by a derivation. If ψ: g → der h is defined by ψ(μδ)(H) = μδ(H), then ψ is a Lie algebra homomorphism into der h. Hence this construction is a special case of a semidirect sum, for when starting from ψ and using the construction in the preceding section, the same Lie brackets result.
By 2-cocycle If ε is a 2-cocycle (background) on a Lie algebra g and h is any one-dimensional vector space, let e = h ⊕ g (vector space direct sum) and define a Lie bracket on e by
[ μ H + G 1 , ν H + G 2 ] = [ G 1 , G 2 ] + ε ( G 1 , G 2 ) H , μ , ν ∈ F . {\displaystyle [\mu H+G_{1},\nu H+G_{2}]=[G_{1},G_{2}]+\varepsilon (G_{1},G_{2})H,\quad \mu ,\nu \in F.}
Here H is an arbitrary but fixed element of h. Antisymmetry follows from antisymmetry of the Lie bracket on g and antisymmetry of the 2-cocycle. The Jacobi identity follows from the corresponding properties of g and of ε. Thus e is a Lie algebra. Put G1 = 0 and it follows that μH ∈ Z(e). Also, it follows with i: μH ↦ (μH, 0) and s: (μH, G) ↦ G that Im i = ker s = {(μH, 0):μ ∈ F} ⊂ Z(e). Hence e is a central extension of g by h. It is called extension by a 2-cocycle.
Theorems Below follows some results regarding central extensions and 2-cocycles. Theorem Let φ1 and φ2 be cohomologous 2-cocycles on a Lie algebra g and let e1 and e2 be respectively the central extensions constructed with these 2-cocycles. Then the central extensions e1 and e2 are equivalent extensions. Proof By definition, φ2 = φ1 + δf. Define
ψ : G + μ c ∈ e 1 ↦ G + μ c + f ( G ) c ∈ e 2 . {\displaystyle \psi :G+\mu c\in {\mathfrak {e}}_{1}\mapsto G+\mu c+f(G)c\in {\mathfrak {e}}_{2}.}
It follows from the definitions that ψ is a Lie algebra isomorphism and (2) holds. Corollary A cohomology class [Φ] ∈ H2(g, F) defines a central extension of g which is unique up to isomorphism. The trivial 2-cocycle gives the trivial extension, and since a 2-coboundary is cohomologous with the trivial 2-cocycle, one has Corollary A central extension defined by a coboundary is equivalent with a trivial central extension. Theorem A finite-dimensional simple Lie algebra has only trivial central extensions. Proof Since every central extension comes from a 2-cocycle φ, it suffices to show that every 2-cocycle is a coboundary. Suppose φ is a 2-cocycle on g. The task is to use this 2-cocycle to manufacture a 1-cochain f such that φ = δf. The first step is to, for each G1 ∈ g, use φ to define a linear map ρG1:g → F by ρ G 1 ( G 2 ) ≡ φ ( G 1 , G 2 ) {\displaystyle \rho _{G_{1}}(G_{2})\equiv \varphi (G_{1},G_{2})} . These linear maps are elements of g∗. Let ν:g∗ →g be the vector space isomorphism associated to the nondegenerate Killing form K, and define a linear map d:g → g by d ( G 1 ) ≡ ν ( ρ G 1 ) {\displaystyle d(G_{1})\equiv \nu (\rho _{G_{1}})} . This turns out to be a derivation (for a proof, see below). Since, for semisimple Lie algebras, all derivations are inner, one has d = adGd for some Gd ∈ g. Then
φ ( G 1 , G 2 ) ≡ ρ G 1 ( G 2 ) = K ( ν ( ρ G 1 ) , G 2 ) ≡ K ( d ( G 1 ) , G 2 ) = K ( a d G d ( G 1 ) , G 2 ) = K ( [ G d , G 1 ] , G 2 ) = K ( G d , [ G 1 , G 2 ] ) . {\displaystyle \varphi (G_{1},G_{2})\equiv \rho _{G_{1}}(G_{2})=K(\nu (\rho _{G_{1}}),G_{2})\equiv K(d(G_{1}),G_{2})=K(\mathrm {ad} _{G_{d}}(G_{1}),G_{2})=K([G_{d},G_{1}],G_{2})=K(G_{d},[G_{1},G_{2}]).}
Let f be the 1-cochain defined by
f ( G ) = K ( G d , G ) . {\displaystyle f(G)=K(G_{d},G).}
Then
δ f ( G 1 , G 2 ) = f ( [ G 1 , G 2 ] ) = K ( G d , [ G 1 , G 2 ] ) = φ ( G 1 , G 2 ) , {\displaystyle \delta f(G_{1},G_{2})=f([G_{1},G_{2}])=K(G_{d},[G_{1},G_{2}])=\varphi (G_{1},G_{2}),}
showing that φ is a coboundary.
The observation that one can define a derivation d, given a symmetric non-degenerate associative form K and a 2-cocycle φ, by
K ( ν ( ρ G 1 ) , G 2 ) ≡ K ( d ( G 1 ) , G 2 ) , {\displaystyle K(\nu (\rho _{G_{1}}),G_{2})\equiv K(d(G_{1}),G_{2}),}
or using the symmetry of K and the antisymmetry of φ,
K ( d ( G 1 ) , G 2 ) = − K ( G 1 , d ( G 2 ) ) , {\displaystyle K(d(G_{1}),G_{2})=-K(G_{1},d(G_{2})),}
leads to a corollary. Corollary Let L:'g × g: → F be a non-degenerate symmetric associative bilinear form and let d be a derivation satisfying
L ( d ( G 1 ) , G 2 ) = − L ( G 1 , d ( G 2 ) ) , {\displaystyle L(d(G_{1}),G_{2})=-L(G_{1},d(G_{2})),}
then φ defined by
φ ( G 1 , G 2 ) = L ( d ( G 1 ) , G 2 ) {\displaystyle \varphi (G_{1},G_{2})=L(d(G_{1}),G_{2})}
is a 2-cocycle. Proof The condition on d ensures the antisymmetry of φ. The Jacobi identity for 2-cocycles follows starting with
φ ( [ G 1 , G 2 ] , G 3 ) = L ( d [ G 1 , G 2 ] , G 3 ) = L ( [ d ( G 1 ) , G 2 ] , G 3 ) + L ( [ G 1 , d ( G 2 ) ] , G 3 ) , {\displaystyle \varphi ([G_{1},G_{2}],G_{3})=L(d[G_{1},G_{2}],G_{3})=L([d(G_{1}),G_{2}],G_{3})+L([G_{1},d(G_{2})],G_{3}),}
using symmetry of the form, the antisymmetry of the bracket, and once again the definition of φ in terms of L. If g is the Lie algebra of a Lie group G and e is a central extension of g, one may ask whether there is a Lie group E with Lie algebra e. The answer is, by Lie's third theorem affirmative. But is there a central extension E of G with Lie algebra e? The answer to this question requires some machinery, and can be found in Tuynman & Wiegerinck (1987, Theorem 5.4).
Applications The "negative" result of the preceding theorem indicates that one must, at least for semisimple Lie algebras, go to infinite-dimensional Lie algebras to find useful applications of central extensions. There are indeed such. Here will be presented affine Kac–Moody algebras and Virasoro algebras. These are extensions of polynomial loop-algebras and the Witt algebra respectively.
Polynomial loop algebra Let g be a polynomial loop algebra (background),
g = C [ λ , λ − 1 ] ⊗ g 0 , {\displaystyle {\mathfrak {g}}=\mathbb {C} [