CENTRAL EXTENSIONS AND INFINITE-DIMENSIONAL LIE ALGEBRAS JOSEPH FYFIELD SUPERVISOR: JØRGEN RASMUSSEN

Contents 1. Introduction 2. Central extensions 2.1. Definitions and notation 2.2. Central extensions and cocycles 2.3. Split extensions and equivalence 2.4. Example: The infinite Heisenberg algebra 3. Cohomology 3.1. Equivalence classes of cocycles as a quotient space 3.2. Definitions in cohomology 3.3. H 1 (g, C) and H 2 (g, C) 4. Affine Lie algebras 4.1. Generalised Cartan matrices 4.2. The Weyl group and root system 4.3. Realisation by extensions 4.4. The Killing form 4.5. Uniqueness of the central extension 5. The Schr¨ odinger algebra 5.1. Definition and structure 5.2. Bilinear forms on s(d) 5.3. Cocycles on Ls(d) 5.4. Extension by a derivation 6. Conclusion References

1 2 2 2 4 6 7 7 8 9 10 10 11 12 13 13 17 17 18 19 23 23 24

1. Introduction In this document, we study infinite-dimensional Lie algebras and their various constructions, in particular via loop algebras and their extensions. We first detail the general theory of central extensions and the related cohomology of Lie algebras before studying the affine Lie algebras of the simple 1

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JOSEPH FYFIELD SUPERVISOR: JØRGEN RASMUSSEN

finite-dimensional Lie algebras. These form a class of examples of extensions and are important in their own right. Then we consider a particular family of finite-dimensional, non-semisimple Lie algebras known as the Schr¨odinger algebras. We will investigate similar constructions as give rise to affine Lie algebras, noting the differences from the semisimple theory. 2. Central extensions A central extension is one way of constructing a new Lie algebra from a given one. The theory of central extensions relies upon cocycles and coboundaries, which we will later see as objects from cohomology. For this section, we refer to Schottenloher [Sch08]. 2.1. Definitions and notation. Throughout, g is an arbitrary Lie algebra (possibly of infinite dimension), a is abelian, and all Lie algebras are over C. Definition 2.1. We call a Lie algebra e a central extension of g by a if there is a short exact sequence 0

a

i

e

π

0

g

and if we have [i(a), e] = 0. We identify a with the corresponding subalgebra of e. Exactness of the sequence and the isomorphism theorems show that a is an ideal of e and g ∼ = e/a. So the map π identifies g with e/a. Definition 2.2. A map Θ : g × g → a which is bilinear, alternating, and satisfies (1)

Θ(x, [y, z]) + Θ(y, [z, x]) + Θ(z, [x, y]) = 0,

x, y, z ∈ g

is called a cocycle. A cocycle of the form Θ(x, y) = µ([x, y]),

x, y ∈ g

(where µ is a linear map from g to a) is called a coboundary. Properly speaking, Θ is called a 2-cocycle on g with values in a, but since these are the cocycles of primary interest here, we will just call them cocycles. 2.2. Central extensions and cocycles. Every cocycle Θ : g×g → a gives rise to a central extension of g by a by equipping the vector space e = g ⊕ a with the Lie bracket (2)

[x ⊕ a, y ⊕ b]e := [x, y]g + Θ(x, y),

x, y ∈ g, a, b ∈ a.

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The properties of Θ and of the bracket in g combine to ensure that this is indeed a Lie bracket. In particular, condition (1) ensures that the Jacobi identity is satisfied. The next lemma will show that we can deal with arbitrary central extensions entirely using the form of (2). First we must define equivalence of central extensions. Definition 2.3. Two central extensions e and e0 are called equivalent if there exists a Lie algebra isomorphism ψ : e → e0 such that the following diagram commutes. e (3)

0

a

π

g

ψ

0

π0

e0

This defines an equivalence relation on the set of central extensions. We will later see a notion of equivalence of cocycles which will turn out to coincide with equivalence of central extensions. Lemma 2.1. Any central extension of g by a is equivalent to one arising from a cocycle Θ : g × g → a in the manner of (2). Proof. Fix a central extension 0

a

e

π

g

0.

By surjectivity of π we can find an injective linear map β : g → e such that π ◦ β = idg (such a map is called a section). Then we define a cocycle by Θ(x, y) = [β(x), β(y)] − β([x, y]),

x, y ∈ g.

A priori this is a map into e. But because π is a Lie algebra homomorphism with π ◦ β = idg , we have π(Θ(x, y)) = π([β(x), β(y)] − π(β([x, y])) = [x, y]g − [x, y]g = 0. So Θ(x, y) ∈ ker π, and exactness of the sequence means then that Θ(x, y) ∈ a. That Θ is bilinear and alternating follows from the properties of the Lie brackets, and Θ also satisfies the Jacobi-like identity (1) because of linearity and the Jacobi identity applied in e and g. To see this, we first use the definition of Θ: Θ(x, [y, z]) + Θ(y, [z, x]) + Θ(z, [x, y]) =

[β(x), β([y, z])] − β([x, [y, z]]) +[β(y), β([z, x])] − β([y, [z, x]]) +[β(z), β([x, y])] − β([z, [x, y]])

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JOSEPH FYFIELD SUPERVISOR: JØRGEN RASMUSSEN

Three of these terms vanish by the Jacobi identity in g and linearity of β. For the remaining three terms, note that β([x, y]) = [β(x), β(y)] − Θ(x, y), but Θ(x, y) ∈ a and so inside a Lie bracket we may replace β([x, y]) with [β(x), β(y)]. Hence Θ(x, [y, z]) + Θ(y, [z, x]) + Θ(z, [x, y]) =

[β(x), [β(y), β(z)]] +[β(y), [β(z), β(x)]] +[β(z), [β(x), β(y)]]

=

0.

Now, we consider the vector space g ⊕ a with the Lie bracket described by this cocycle in the sense of (2), and we define the linear map ψ : g ⊕ a → e,

x ⊕ a 7→ β(x) + a.

We show that ψ is a vector space isomorphism and indeed a Lie algebra homomorphism (hence isomorphism). For injectivity, note that if x ⊕ a ∈ ker ψ, we have β(x) + a = 0. Application of π shows that x = 0, hence β(x) = 0, hence a = 0. For surjectivity, note that if x 6= y, β(x) falls in a different coset of a than β(y). Cosets of a exhaust e, and the element a ∈ a of β(x) + a is arbitrary. So im ψ = e. To check respect for the Lie bracket, we note ψ([x ⊕ a, y ⊕ b]) = ψ([x, y]g + Θ(x, y)) = β([x, y]g ) + Θ(x, y) = [β(x), β(y)] = [β(x) + a, β(y) + b] = [ψ(x ⊕ a), ψ(y ⊕ b)]. With the natural injection and projection, g ⊕ a is then not only a central extension, but is equivalent to the given one e.  2.3. Split extensions and equivalence. We would like to have a way to determine if two extensions are equivalent, and in particular identify any which are essentially trivial, the natural example being the direct sum of Lie algebras g ⊕ a. Formally, we make the following definition. Definition 2.4. Let e be a central extension of g by a. We call e a split extension if it is equivalent to the direct sum g ⊕ a (as Lie algebras) of g and a. If two extensions are equivalent in the sense of Definition 2.3, this is reflected by their corresponding cocycles in the following way.

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Lemma 2.2. Suppose e and e0 are two central extensions of g by a. Then e and e0 are equivalent extensions if and only if their corresponding cocycles Θ and Θ0 (in the sense of Lemma 2.1) differ by a coboundary. Proof. Suppose the diagram (3) in the definition of equivalence does commute, and fix notation accordingly. Let β : g → e and β 0 : g → e0 be two corresponding sections. Then the following cocycles give rise to the extensions e and e0 respectively: Θ(x, y) := [β(x), β(y)] − β([x, y]) Θ0 (x, y) := [β 0 (x), β 0 (y)] − β 0 ([x, y]). The commutativity of the diagram gives that π 0 ◦ ψ = π. Hence π 0 ◦ ψ ◦ β = idg , and we have π 0 ◦ (ψ ◦ β − β 0 ) = 0. So ψ ◦ β − β 0 maps to a ⊆ e0 and we may replace (ψ ◦ β)(x) with β 0 (x) in any bracket in e. Note that ψ◦Θ is a cocycle into e0 by the same arguments as in Lemma 2.1, as ψ ◦ Θ(x, y) = [ψ ◦ β(x), ψ ◦ β(y)] − ψ ◦ β([x, y]), So as to work with maps with images in the same space, we show that ψ ◦ Θ and Θ0 differ by a coboundary. We calculate that ψ ◦ Θ(x, y) = [ψ ◦ β(x), ψ ◦ β(y)] − ψ ◦ β([x, y]) = [β 0 (x), β 0 (y)] − ψ ◦ β([x, y]) = [β 0 (x), β 0 (y)] − β 0 ([x, y]) + (β 0 − ψ ◦ β)([x, y]) = Θ0 (x, y) + (β 0 − ψ ◦ β)([x, y]). The latter term here truly is a coboundary since it maps into ker π 0 = a π 0 ◦ (β 0 − ψ ◦ β) = idg − idg = 0. Conversely, suppose e and e0 are two central extensions with their Lie brackets [·, ·] and [·, ·]0 defined (in the manner of (2)) by Θ and Θ0 = Θ + µ([·, ·]) respectively. As vector spaces, both are g ⊕ a. Then ϕ : e → e0 ,

x ⊕ a 7→ x ⊕ (a + µ(x))

is a Lie algebra isomorphism. For injectivity, note that if x ⊕ a ∈ ker ϕ, we have x ⊕ (a + µ(x)) = 0. Applying π 0 gives that x = 0, hence µ(x) = 0, so a = 0. For surjectivity, note that x ⊕ a is the image of x ⊕ (a − µ(x)). As

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JOSEPH FYFIELD SUPERVISOR: JØRGEN RASMUSSEN

for the Lie bracket, ϕ([x ⊕ a, y ⊕ b]) = ϕ([x, y]g + Θ(x, y)) = [x, y]g + Θ(x, y) + µ([x, y]) = [x, y]g + Θ0 (x, y) = [x, y]0 = [x ⊕ (a + µ(x)), y ⊕ (b + µ(y))]0 = [ϕ(x ⊕ a), ϕ(y ⊕ b)]0 . Hence the two extensions are equivalent.



For a single central extension, different choices of a section give rise to two different cocycles by Lemma 2.1. These cocycles both describe the same extension and so they differ by a coboundary. This classifies split extensions as a corollary — consider the direct sum of Lie algebras g ⊕ a. It is trivially a central extension of g by a, with one corresponding cocycle being the zero map. So we have that a central extension splits if and only if its corresponding cocycle is a coboundary. So there is a correspondence between equivalence classes of central extensions and equivalence classes of cocycles. We will soon see that what we have been calling cocycles and coboundaries arise in a wider theory called cohomology. 2.4. Example: The infinite Heisenberg algebra. The infinite Heisenberg algebra is a central extension of the (commutative) algebra of Laurent polynomials in t, denoted C[t, t−1 ], by a single nonzero central element Z. As a vector space, the Heisenberg algebra is H := C[t, t−1 ] ⊕ CZ. The Lie bracket is defined by [f ⊕ λZ, g ⊕ µZ] :=

X

kfk g−k Z,

k∈Z

P P for f, g ∈ C[t, t−1 ] written f = n∈Z fn tn and g = n∈Z gn tn , with λ, µ ∈ C. The sum over k ∈ Z is well-defined since f and g have finitely many terms. There are two Lie algebra homomorphisms i : C → H,

λ 7→ λZ,

pr1 : H → C[t, t−1 ],

f ⊕ λZ 7→ f,

which are injective and surjective respectively. So we have the following exact sequence corresponding to this central extension. 0

C

i

H

pr1

C[t, t−1 ]

0

CENTRAL EXTENSIONS AND INFINITE-DIMENSIONAL LIE ALGEBRAS

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The corresponding cocycle is Θ : C[t, t−1 ] × C[t, t−1 ] → C,

(f, g) 7→

X

kfk g−k .

k∈Z

To see that it is alternating, make the change of variable ` = −k in the summation. The identity (1) holds automatically because the Lie bracket in C[t, t−1 ] is zero. There is an alternate definition of the cocycle as Θ(f, g) = − resz=0 f (z)g 0 (z), which holds since ! f g 0 (t) =

X X n∈Z

(n − k + 1)fk gn−k+1

tn

k∈Z

in which the residue is given by choosing n = −1 in this sum. This gives X resz=0 f (z)g 0 (z) = −kfk g−k , k∈Z

We note here an alternate definition of H. If we write an = tn for a ∈ C, the Lie bracket can be defined by (4)

[am , an ] = mδm+n Z;

[Z, am ] = 0. P P Taking the Lie bracket of two Laurent polynomials i∈Z λi ai and j∈Z µj aj gives   X X X X  λi ai , µj bj  = λi µj [ai , aj ] = iλi µ−i , i∈Z

j∈Z

i,j∈Z

i∈Z

as in the previous definition — the δm+n appearing in Equation (4) ensure that the terms which appear in the sum are products of coefficients of opposite powers of t. 3. Cohomology In this section, we define the Chevalley-Eilenberg complex of a Lie algebra and explore the meaning of the first two cohomology groups. We refer to Knapp [Kna88], Etingof et. al. [EGH+ 11], and Weibel [Wei94] for additional details. 3.1. Equivalence classes of cocycles as a quotient space. We make the following definitions: Z 2 (g, a) := {2-cocycles on g with values in a} B 2 (g, a) := {2-coboundaries on g with values in a} H 2 (g, a) := Z 2 (g, a)/B 2 (g, a).

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JOSEPH FYFIELD SUPERVISOR: JØRGEN RASMUSSEN

The sets Z 2 (g, a) and B 2 (g, a) form vector spaces, so the quotient is taken as a quotient of vector spaces. It is no coincidence that this corresponds to our theory of central extensions. The object H 2 (g, a) is called the second cohomology, and its dimension counts the number of inequivalent central extensions of g by a. 3.2. Definitions in cohomology. We first note that there are many theories of varying generality which fall under the umbrella of cohomology. In this section we introduce only the concepts necessary for the application of cohomology to our considerations on Lie algebras. Definition 3.1. A complex is a sequence of vector spaces Ck and linear maps dk : Ck → Ck+1 such that im dk ⊆ ker dk+1 . The elements of ker dk are called k-cocycles, and elements of im dk−1 are called k-coboundaries. Elements of Ck are called k-cochains. The maps dk are called differentials or coboundary operators. To a complex we associate its cohomology which is the sequence of objects H k := ker dk / im dk−1 . The condition that im dk ⊆ ker dk+1 is equivalent to requiring that dk+1 ◦ dk = 0, and is weaker than requiring exactness of the sequence of maps dk . The complex of interest in our case is called the Chevalley-Eilenberg complex of g with values in a g-module V . The objects Ck are the vector spaces of alternating multilinear maps Ck (g, V ) := Hom(∧k g, V ),

k≥0

Note that ∧0 g = C, so an element of C0 (g, V ) is defined by a choice of v ∈ V (as the image of 1 ∈ C). The differentials admit a general definition and so will just be called d from now on. They are defined by the action of their image: Let α be a k-cochain. Then (dα) is a (k + 1)-cochain which acts in the following way: k+1 X

(dα)(x1 , . . . , xk+1 ) =

(−1)i+1 xi · α(x1 , . . . , x ˆi , . . . , xk+1 )

i=1

+

X

(−1)i+j α([xi , xj ], x1 , . . . , x ˆi , . . . , x ˆj , . . . , xk+1 )

i
where x · v is the action of g on the module V , and a hat signifies omission of that argument. That the image of this map is an alternating bilinear map follows from the properties of α and of the Lie bracket. The fact that d ◦ d = 0 can be verified directly, see [Kna88].

CENTRAL EXTENSIONS AND INFINITE-DIMENSIONAL LIE ALGEBRAS

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In order to examine the first few differentials in detail, let α, β, and γ be 0-, 1-, and 2-cochains respectively. Then (dα)(x) = x · α (dβ)(x1 , x2 ) = x1 · β(x2 ) − x2 · β(x1 ) − β([x1 , x2 ]) (dγ)(x1 , x2 , x3 ) = x1 · γ(x2 , x3 ) − x2 · γ(x1 , x3 ) + x3 · γ(x1 , x2 ) − γ([x1 , x2 ], x3 ) + γ([x1 , x3 ], x2 ) − γ([x2 , x3 ], x1 ) In our case, an arbitrary abelian Lie algebra a can be considered a g-module with trivial (zero) action. The first few differentials then reduce to (dα)(x) = 0 (dβ)(x1 , x2 ) = −β([x1 , x2 ]) (dγ)(x1 , x2 , x3 ) = −γ([x1 , x2 ], x3 ) + γ([x1 , x3 ], x2 ) − γ([x2 , x3 ], x1 ). The Chevalley-Eilenberg complex and its cohomology are collectively referred to as the cohomology of g with values in V . 3.3. H 1 (g, C) and H 2 (g, C). Consider the (Chevalley-Eilenberg) cohomology of g with values in C (a g-module with trivial action). We can compute H 1 (g, C) by comparing the “cocycle condition” with the “coboundary condition” for 1-cochains β. The cocycle condition is that β([x1 , x2 ]) = 0 (ie. β ∈ ker d1 ). That is, β vanishes on [g, g]. The coboundary condition is that β(x) = 0 (ie. β ∈ im d0 ). So elements of H 1 (g, C) are linear maps g → C which vanish on [g, g]. We can also realise this as the quotient H 1 (g, C) ∼ = (g/[g, g])∗ . For more general modules V , the space H 1 (g, V ) describes the space of derivations g → V (maps β : g → V for which β([x, y]) = xβ(y) − yβ(x)) modulo inner derivations (those given by β(x) = xv for some fixed v ∈ V ). A derivation which is not inner is called outer, so H 1 (g, V ) classifies outer derivations up to equivalence by inner derivations. Similarly, considering 2-cocycles (satisfying (dγ) = 0) modulo 2-coboundaries (those for which γ = dβ) yields the same H 2 (g, C) discussed earlier and thereby classifies one-dimensional central extensions. A key result on the cohomology of simple finite-dimensional Lie algebras is that their low-dimensional cohomologies are trivial [Wei94]. Theorem 3.1 (Whitehead’s lemmas). If V is a finite-dimensional g-module, then H 1 (g, V ) = 0 and H 2 (g, V ) = 0.

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JOSEPH FYFIELD SUPERVISOR: JØRGEN RASMUSSEN

So not only are all derivations of g on a module V inner derivations, but also H 2 (g, C) = 0 and so a finite-dimensional simple Lie algebra over C admits no nontrivial one-dimensional central extensions.

4. Affine Lie algebras In the late 1960s, Kac and Moody independently generalised the welldeveloped theory of semisimple Lie algebras, giving rise to a new class of Lie algebras which includes the finite-dimensional semisimple ones as well as many infinite-dimensional families. Of these Kac-Moody algebras, the most important for our purposes are those of “affine type”. These admit an abstract construction as well as a concrete realisation as extensions of a loop algebra. As in the semisimple theory, the root system and Weyl group play a major role, so we study these first. Then we will describe the realisation, as it provides a class of examples of the concepts studied above. We refer to Carter [Car05] for details of this section.

4.1. Generalised Cartan matrices. Let A be an n × n generalised Cartan matrix (GCM) which is indecomposable and of affine type. While the general theory of Kac-Moody algebras is far less restrictive, this is the case of interest here. Firstly, we list some properties of A to fix our notation. Most importantly, n = rank A + 1, so we may index the elements of A by values 0 through ` = rank A. There exist vectors a = (a0 , a1 , . . . , a` ) and c = (c0 , c1 , . . . , c` ) such that Aat = 0 and cA = 0, and these vectors may be chosen so that their components are all positive integers with no common factor. We also have A = DB where D = diag(d0 , d1 , . . . , d` ), di = ai /ci , and B is symmetric. A realisation of A is a triple (h, Π, Π∨ ), where h is a complex vector space, Π = {α0 , . . . , α` } ⊂ h∗ is the set of simple roots, and Π∨ = {h0 , . . . , h` } ⊂ h is the set of simple coroots, with the condition that αi (hj ) = Aji . The minimal dimension of h so that this is possible is 2n − rank A = ` + 2, and indeed every GCM has a realisation of this dimension. So we consider only minimal realisations from this point. The realisation is the starting point for the abstract construction of the Kac-Moody algebra g(A) = g, in which h will be a Cartan subalgebra. So the study of the root system and Weyl group primarily requires an understanding of the spaces in the triple (h, Π, Π∨ ). P A nonzero element α = `i=0 ki αi of h∗ is called a root if gα 6= 0, where gα is the root space: gα = {x ∈ g | [h, x] = α(h)x ∀h ∈ h}.

CENTRAL EXTENSIONS AND INFINITE-DIMENSIONAL LIE ALGEBRAS

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We denote the set of roots by Φ. Crucially, all roots are either positive (ki ∈ N0 for all i = 0, . . . , `) or negative (−ki ∈ N0 for all i = 0, . . . , `), so we have the respective disjoint subsets Φ+ , Φ− ⊂ Φ, the union of which is Φ. With reference to the decomposition A = DB and after a choice of a complementary subspace h00 to h0 = span{h0 , . . . , h` } in h, we can define a nondegenerate, invariant, symmetric bilinear form on h by hhi , hj i = di dj Bij ,

hhi , xi = di αi x,

hx, yi = 0,

for i, j = 0, . . . , ` and x, y ∈ h00 . Note that h00 will have dimension one, since dim h = ` + 2. An inductive procedure then extends this to such a form on g. Since the form is nondegenerate, it induces an isomorphism h∗ → h given by λ 7→ h0λ , where h0λ is uniquely defined as the element for which hh0λ , hi = λ(h) h ∈ h. We note that the isomorphism identifies hi with di αi , so roots and coroots are closely related. The isomorphism then induces a bilinear form on h∗ by hλ, µi = hh0λ , h0µ i. 4.2. The Weyl group and root system. The Weyl group of a Lie algebra is a group of symmetries of both the root space and coroot space. While defined as a reflection group of h, in the affine case it admits a useful description as a group of transformations on the real span of the coroots of an “underlying” simple Lie algebra. This section defines the Weyl group and explains that action. The Weyl group W of a Lie algebra g is the group generated by the fundamental reflections si : h → h defined by si (x) = x − αi (x)hi

x ∈ h i = 0, . . . , `.

The group W also acts on h∗ by (wλ)(x) = λ(w−1 x) for w ∈ W , λ ∈ h∗ , and x ∈ h. Under this action and the isomorphism described earlier, the action of the fundamental reflections on h∗ is given by si (λ) = λ − λ(hi )αi , and, notably, permutes the roots. We now partition the root system Φ yet again, this time into real roots ΦRe and imaginary roots ΦIm . The former are those roots α such that α = w(αi ) for some w ∈ W , and the latter are the roots for which this does not hold. Affine Lie algebras have an imaginary root given by δ = a0 α0 + · · · + a` α` , and the entire set of imaginary roots is ΦIm = {kδ | k ∈ Z, k 6= 0}. In order to consider another action of W , first note that the submatrix of A given by (Aij )`i,j=1 is an ` × ` matrix of finite type. So for i = 1, . . . , `, the roots αi and coroots hi are those of a simple Lie algebra g0 which we refer

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JOSEPH FYFIELD SUPERVISOR: JØRGEN RASMUSSEN

to as the underlying Lie algebra of g(A). The real span of the underlying coroots h0R admits an action of W — the derivation of this action requires a detailed study of h and its dual space, but we record the resultant highly geometric description. The fundamental reflections s1 , . . . , s` act on the underlying Cartan subalgebra as defined earlier, as this coincides with the semisimple theory. The action of s0 on h0R is as reflection in the hyperplane Lθ,1 = {h ∈ h0R | θ(h) = 1}, where θ is the highest root of the underlying Lie algebra, given by θ = a1 α1 + · · · + a` α` . The resultant infinite group of symmetries of h0R is the semidirect product of the underlying classical Weyl group generated by s1 through s` , and the group of translations by a certain lattice in h0R . 4.3. Realisation by extensions. In this section we outline the way in which many affine Lie algebras of the kind studied above can be realised by extensions of a loop algebra of a simple Lie algebra. In the classification of affine Lie algebras, those which arise in this way are referred to as untwisted. Let g0 be a finite-dimensional simple Lie algebra. We first form the loop algebra Lg0 by taking the vector space g0 ⊗ C[t, t−1 ] adorned with the Lie bracket [xtm , ytn ] = [x, y]g0 tm+n , x, y ∈ g0 , m, n ∈ Z extended by linearity. Definition 4.1. The Killing form κ of a Lie algebra g is the bilinear form on g defined by κ(x, y) = tr(ad x ◦ ad y), x, y ∈ g. The Killing form is symmetric and satisfies κ([x, y], z) = κ(x, [y, z]). Symmetric bilinear forms for which this holds are called invariant. A cocycle Θ : Lg0 × Lg0 → C can be defined by Θ(xtm , ytn ) = mδm+n κ(x, y). That this is a cocycle on Lg0 follows from properties of the Killing form: Θ(xt` , [ytm , ztn ]) + Θ(ytm , [ztn , xt` ]) + Θ(ztn , [xt` , ytm ]) = `δ`+m+n,0 κ(x, [y, z]) + mδ`+m+n,0 κ(y, [z, x]) + nδ`+m+n,0 κ(z, [x, y]) = (−m − n)κ(x, [y, z]) + mκ(x, [y, z]) + nκ(x, [y, z]) = 0. ˜ 0 is the corresponding central extension. As a vector The Lie algebra Lg ˜ 0 = Lg0 ⊕ CZ, and the Lie bracket is space, Lg [xtm , ytn ] = [x, y]g0 tm+n + mδm+n κ(x, y)Z, with Z central.

x, y ∈ g0 , m, n ∈ N

CENTRAL EXTENSIONS AND INFINITE-DIMENSIONAL LIE ALGEBRAS

13

˜ 0 defined by ∆ = t d . We further extend the algebra by a derivation on Lg dt As a vector space, we define g = Lg0 ⊕ CZ ⊕ C∆, with the same Lie bracket ˜ 0 except for as Lg [∆, xtm ] = ∆(xtm ) = mxtm ,

[∆, Z] = 0,

x ∈ g0 , m ∈ Z

That the Jacobi identity is satisfied follows from the fact that ∆ is a derivation. It is this algebra g which we call the affine Lie algebra corresponding to g0 . 4.4. The Killing form. Given the importance of a bilinear form in the construction of affine Lie algebras, we now turn to the study of invariant symmetric bilinear forms on simple Lie algebras. We prove that the Killing form is the unique such form up to a scalar factor, which will allow us to then show that the above central extension is unique. Theorem 4.1. Let g be a finite-dimensional simple Lie algebra over C. Then the Killing form κ of g is the unique invariant symmetric bilinear form on g, up to a scalar factor. Proof. Firstly, note that since g is simple, it is an irreducible finite-dimensional g-module. So Schur’s lemma gives that the space of intertwiners Homg (g, g) is one-dimensional, that is, Homg (g, g) ∼ = C. Secondly, consider the dual space g∗ of g as a g-module with the action given by (x · ϕ)(y) = −ϕ([x, y]) for x, y ∈ g, ϕ ∈ g∗ . Then g and g∗ are isomorphic as g-modules, with the intertwiner given by f : y 7→ κ(y, ·). To check this, we calculate the action of f ◦ x and x ◦ f on y ∈ g as follows: (f ◦ x)(y) = f ([x, y]) = κ([x, y], ·) = −κ(y, [x, ·]) (x ◦ f )(y) = x · κ(y, ·) = −κ(y, [x, ·]). The above two facts combine to give that Homg (g, g∗ ) ∼ = C. Letting B be any symmetric bilinear invariant form on g, the same calculations show that x 7→ B(x, ·) is an intertwiner between g and g∗ . But the space of intertwiners is one-dimensional, and hence B = ακ for some α ∈ C.  As a corollary of this, we have that if V is a finite-dimensional vector space and B is a symmetric bilinear invariant form on g with image in V , then there exists some v ∈ V such that B(x, y) = κ(x, y)v for all x, y ∈ g. 4.5. Uniqueness of the central extension. We follow Wilson [Wil82] to prove that the central extension described above is the essentially unique central extension of the loop algebra of a simple Lie algebra. The proof relies on the fact that given a section of the map π, a cocycle corresponding to the central extension can be constructed. After proving the

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JOSEPH FYFIELD SUPERVISOR: JØRGEN RASMUSSEN

existence of a cocycle with a certain property, that cocycle can be written in terms of the Killing form on g, and will turn out to be the familiar cocycle mδm+n κ of Section 4.3. Lemma 4.1. Let g be a simple Lie algebra and e a central extension of Lg: 0

C

e

π

Lg

0.

Let S denote the set of linear maps β : Lg → e such that π ◦ β = idg . Define β ∗ : Lg × Lg → e by β ∗ (xtm , ytn ) = [β(xtm ), β(ytn )] − β([x, y]tm+n ). Then β ∗ is a cocycle on Lg. Furthermore, let S0 = {β ∈ S | β ∗ (xtm , yt0 ) = 0 ∀x, y ∈ g, m ∈ Z}. Then the set S0 is not empty. Proof. The proof of Lemma 2.1 shows that β ∗ , defined in terms of a section β, is indeed a cocycle. Take β ∈ S. Then the properties of a cocycle give (5)

β ∗ ([xtm , yt0 ], zt0 ) − β ∗ (xtm , [y, z]t0 ) + β ∗ ([z, x]tm , yt0 ) = 0.

Define f m : g → Hom(g, C) by f m (y) = fym where fym (x) = β ∗ (xtm , y). Then Equation 5 becomes m f[y,z] (x) = fym ◦ adz (x) − fzm ◦ ady (x)

= −(z · fym )(x) + (y · fzm )(x) where · is the action of g on Hom(g, C) defined by x : f 7→ −f ◦ adx . So f m is a derivation on g with coefficients in Hom(g, C). By Whitehead’s Lemma, then, there exists g m ∈ Hom(g, C) such that fym = −y · g m = g m ◦ ady . Define g ∈ Hom(Lg, C) by g(xtm ) = g m (x). Then set ϕ = β − g and observe that since g(xtm ) is always central, ϕ∗ (xtm , yt0 ) = [(β − g)(xtm ), (β − g)(yt0 )] − (β − g)[xtm , yt0 ] = β ∗ (xtm , yt0 ) + g([x, y]tm ) = β ∗ (xtm , yt0 ) − g m ◦ ady (x) = β ∗ (xtm , yt0 ) − fym (x), but since fym (x) was defined as β ∗ (xtm , yt0 ), this expression is zero and hence ϕ ∈ S0 . 

CENTRAL EXTENSIONS AND INFINITE-DIMENSIONAL LIE ALGEBRAS

15

∗ Lemma 4.2. Let β ∈ S0 . Then for each m, n ∈ Z, the map βm,n : (x, y) 7→ ∗ m n ∗ β (xt , yt ) is an invariant bilinear form on g. Hence βm,n = λm,n κ for some λm,n ∈ C and κ the Killing form of g.

Proof. Since β ∗ is a cocycle, for any x, y, z ∈ g we have β ∗ ([xtm , yt0 ], ztn ) + β ∗ ([yt0 , ztn ], xtm ) + β ∗ ([ztn , xtm ], yt0 ) = 0. But since β ∈ S0 , only two terms are nonzero. Using the antisymmetry of cocycles we have β ∗ ([x, y]tm , ztn ) = β ∗ (xtm , [y, z]tn ). ∗ ([x, y], z) = β ∗ (x, [y, z]). So by Theorem 4.1, β ∗ That is, βm,n m,n m,n is a multiple of the Killing form on g. 

Lemma 4.3. Let β ∈ S0 , with λm,n as per Lemma 4.2. Then for all `, m, n ∈ Z, we have (6)

λm,0 = 0

(7)

λm,n = −λn,m

(8)

λ`+m,n + λm+n,` + λn+`,m = 0

(9)

λm,n = mδm+n,0 λ1,−1

∗ (x, y) = 0. Choosing Proof. For (6), recall that since β ∈ S0 we have βm,0 x and y such that κ(x, y) 6= 0 (which we may since the Killing form is nondegenerate), we have 0 = λm,0 κ(x, y). Hence λm,0 = 0. Similarly (7) can be shown, using the symmetry of the Killing form. For (8), we use symmetry and invariance of the Killing form to obtain:

0 = β ∗ ([xt` , ytm ], ztn ) + β ∗ ([ytm , ztn ], xt` ) + β ∗ ([ztn , xt` ], ytm ) = λ`+m,n κ([x, y], z) + λm+n,` κ([y, z], x) + λn+`,m κ([z, x], y) = (λ`+m,n + λm+n,` + λn+`,m )κ(x, [y, z]) Then by nondegeneracy, for any [y, z] we may choose x such that κ(x, [y, z]) 6= 0 and so (8) holds. For (9) we first show that λa,−a = aλ1,−1 by induction, and then show that λa,b = 0 if a + b 6= 0. For the induction, note that it holds for a = 1 and assume that λa−1,−(a−1) = (a − 1)λ1,−1 for some a ∈ N. Then setting ` = a − 1, m = 1, and n = −a in (8) gives λa,−a + λ−a+1,a−1 + λ−1,1 = 0, which may be rearranged to give λa,−a = λa−1,−(a−1) + λ1,−1 . Then by the inductive hypothesis λa,−a = aλ1,−1 .

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JOSEPH FYFIELD SUPERVISOR: JØRGEN RASMUSSEN

Now let s 6= 0 and take m = s − ` − 1, n = 1 in (8) with ` free. This gives λs−1,1 = −λ`+1,s−(`+1) + λ`,s−` .

(10)

Summing over ` in this expression gives a telescoping sum. In particular, if s > 0, summing this expression from ` = 0 to s − 1 gives the following, and if s < 0, summing from ` = s to −1 gives the same expression, which is zero by (6): sλs−1,1 = λ0,s − λs,0 = 0. The different ranges of summation ensure that the sum is nonempty. Since s 6= 0 we have λs−1,1 = 0. From (10), then, we have λ`,s−` = λ`+1,s−(`+1) . Applying this equality k times gives λ`,s−` = λ`+k,s−(`+k) , that is, the value of λa,b depends only on the sum of the indices a + b rather than a and b independently, as long as s = a + b 6= 0. But we have seen that λs−1,1 = 0 for all s 6= 0. So for all a + b 6= 0 we have λa,b = 0. Hence we may write λm,n = mδm+n,0 λ1,−1 .  So given the arbitrary central extension, we have been able to write a corresponding cocycle β ∗ in the form β ∗ (xtm , ytn ) = mδm+n,0 λ1,−1 κ(x, y). This will allow us to relate it to the central extension of Section 4.3. ˜ constructed in SecTheorem 4.2. The centrally extended Lie algebra Lg tion 4.3 is the unique nontrivial central extension of Lg up to isomorphism. Proof. Let e be a one-dimensional central extension of Lg with central element Z 0 such that [e, e] = e (so that the central extension is not split). Let ˜ be as in Section 4.3, and by Lemma 4.1 take β ∈ S0 and λm,n as in Lg Lemma 4.2 ˜ → e by Define φ : Lg φ(xtm + aZ) = β(xtm ) + aλ1,−1 Z 0 ,

x, y ∈ g, a, b ∈ C

Then φ([xtm + aZ, ytn + bZ]) = φ([x, y]tm+n + mδm+n,0 κ(x, y)Z) = β([x, y]tm+n ) + mδm+n,0 κ(x, y)λ1,−1 Z 0 = β ∗ (xtm , ytn ) + β([xtn , ytm ]) = [β(xtm ), β(ytn )] = [β(xtm ) + aλ1,−1 Z 0 , β(ytn ) + bλ1,−1 Z 0 ] = [φ(xtm + aZ), φ(ytn + bZ)]

CENTRAL EXTENSIONS AND INFINITE-DIMENSIONAL LIE ALGEBRAS

17

Since e is not a split extenson, λ1,−1 6= 0 (otherwise the cocycle is trivial) and so φ is surjective. Injectivity of φ follows from that of β by the same ˜ arguments as in Lemma 2.1. Hence e is isomorphic to Lg. 

¨ dinger algebra 5. The Schro In this section we focus on a family of Lie algebras indexed by d ∈ N, called the Schr¨ odinger algebras. We will denote the Schr¨odinger algebra of spatial dimension d by s(d). These are finite-dimensional but not semisimple, however, their structure is such that they are amenable to some careful study as described in [DDM97]. 5.1. Definition and structure. The constituents of the Schr¨odinger algebra are referred to by their roles in the corresponding physical theory. A basis of s(d) is given by the rotation generators Jij (where Jij = −Jji ), the space translations Pi , the special Galilei transformations Gi (also known as Galilean boosts), the time translation PT , a dilation D, and a Galilean conformal transformation K, where i, j = 1, . . . , d. Their nontrivial commutation relations, as given in [DDM97], are

(11)

[Jij , Jkl ] = δij Jjl + δjl Jik − δil Jjk − δjk Jil

(12)

[Jij , Pk ] = δik Pj − δjk Pi

(13)

[Jij , Gk ] = δik Gj − δjk Gi

(14)

[PT , Gi ] = Pi

(15)

[D, Gi ] = Gi

(16)

[D, Pi ] = −Pi

(17)

[D, PT ] = −2PT

(18)

[D, K] = 2K

(19)

[K, Pi ] = −Gi

(20)

[K, PT ] = −D

Note that the Jij form an so(d) subalgebra and the elements {PT , −D, −K} form an sl(2) subalgebra (they correspond to {e, h, f } respectively). The Schr¨ odinger algebra has a Levi decomposition which expresses it as a semidirect sum of the semisimple component and the abelian subalgebra a (spanned by the Pi and Gi ) on which the semisimple component acts: s(d) = (so(d) ⊕ sl(2)) n a.

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JOSEPH FYFIELD SUPERVISOR: JØRGEN RASMUSSEN

Furthermore, s(d) is a graded Lie algebra with D the grading operator if we define deg K = 2 deg Gi = 1 deg D = deg Jij = 0 deg Pi = −1 deg PT = −2 There are some further subalgebras of physical interest: each of the sets {Jij , Pi , Gi , PT } and {Jij , Pi , Gi , K} generate a Galilei subalgebra, which are related by an automorphism of s(d). Since the Galilei algebra admits multiple central extensions [Mru97], it could be hoped that these also define central extensions of s(d) by way of the Galilei subalgebras present. However, most of these central extensions are not admissible as they do not preserve the Jacobi identity in s(d). In the case of s(1), each of the Galilei subalgebras suggest one additional extension, but calculating the Jacobi identity as in [DDM97] shows that they do not define extensions of s(1). We can make a similar calculation in the case of s(2) — as per [DM95], the Galilei subalgebras defined earlier admit a central extension with the additional bracket [G1 , G2 ] = m0 . But calculating the Jacobi identity in s(2) with D, G1 , and G2 gives 0 = [[D, G1 ], G2 ]+[[G1 , G2 ], D]+[[G2 , D], G1 ] = [G1 , G2 ]+[−G2 , G1 ] = 2m0 , and hence m0 = 0. In fact, the finite-dimensional Lie algebra s(d) admits only a single central extension [Mru97] defined by the additional relation [Pi , Gj ] = δij m, and the extended algebra is denoted sˆ(d). 5.2. Bilinear forms on s(d). We have seen that the affinisation of a simple Lie algebra relies on an invariant symmetric bilinear form. As a first step in affinising the Schr¨ odinger algebra, then, we should consider the space of such forms on s(d). We can in fact classify these: Lemma 5.1. The space of invariant symmetric bilinear forms on s(d) has dimension 1 if d = 1, dimension 3 if d = 4, and dimension 2 otherwise. Proof. Let B(·, ·) be such a form. Defining B amounts to choosing a value for B on every pair of basis elements. By restriction, B defines an invariant form on the sl(2) subalgebra. So on PT , D, and K, the form B must agree with the Killing form of sl(2) up to a scalar factor. Similar consideration of the so(d) subalgebra restricts the values of B on pairs of elements Jij , Jk` to values chosen from the space of such forms on so(d) (which is trivial when

CENTRAL EXTENSIONS AND INFINITE-DIMENSIONAL LIE ALGEBRAS

19

d = 1, has dimension 2 when d = 4 (since so(4) = so(3) ⊕ so(3)), and which has dimension 1 otherwise). In the appendix of [DDM97], it is shown that all of the elements Pi and Gj are degenerate with respect to any such form on sˆ(d), but their argument, which relies on computing invariance conditions on B, also applies in the case of s(d). For the other elements, writing down particular invariance conditions on B and evaluating each side shows that all of B(Jij , PT ), B(Jij , D), and B(Jij , K) are zero: −2B(Jij , PT ) = B(Jij , [D, PT ]) = B([Jij , D], PT ) = 0, 2B(Jij , K) = B(Jij , [D, K]) = B([Jij , D], K) = 0, B(Jij , D) = B(Jij , [PT , K]) = B([Jij , PT ], K) = 0. So B takes the value zero except where it agrees with a Killing form of a simple subalgebra as described above. That B does take nonzero values on the so(d) and sl(2) subalgebras does not contradict any of the invariance conditions, because none of the elements Jij , PT , D, or K can be achieved by Lie brackets of any elements from outside their respective subalgebras (so there are no calculations like the above which will eliminate B on these elements). Counting the dimensions of the space of forms on the semisimple Lie algebras so(d) and sl(2) gives the result.  These bilinear forms will give rise to central extensions of the loop-Schr¨odinger algebra in the manner of Section 4.3. However, it will turn out that not all of the central extensions are defined in this way. 5.3. Cocycles on Ls(d). We now consider the loop-Schr¨odinger algebra Ls(d) = s(d) ⊗ C[t, t−1 ]. The affinisation of a finite-dimensional simple Lie algebra was an extension of a loop algebra by both a central element and a derivation, so we now investigate similar extensions of Ls(d). The first step will be to classify cocycles on Ls(d) (and thereby, one-dimensional central extensions), and then we will determine which of these permit the same derivation as in the case of the affinisation of simple Lie algebras. Note that we can already identify a subspace of H 2 (Ls(d), C), given by cocycles of the form Θ(xtm , ytn ) = mδm+n B(x, y) where B is chosen from the space of invariant bilinear forms determined previously. That these are indeed cocycles follows from invariance, and that they are nontrivial follows from the fact that they are nontrivial on their respective subalgebras. Now we consider an arbitrary cocycle on Ls(d) and utilise the cocycle and coboundary conditions to identify any trivial components of the cocycle. Our calculations make use of the basis described in Section 5.1, as an arbitrary

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JOSEPH FYFIELD SUPERVISOR: JØRGEN RASMUSSEN

cocycle Θ is defined by the values Θ(xtm , ytn ), where x and y are basis elements of s(d). Since Jij tm , Dtm , Ktm , PT tm , form a basis for the (loop-) semisimple component, any cocycle which is nonzero on these elements must have a component arising from bilinear forms in the manner described earlier, because extensions of these subalgebras are essentially unique (by Section 4.5). We now see that any cocycle is zero on pairings with one element from the semisimple component and one element from the abelian component span{Pi tm , Gj tm }. Henceforth, let Θ be an arbitrary cocycle on Ls(d). Lemma 5.2. The cocycle Θ takes value zero on the following pairs of basis elements: (Pi tm , Pj tn ), (Gi tm , Gj tn ), (Pi tm , PT tn ), (Gi tm , Ktn ). Proof. We apply the cocycle condition on Θ with the triple of elements Pi tm , Pj tn , and Dt0 . 0 = Θ(Pi tm , [Pj tn , D]) + Θ(Pj tn , [D, Pi tm ]) + Θ(D, [Pi tm , Pj tn ]) = Θ(Pi tm , Pj tn ) − Θ(Pj tn , Pi tn ) = 2Θ(Pi tm , Pj tn ). The same calculation with Gi , Gj in place of Pi , Pj shows that Θ(Gi tm , Gj tn ) = 0. To see that Θ(Pi tm , PT tn ) = 0, we apply the same with the elements D, Pi t m , PT t n : 0 = Θ(D, [Pi tm , PT tn ]) + Θ(Pi tm , [PT tn , D]) + Θ(PT tn , [D, Pi tm ]) = 0 + Θ(Pi tm , 2Pt tn ) + Θ(PT tn , −Pi tm ) = 3Θ(Pi tm , PT tm ) The same calculation with Gi and K in place of Pi and PT shows that Θ(Gi tm , PT tn ) = 0.  In order to deal with nonzero coboundaries, we first show a technicality. Lemma 5.3. Suppose x, y ∈ s(d) are not oppositely graded by D. Then Θ(xtm , ytn ) depends only on m + n rather than m and n independently. Proof. We expand the cocycle condition with the elements D, xtm , and ytn , writing [D, x] = (deg x)x: 0 = Θ(D, [x, y]tm+n ) + Θ(xtm , [y, D]tn ) + Θ(ytn , [D, x]tm ) = Θ(D, [x, y]tm+n ) + (− deg(y) − deg(x))Θ(xtm , ytn ).

CENTRAL EXTENSIONS AND INFINITE-DIMENSIONAL LIE ALGEBRAS

21

Provided that deg(x) + deg(y) 6= 0, we may write this as (21)

Θ(xtm , ytn ) =

1 Θ(D, [x, y]tm+n ). deg(x) + deg(y)

The value of (21) depends only on m + n. So if m + n = m0 + n0 , we have 0 0 Θ(xtm , ytn ) = Θ(xtm , ytn ).  It turns out that the value of a cocycle on many other pairs of basis elements must always be trivial in the following sense. Lemma 5.4. The cocycle Θ is coboundary-equivalent to a cocycle which takes value zero on the following pairs of basis elements: (Jij tm , Pk tn ), (Jij tm , Gk tn ), (Pi tm , Ktn ), (Pi tm , Dtn ), (Gi tm , PT tn ), (Gi tm , Dtn ). Proof. In order to prove this, we need to understand the nature of coboundaries on Ls(d). Recall that Θ is called a coboundary if Θ(·, ·) = µ([·, ·]) where µ is a linear map to C. In a slight abuse of notation, we will also use µ(·, ·) to denote to the coboundary defined by µ([·, ·]). We want to show that Θ(·, ·) = µ([·, ·]) on the pairings above, for some well-chosen linear map µ : Ls(d) → C. Recall the commutation relations (12)–(16), (19) of Section 5.1, which describe the action of the semisimple component on the Pi and Gj . Now we need to show that for an arbitrary cocycle Θ, both of the following hold. (22)

Θ(Jkj tm , Pk tn ) = Θ(Pj tm , Dtn ) = Θ(PT tm , Gj tn )

(23)

Θ(Jkj tm , Gk tn ) = Θ(Pj tm , Ktn ) = Θ(Dtm , Gj tn )

Then we may define µ : Ls(d) → C by letting µ(Pj tm+n ) be the shared value of (22), and µ(Gj tm+n ) be the shared value of (23). This is welldefined because each term in (22) (resp. (23)) proposes to define the same value of µ, and additionally the previous lemma applies so that the value of (22) only depends on m + n (and hence is well-defined for each m + n). By the commutation relations recalled above, we will then have Θ(·, ·) = µ([·, ·]) on the pairs specified. Applying the cocycle condition with the elements Jkj tm , Pk t0 , and Dtn gives Θ(Jkj tm , Pk tn ) = Θ(Pj tm , Dtn ). Similar direct application of the cocycle condition with elements Jkj tm , Gk t0 , and PT tn gives Θ(Pj tm , Dtn ) = Θ(PT tm , Gj tn ) and hence Equation 22. A similar calculation, using elements Jkj tm , Pk t0 , Ktn gives the equality Θ(Jkj tm , Gk tn ) = Θ(Pj tm , Ktn ). Then, taking the elements Jkj tm , Gk t0 , and Dtn , the cocycle condition gives 0 = Θ(Jkj tm , −Gk tn ) + Θ(Dtn , Gj tm ).

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JOSEPH FYFIELD SUPERVISOR: JØRGEN RASMUSSEN

Since Lemma 5.3 applies, these values of Θ depend on m + n only, so we may exchange m and n in the second term to show the following equality and hence Equation 23. Θ(Jkj tm , Gk tn ) = Θ(Dtm , Gj tn ). Now that we have shown (22) and (23), we may define µ such that Θ(·, ·) = µ([·, ·]) on the pairs specified. Hence, as a map on the entirety of Ls(d) we have Θ = Θ0 + µ, where Θ0 takes value zero on the pairs in question.  So we have seen that an arbitrary cocycle on Ls(d) is defined on the semisimple component by bilinear forms, and on most other pairs of basis elements by a coboundary. It remains to describe possible cocycles on pairs of elements (Pi tm , Gj tn ). Indeed, it is here where other nontrivial cocycles arise. Note that a coboundary will take value zero on (Pi tm , Gj tn ) because [Pi tm , Gj tn ] = 0, so any cocycle which is nonzero on such pairs of elements is automatically nontrivial. Lemma 5.5. The value of Θ(Pi tm , Gi tn ) depends only on m+n rather than m and n independently. Furthermore, Θ(Pi tm , Gi tn ) = Θ(Pj tm , Gj tn ). We also have Θ(Pi tm , Gj tn ) = 0 for i 6= j. Proof. Applying the cocycle condition with Dt` , Pi tm , and Gj tn yields 0 = Θ(Pi , −Gj tn+` ) + Θ(Gj tn , −Pi tm+` ). Then rearranging and setting ` = m0 − m − n gives 0

0

Θ(Pi tm , Gj tm −m ) = Θ(Pi tm −n , Gj tn ). So Θ(Pi tm , Gj tn ) depends only on m + n. To see that Θ(Pi tm , Gi tn ) = Θ(Pj tm , Gj tn ), note that if d = 1 this is trivial. For d ≥ 2, applying the cocycle condition to Jij , Pi tm and Gj tn immediately gives the result. To see that Θ(Pi tm , Gj tn ) = 0 for i 6= j, suppose i 6= j and note the following. The cocycle condition applied to K, Pi tm , and Pj tn and then exchanging indices gives Θ(Pi tm , Gj tn ) = Θ(Pj tn , Gi tm ) = Θ(Pj tm , Gi tn ). Applying the same to Jij , Pi tm , and Gi tn and then applying the alternating property of cocycles gives Θ(Pi tm , Gj tn ) = Θ(Gi tn , Pj tm ) = −Θ(Pj tm , Gi tn ). Combining the above gives that Θ(Pi tm , Gj tn ) = 0. Note that the argument does not apply for i = j because of the use of the element Jij which is zero if i = j. 

CENTRAL EXTENSIONS AND INFINITE-DIMENSIONAL LIE ALGEBRAS

23

In fact, we have incorporated all conditions on Θ(Pi tm , Gj tn ). So for any choice of λn ∈ C, a cocycle may be defined by (24)

Θ(Pi tm , Gj tn ) = δij λm+n .

Recalling the relationships between cocycles, coboundaries, and one-dimensional central extensions from Section 2, we have shown the following. Theorem 5.1. The nontrivial one-dimensional central extensions of Ls(d) are linear combinations of those arising from invariant symmetric bilinear forms on s(d) and those arising on span{Pi , Gi } in the manner of (24). d 5.4. Extension by a derivation. Recall the derivation ∆ = t dt which is introduced in the affine Lie algebra corresponding to a simple Lie algebra. That this was indeed a derivation followed from the definition of the central d extension of the loop algebra. In the case of the Schr¨odinger algebra, t dt turns out to be a derivation for only some of the central extensions determined above. Now consider the algebra Ls(d)⊕CM , with central extension defined by a cocycle of the form described in Theorem 5.1: Θ(xtm , ytn ) = mδm+n B(x, y)+ δij λm+n , with central element denoted M . Supposing that ∆ is a derivation, we are able to derive conditions on λm+n . The definition of a derivation applied to Pi tm and Gj tn is

∆[Pi tm , Gj tn ] = [∆Pi tm , Gj tn ] + [Pi tm , ∆Gj tn ] and gives 0 = (m + n)λm+n . d So in order for Ls(d) ⊕ CM to admit t dt as a derivation, we must have λm+n = λδm+n for some λ ∈ C. If the cocycle Θ is defined as above, with λn = 0 unless n = 0, then in analogy to the affine Kac-Moody Lie algebras of the simple Lie algebras we may now form the Lie algebra

s˜(d) = Ls(d) ⊕ CM ⊕ C∆. So while there are finitely many inequivalent central extensions of Ls(d) d which admit t dt as a derivation, not all of these central extensions arise from bilinear forms on s(d). 6. Conclusion We have considered the general theory of central extensions as well as some of the underlying theory of cohomology, in particular the ChevalleyEilenberg cohomology of a Lie algebra g which classifies the outer derivations and central extensions of g up to respective equivalence. As examples of

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JOSEPH FYFIELD SUPERVISOR: JØRGEN RASMUSSEN

extensions, and interesting objects in their own right, we have considered the untwisted affine Lie algebras which arise as examples of the more general Kac-Moody algebras. We then investigated the family of Schr¨odinger algebras as an example of finite-dimensional non-semisimple Lie algebras whose central extensions have been studied. In our attempt to “affinise” the Schr¨odinger algebra we made use of the framework of central extensions in terms of cocycles and coboundaries, eventually arriving at a classification of the ways in which the algebra Ls(d) ⊕ CM ⊕ C∆ may be formed. It is expected that the expression of the Schr¨odinger algebra in terms of a semisimple subalgebra acting on an abelian subalgebra, given by the Levi decomposition, plays a more prominent role in these results than was made use of. It may be the case that some of these results follow from more general properties of this Levi decomposition not yet investigated. References [Car05]

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