The Segal–Sugawara Construction

Home , Casimir element, Virasoro algebra

The Segal–Sugawara construction

Peter J. Haine January 5, 2020

Contents

1 Introduction 1 1.1 Reminders from previous talks ...... 1 1.2 Talk overview ...... 2

2 Loop algebras & Kac–Moody algebras 2 2.1 Loop algebras ...... 3 2.2 Recollection on bilinear forms & semisimplicity ...... 3 2.3 Kac–Moody algebras ...... 4

3 The Segal–Sugawara construction 5 3.1 Motivating case: the Heisenberg algebra ...... 6 3.2 The Casimir element ...... 8 3.3 The general case ...... 9

References 11

1 Introduction 1.1 Reminders from previous talks 1.1.1 Definition. The (complex) Witt algebra is the complex Lie algebra Witt of poly- 1 푖푚휃 d nomial vector fields on S . Explicitly, Witt has generators 퐿푚 ≔ 푖푒 d휃 for 푚 ∈ 퐙 with Lie bracket [퐿푚, 퐿푛] ≔ (푚 − 푛)퐿푚+푛 for all 푚, 푛 ∈ 퐙.

1.1.2. Ignoring regularity issues, the Witt algebra is the complexification of the Lie algebra of the group Diff +(S1) of orientation-preserving diffeomorphisms of the circle.

2 1.1.3. We have that HLie(Witt; 퐂) ≅ 퐂, so there is a 1-dimensional space of central extensions of the Witt algebra.

1 1.1.4 Definition. The (complex) Virasoro algebra Vir is the central extension

1 퐂chg Vir Witt 1 of Witt with generators 퐿푚 for 푚 ∈ 퐙 and a central element chg, and nontrivial Lie bracket given by 푚3 − 푚 [퐿 , 퐿 ] ≔ (푚 − 푛)퐿 + 훿 chg 푚 푛 푚+푛 푚,−푛 12 for all 푚, 푛 ∈ 퐙. We call the central element chg ∈ Vir the central charge. 1.1.5. Again, ignoring regularity issues, the Virasoro algebra is the complexification of the Lie algebra of the Virasoro group Diff̃ +(S1).

1.2 Talk overview 1.2.1. Let 퐺 be a simply-connected, simple, compact Lie group with Lie algebra 픤. Last time we looked at central extensions

1 S1 L̃퐺 L퐺 1 of the loop group L퐺 ≔ C∞(S1, 퐺). The group Diff +(S1) of orientation-preserving diffeomorphisms of the circle acts on L퐺 by precomposition. So we might expect an action of the Virasoro group Diff̃ +(S1) on L̃퐺. Last time we saw that even though there is not an action of Diff̃ +(S1) on L̃퐺, roughly, the Virasoro group acts on any positive energy representation of L̃퐺. However, the Virasoro action on positive energy representation of L̃퐺 is very inexplicit, and we can only guarantee the existence of the Virasoro action up to ‘essential equivalence’, which is not actually an equivalence relation. In particular, the Pressley–Segal Theorem [8, Theorem 13.4.3] does not explicitly explain how the central circle S1 ⊂ L̃퐺 acts. 1.2.2 Goal. The goal of this talk is to explain the Lie algebra version of the Pressley– Segal Theorem, which gives an explicit representation of the Virasoro algebra on any ‘positive energy’ representation of the Kac–Moody algebra L̃픤 associated to a simple Lie algebra 픤 (over the complex numbers). We’ll be able to do this by writing down explicit universal formulas for ‘elements’ of the universal enveloping algebra U(L̃픤) that satisfy the Virasoro relations. The catch is that these universal formulas involve infinite sums, so they do not actually make sense as elements of U(L̃픤), but they do make sense whenever we act on a representation where only finitely many of the terms don’t act by zero (this is the ‘positive energy’ condition).

2 Loop algebras & Kac–Moody algebras

The first thing we need to explain in order to state the Segal–Sugawara construction is what the Kac–Moody algebra L̃픤 is. As the notation suggests, L̃픤 is the Lie algebra analog of the central extension L̃퐺 of the loop group L퐺 (with suitable finiteness hypotheses). Before talking about Kac–Moody algebras, we need to talk about loop algebras.

2 2.1 Loop algebras 2.1.1 Recollection. Let 픤 be a Lie algebra over a ring 푅, and let 푆 be an 푅-algebra. The basechange 픤 ⊗푅 푆 of 픤 to 푆 is the Lie algebra over 푆 with underlying 푆-module the basechange 픤 ⊗푅 푆 of the underlying 푅-module of 픤 to 푆 with Lie bracket extended from pure tensors from the formula

[푋 ⊗ 푠 , 푋 ⊗ 푠 ] ≔ [푋 , 푋 ] ⊗ 푠 푠 . 1 1 2 2 픤⊗푅푆 1 2 픤 1 2 2.1.2 Definition. Let 픤 be a complex Lie algebra. The loop algebra L픤 of 픤 is the Lie algebra ±1 L픤 ≔ 픤 ⊗퐂 퐂[푡 ] , regarded as a Lie algebra over 퐂 (rather than 퐂[푡±1]).

2.1.3 Notation. Let 픤 be a complex Lie algebra, 푋 ∈ 픤, and 푚 an integer. We write

푋⟨푚⟩ ≔ 푋 ⊗ 푡푚 ∈ L픤 .

2.1.4. If {푢푖}푖∈퐼 is a Lie algebra basis for 픤, then

{푢푖⟨푚⟩}(푖,푚)∈퐼×퐙 is a basis for L픤. 2.1.5 Remark. The loop algebra functor

L∶ Lie퐂 → Lie퐂 preserves finite products. 2.1.6 Recollection. A finite dimensional Lie algebra 픤 is simple if 픤 is not abelian and the only ideals of 픤 are 픤 and 0.

2.1.7 Theorem (Garland [3, §§1 & 2]). If 픤 is a simple Lie algebra over 퐂, then

2 HLie(L픤; 퐂) ≅ 퐂 .

In particular, if 픤 is simple there is a 1-dimensional space of central extensions of L픤.

2.2 Recollection on bilinear forms & semisimplicity

2.2.1 Notation. Let 픤 be a complex Lie algebra. We write ad∶ 픤 → End퐂(픤) for the adjoint representation, defined by

ad(푋) ≔ [푋, −] .

2.2.2 Example. A Lie algebra 픤 is abelian if and only if the adjoint representation of 픤 is trivial.

3 2.2.3 Recollection (Killing form). Let 픤 be a finite dimensional Lie algebra. The Killing form on 픤 is the bilinear form

K픤 ∶ 픤 × 픤 → 퐂 (푋, 푌) ↦ tr(ad(푋) ∘ ad(푌)) .

The Killing form is symmetric and invariant in the sense that

K픤([푋, 푌], 푍) = K픤(푋, [푌, 푍]) for all 푋, 푌, 푍 ∈ 픤. 2.2.4 Example. If 픤 is a simple Lie algebra, then every invariant symmetric bilinear form on 픤 is a 퐂-multiple of the Killing form K픤. See [2] for a nice exposition of this fact. 2.2.5 Example. Let 픞 be a finite dimensional abelian Lie algebra over 퐂. Then since the adjoint representation of 픞 is trivial, the Killing form of 픞 is identically zero. Also note that every bilinear form on the underlying vector space of 픞 is an invariant bilinear form on 픞. 2.2.6 Proposition ([9, Chapter ii, Theorems 2 & 4]). Let 픤 be a finite dimensional complex Lie algebra. The following conditions are equivalent: (2.2.6.1) The center of 픤 is trivial. (2.2.6.2) The only abelian ideal in 픤 is 0. (2.2.6.3) The Lie algebra 픤 is isomorphic to a product of simple Lie algebras. (2.2.6.4) Cartan–Killing criterion: the Killing form of 픤 is nondegenerate. If the equivalent conditions (2.2.6.2)–(2.2.6.4) are satisfied, we say that 픤 is semisimple.

2.3 Kac–Moody algebras Now we define the Lie algebra analogue of the central extension L̃퐺 of the loop group L퐺. In addition to the Lie algebra 픤, we need to input an invariant symmetric bilinear form on 픤; the Killing form provides a canonical choice. 2.3.1 Definition. Let 픤 be a Lie algebra over 퐂 with invariant symmetric bilinear form 퐵∶ 픤 × 픤 → 퐂. The Kac–Moody algebra of 픤 with respect to the form 퐵 is the central extension ̃ 1 퐂푐 L퐵픤 L픤 1 with central element 푐 and with Lie bracket extended from the relation

[푋⟨푚⟩, 푌⟨푛⟩]̃ ≔ [푋⟨푚⟩, 푌⟨푛⟩] + 훿 푚퐵(푋, 푌)푐 L퐵픤 L픤 푚,−푛

= [푋, 푌]픤⟨푚 + 푛⟩ + 훿푚,−푛푚퐵(푋, 푌)푐 for all 푋, 푌 ∈ 픤.

4 2.3.2. If {푢푖}푖∈퐼 is a Lie algebra basis for 픤, then

{푢푖⟨푚⟩}(푖,푚)∈퐼×퐙 ∪ {푐} ̃ is a basis for L퐵픤. 2.3.3 Remark. The Kac–Moody algebra L̃픤 is usually denoted by 픤̂ and is also known as the affine Lie algebra of 픤.

2.3.4 Remark. Let 픤1 and 픤2 be complex Lie algebras equipped with invariant symmetric bilinear forms

퐵1 ∶ 픤1 × 픤1 → 퐂 and 퐵2 ∶ 픤2 × 픤2 → 퐂 .

Write 퐵 for the bilinear form on the product Lie algebra 픤1 × 픤2 defined by

퐵((푥1, 푥2), (푦1, 푦2)) ≔ 퐵1(푥1, 푦1) + 퐵(푥2, 푦2) . Then we have a canonical isomorphism

L̃ (픤 × 픤 ) ≅ L̃ 픤 × L̃ 픤 . 퐵 1 2 퐵1 1 퐵2 2

3 The Segal–Sugawara construction

We now have enough of of the background on Lie algebras to give a vague statement of the Segal–Sugawara construction. 3.0.1 Definition. Let 픤 be a Lie algebra over 퐂 and 퐵 an invariant symmetric bilinear ̃ form on 픤. A representation 휌∶ L퐵픤 → End퐂(푉) has positive energy if for all 푣 ∈ 푉 and 푋 ∈ 픤 there exists an integer 푚 > 0 such that

휌(푋⟨푚⟩)푣 = 0 .

3.0.2 Remark. In the theory of Kac–Moody algebras, positive energy representations are more often called admissible. We have chosen the term ‘positive energy’ to align with the loop group terminology. 3.0.3 Theorem (Segal–Sugawara construction, vague formulation). Let 픤 be an abelian or simple Lie algebra over 퐂 and let 퐵∶ 픤 × 픤 → 퐂 be a nondegenerate invariant symmetric bilinear form on 픤. Write Cas퐵(픤) ∈ U(픤) for the Casimir element of 픤 with respect to the bilinear form 퐵. Let ̃ 휌∶ L퐵픤 → End퐂(푉) ̃ be a positive energy representation of L퐵픤 such that ̃ (3.3.4.1) the central element 푐 ∈ L퐵픤 acts by multiplication by a complex number ℓ, (3.3.4.2) and the complex number −ℓ is not equal to tr(ad(Cas (픤))) 휆 (퐺) ≔ 퐵 . 퐵 2 dim(픤)

5 Then there is an explicit action of the Virasoro algebra on 푉 where the central charge chg ∈ Vir acts by multiplication by ℓ dim(픤) . ℓ + 휆퐵(픤) As special cases:

(3.3.4.3) If 픤 is abelian, then 휆퐵(픤) = 0 for any nondegenerate invariant symmetric bilinear form 퐵, and central charge chg ∈ Vir acts by multiplication by dim(픤). (3.3.4.4) If 픤 is simple and 퐵 is the normalization of the Killing form such that the long roots of 픤 have square length 2, then 휆퐵(픤) is a positive integer known as the dual Coxeter number of 픤. 3.0.4. The complex number ℓ in Theorem 3.3.4 is known as the level of the positive energy representation 휌. 3.0.5 Goal. The goal for the rest of the talk is to explain this construction, the Casimir element Cas퐵(픤), and give a better description of the normalized trace 휆퐵(픤) as an eigenvalue of ad(Cas퐵(픤)).

3.1 Motivating case: the Heisenberg algebra As motivation for the Segal–Sugawara construction, we start with the most simple case, where 픤 is the 1-dimensional abelian Lie algebra. Since the constant 휆퐵(픤) will be zero in this case, we can do this without yet introducing the Casimir element. 3.1.1 Definition. The Heisenberg algebra is the Kac–Moody algebra Heis ≔ L̃퐂 of the 1-dimensional abelian Lie algebra 퐂 with respect to the bilinear form 퐂 × 퐂 → 퐂 given by multiplication. 3.1.2. Write 푢 ∈ 퐂 for the element 1, which we regard as a basis for 퐂 as a 1-dimensional abelian Lie algebra. Then the Heisenberg algebra has generators {푐} ∪ {푢⟨푚⟩}푚∈퐙, where 푐 is central and the nontrivial bracket relation is given by

[푢⟨푚⟩, 푢⟨푛⟩] ≔ 훿푚,−푛푚푐 . 3.1.3 Definition. Let 휇, ℏ ∈ 퐂. Write 푢 ∈ 퐂 for the element 1, which we regard as a basis for 퐂 as a 1-dimensional abelian Lie algebra. The Fock representation Fock(휇, ℏ) is the representation of the Heisenberg algebra on the polynomial ring

Fock(휇, ℏ) ≔ 퐂[푥1, 푥2, …] in infinitely many variables, where 푐 ↦ ℏ id 휕 , 푛 > 0 { 휕푥푛 푢⟨푛⟩ ↦ −ℏ푥 , 푛 < 0 { −푛 {휇 id , 푛 = 0 .

6 The following fact about the irreducibility of Fock representations is easy: 3.1.4 Lemma ([6, Lemma 2.1]). Let 휇, ℏ ∈ 퐂. If ℏ ≠ 0, then the Heis-representation Fock(휇, ℏ) is irreducible. 3.1.5. If ℏ = 0, then the constants 퐂 ⊂ Fock(휇, 0) are invariant. 3.1.6 Properties. The following are some important properties of the Fock representations of the Hiesenberg algebra. (3.1.6.1) The elements 푢(0) and 푐 of Heis act by multiplication. (3.1.6.2) For every polynomial 푝 ∈ Fock(휇, ℏ), there exists an integer 푛 ≫ 0 such that 푢⟨푛⟩푝 = 0: let 푛 be any positive such that the variable 푥푛 does not appear in 푝. That is, the Fock representation Fock(휇, ℏ) is ‘positive energy’ in the sense of Definition 3.0.1. (3.1.6.3) For each integer 푛 > 0, the element 푢⟨푛⟩ ∈ Heis acts locally nilpotently on Fock(휇, ℏ). Now we can give the Segal–Sugawara construction for the Fock representations of the Heisenberg algebra. 3.1.7 Construction (Virasoro action of Fock representations). For each integer 푚 ∈ 퐙, define an infinite sum of elements of U(Heis) by

푆 1 퐿푚 ≔ ∑ ∶푢⟨−푗⟩푢⟨푗 + 푚⟩∶ . 2 푗∈퐙

Here, ∶푢⟨−푗⟩푢⟨푗 + 푚⟩∶ denotes the normal ordering on 푢⟨−푗⟩푢⟨푗 + 푚⟩, defined by

푢⟨−푗⟩푢⟨푗 + 푚⟩ , −푗 ≤ 푗 + 푚 ∶푢⟨−푗⟩푢⟨푗 + 푚⟩∶≔ { 푢⟨푗 + 푚⟩푢⟨−푗⟩ , −푗 ≥ 푗 + 푚 .

Explicitly, 1 2 { 푢⟨푛⟩ + ∑ 푢⟨푛 − 푗⟩푢⟨푛 + 푗⟩ , 푚 = 2푛 푆 {2 푗>0 퐿푚 = { {∑ 푢⟨푛 + 1 − 푗⟩푢⟨푛 + 푗⟩ , 푚 = 2푛 + 1 . {푗>0 푆 The operators 퐿푚 are not well-defined elements of U(Heis), but since the Fock repre- 푆 sentations of Heis are positive energy (3.1.6.2), the operators 퐿푚 make sense as operators on Fock(휇, ℏ). 3.1.8 Theorem (Segal–Sugawara for Fock(휇, 1) [6, Proposition 2.3]). Under the repre- 푆 sentation of Heis on the Fock space Fock(휇, 1), the operators 퐿푚 on Fock(휇, 1) satisfy the commutation relation 푚3 − 푚 [퐿푆 , 퐿푆 ] = (푚 − 푛)퐿푆 + 훿 . 푚 푛 푚+푛 푚,−푛 12

7 Hence the assignment

Vir → End퐂(Fock(휇, 1)) 푆 퐿푚 ↦ 퐿푚 chg ↦ id is a Vir-representation with central charge 1.

3.1.9 Remark. To derive the Segal–Sugawara action on Fock(휇, ℏ) for ℏ ≠ 0, let 퐿푚 act 1 푆 by ℏ 퐿푚. 3.1.10 Remark. Gordon’s notes [4] give a nice exposition of the Segal–Sugawara construction for Fock representations and the representation theory of the Virasoro algebra.

3.2 The Casimir element In the general case, the idea is to try to mimic the formulas that we wrote down defining the operators on the Fock representations that satisfy the Virasoro relations. First, we need to explain the ‘Casimir element’ and normalized trace 휆퐵(픤) appearing in Theo- rem 3.3.4. 3.2.1 Recollection. Let 픤 be a finite dimensional Lie algebra over 퐂 and let 퐵 be a nondegenerate invariant symmetric bilinear form on 픤. The Casimir element Cas퐵(픤) of 픤 with respect to the form 퐵 is the element of the universal enveloping algebra U(픤) given by the image of id픤 under the composite

∨ ∼ End퐂(픤) ≅ 픤 ⊗퐂 픤 픤 ⊗퐂 픤 T퐂(픤) U(픤) .

∨ Here the isomorphism 픤 ⊗퐂 픤 ⥲ 픤 ⊗퐂 픤 is the identity on the first factor and the iso- ∨ morphism 픤 ⥲ 픤 induced by the form 퐵 on the second factor, and T퐂(픤) is the tensor algebra of 픤 over 퐂. The following are some key properties that we need to know about the Casimir element:

(3.2.1.1) The Casimir element Cas퐵(픤) is a central element of U(픤). 1 푑 (3.2.1.2) If {푢1, …, 푢푑} and {푢 , …, 푢 } are bases of 픤 that are dual with respect to the bilinear form 퐵 in the sense that

푗 퐵(푢푖, 푢 ) = 훿푖,푗 ,

푑 푖 then Cas퐵(픤) = ∑푖=1 푢푖푢 . (3.2.1.3) Assume that 픤 is simple. Then the Casimir element of the Killing form of 픤 acts by the identity in the adjoint representation. Hence for any nondegenerate invariant symmetric bilinear form 퐵 on 픤, the Casimir element Cas퐵(픤) acts by scalar multiplication in the adjoint representation of 픤. If 퐵 is the normalization of the Killing form on 픤 such that long roots have square length 2, then in the adjoint representation Cas퐵(픤) acts by multiplication by an even positive integer.

8 (3.2.1.4) If 픤 is abelian, then since the adjoint representation of 픤 is trivial, for any nondegenerate invariant symmetric bilinear form 퐵 on 픤 we have ad(Cas퐵(픤)) = 0. In particular, in the adjoint representation Cas퐵(픤) acts by scalar multiplication. Even though there are no Lie algebras that are both abelian and simple, it is important for us that both types of Lie algebras have the property that the Casimir element associated to any nondegenerate invariant symmetric bilinear form acts by scalar multiplication in the adjoint representation. In particular, if 픤 is abelian or simple, then ad(Cas퐵(픤)) only has exactly one eigenvalue. 3.2.2 Definition. Let 픤 be a finite dimensional abelian or simple Lie algebra over 퐂 and let 퐵∶ 픤 × 픤 → 퐂 be a nondegenerate invariant symmetric bilinear form on 픤. Define a complex number 휆퐵(픤) by 1 휆 (픤) ≔ (eigenvalue of ad(Cas (픤))) . 퐵 2 퐵 3.2.3. If dim(픤) > 0, then tr(ad(Cas (픤))) 휆 (픤) = 퐵 , 퐵 2 dim(픤) which aligns with the vague formulation of the Segal–Sugawara construction (Theo- rem 3.3.4). 3.2.4 Example. If 픤 is simple and 퐵 is the normalization of the Killing form on 픤 such that long roots have square length 2, then 휆퐵(픤) is a positive integer (3.2.1.3) known as the dual Coxeter number of 픤. 3.2.5 Example. If 픞 is an abelian Lie algebra, then for any nondegenerate invariant symmetric bilinear form 퐵 on 픞, we have 휆퐵(픞) = 0.

3.3 The general case Now let us try ‘the same’ formula to write down a Virasoro action on positive energy representations of L̃픤 as we did for the Heisenberg algebra. The first modification is that we need to sum over a basis of 픤. 3.3.1 Construction. Let 픤 be a finite dimensional Lie algebra over 퐂 and let 퐵 be a nondegenerate invariant symmetric bilinear form on 픤. Given a positive energy repre- ̃ sentation 휌∶ L퐵픤 → End퐂(푉), for each integer 푚 ∈ 퐙 define

푑 휌 1 푖 푇푚 ≔ ∑ ∑ ∶휌(푢푖⟨−푗⟩)휌(푢 ⟨푗 + 푚⟩)∶ ∈ End퐂(푉) . 2 푖=1 푗∈퐙

휌 Note that even though the formula defining 푇푚 involves an infinite sum, since 휌 is a positive energy representation, for each 푣 ∈ 푉, all but finitely many terms in the sum 휌 휌 defining 푇푚 annihilate 푣. Hence 푇푚 is well-defined as an element of End퐂(푉).

9 We used the letter ‘푇’ instead of ‘퐿’ because the commutation relation is not quite right: 3.3.2 Lemma ([6, Theorem 10.1]). Let 픤 be a finite dimensional abelian or simple Lie algebra over 퐂 and let 퐵 be a nondegenerate invariant symmetric bilinear form on 픤. For every ̃ positive energy representation 휌∶ L퐵픤 → End퐂(푉), we have the following commutation relation in End퐂(푉): 휌 휌 휌 [푇푚, 푇푛 ] = (휌(푐) + 휆퐵(픤))(푚 − 푛)푇푚+푛 푚3 − 푚 + 훿 dim(픤) 휌(푐)(휌(푐) + 휆 (픤)) . 푚,−푛 12 퐵 휌 3.3.3 Idea. The naïve guess that the operators 푇푚 satisfy the Virasoro relations is not correct. However, if we could invert 휌(푐) + 휆퐵(픤), then the operators

1 휌 푇푚 휌(푐) + 휆퐵(픤) would satisfy the Virasoro relations. We can do this provided that the central element ̃ 푐 ∈ L퐵픤 acts by a scalar ℓ on 푉, and ℓ ≠ −휆퐵(픤). 3.3.4 Theorem (Segal–Sugawara construction [6, Corollary 10.1]). Let 픤 be a finite dimensional abelian or simple Lie algebra over 퐂 and let 퐵∶ 픤 × 픤 → 퐂 be a nondegenerate invariant symmetric bilinear form. Let ̃ 휌∶ L퐵픤 → End퐂(푉) ̃ be a positive energy representation of L퐵픤 such that ̃ (3.3.4.1) the central element 푐 ∈ L퐵픤 acts by multiplication by a complex number ℓ,

(3.3.4.2) and ℓ ≠ −휆퐵(픤). 1 푑 Choose bases {푢1, …, 푢푑} and {푢 , …, 푢 } of 픤 that are dual with respect to the bilinear form 퐵. Then the assignment

푑 휌 1 푖 퐿푚 ↦ 퐿푚 ≔ ∑ ∑ ∶휌(푢푖⟨−푗⟩)휌(푢 ⟨푗 + 푚⟩)∶ 2(ℓ + 휆퐵(픤)) 푖=1 푗∈퐙 extends to a Vir-representation on 푉 with central charge

ℓ dim(픤) . ℓ + 휆퐵(픤) 휌 That is, in End퐂(푉), the operators 퐿푚 satisfy the commutation relation

3 휌 휌 휌 푚 − 푚 ℓ dim(픤) [퐿푚, 퐿푛] = (푚 − 푛)퐿푚+푛 + 훿푚,−푛 . 12 ℓ + 휆퐵(픤)

10 푑 푖 3.3.5 Remark. For 푎, 푏 ∈ 퐙, the sum ∑푖=1 푢푖(푎)푢 (푏) is independent of the choice of 휌 basis {푢1, …, 푢푑} of 픤. In particular, the operators 퐿푚 are independent of the choice of basis.

3.3.6 Remark. If ℓ = −휆퐵(픤), then the formulas we wrote down for the Segal–Sugawara 휌 operators 퐿푚 do not make sense, and there is a fundamental difficulty in dealing with the ‘critical level’ ℓ = −휆퐵(픤). At the critical level, the theory seems to resemble the positive characteristic situation rather than the classical one; see [5] for some discussion of this point. 3.3.7 Remark. In light of Remark 2.3.4, the Segal–Sugawara construction can be extended to the case where 픤 is reductive, i.e., 픤 decomposes as a product

픤 ≅ 픞 × 픤1 × ⋯ × 픤푟 , where 픞 is an abelian Lie algebra and 픤1, …, 픤푟 are simple Lie algebras. In this case, the central charge of the resulting Vir-representation is

푟 ℓ dim(픤 ) dim(픞) + ∑ 푖 푖 , ℓ + 휆 (픤 ) 푖=1 푖 퐵푖 푖 where the central element of L̃픞 acts by multiplication by a nonzero complex number and the central element of L̃픤 acts by multiplication by ℓ ∈ 퐂, and ℓ ≠ −휆 (픤 ). This is 푖 푖 푖 퐵푖 푖 rather useful as all of the ‘classical’ Lie algebras are reductive [7, Theorem 5.49]; see [6, Remark 10.3] for details. 3.3.8 Remark. The Segal–Sugawara construction is usually stated with the assumptions that 픤 is simple and 퐵 is the normalization of the Killing form such that the long roots of 픤 have square length 2 (so that 휆퐵(픤) is the dual Coxeter number, often denoted by ℎ∨). This is somewhat unfortunate; because the Killing form of an abelian Lie algebra is trivial, to include the abelian case (and the reductive extension) the ‘usual’ statement needs to be modified to include arbitrary nondegenerate invariant symmetric bilinear forms as in Theorem 3.3.4. 3.3.9 Remark. One of the motivations for the formula for the Segal–Sugawara operators 휌 퐿푚 comes from the theory of vertex algebras. See [1, §3], in particular [1, Proposition 3.3.1], for more details on the relation to vertex algebras.

References

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