STRUCTURE AND REPRESENTATIONS OF CONFORMAL ALGEBRAS

ALEXANDER RETAKH

Introduction

0.1. In the last two decades vertex algebras have been an important tool in such diverse subjects as representations of infinite-dimensional algebras and the theory of finite groups [15, 19].

Roughly speaking, a vertex algebra is a space V such that to each element of V there corresponds a formal distribution, i.e. an element of End V [[z^{±1}]]. (Note that any algebraic operation performed on the space of formal distribution will have to involve more than one variable, as the distributions are power series in both z and z^{−1}.) Two distributions a(z) and b(w) must be local, that is, commute outside the diagonal of the zw-plane.

0.2. The first definition of vertex algebras was given in [6] and is rather involved. With time a need for an algebraic formalism for vertex algebras became clear. Since local formal distributions are in some sense meromorphic, it is reasonable to look at the “singular” part first. Such an approach was emphasized in [26, 27] and especially [19], where this theory was fully developed (see also [4, 23] for geometric counterparts).

This setting is quite general; for an algebra A, consider the following operation on formal distributions over A:

(0.1) a(z) ◯^{n} b(z) = Res_{w=0} a(w)b(z)(w − z)^{n}, a(z), b(z) ∈ A[[z^{±1}]], n ∈ ℤ_{≥0}.

Two formal distributions are local if a finite number of their products (0.1) is nonzero. The formalization of this definition leads to the concept of a conformal algebra (Definition 1.1). Then a vertex algebra is defined as having two related structures: that of a Lie conformal algebra and a left symmetric differentiable algebra [2].

0.3. Conformal algebras also have an intriguing connection to Hamiltonian formalism in the theory of nonlinear evolutionary equations [1, 34]. In fact the first appearance in the literature of conformal-like structures predates the discovery of vertex algebras and comes from the calculus of variations [17]. However, this subject is outside the scope of this survey.

0.4. This survey is dedicated to the study of conformal algebras and their representations. Other expository papers on the subject appeared in the past, in particularly, [20] and [35]; however, there have been new developments in the field since their publication.

Partially supported by the NSF.

We begin by defining conformal algebras and discussing their general properties in Chapter 1. It also contains several basic examples. Chapter 2 develops the theory of representations of conformal algebras. The bulk of this survey, Chapter 3, is dedicated to the very important conformal algebras gc_{n} and Cend_{n} and their subalgebras (roughly, they are the analogs of matrix algebras). We conclude with conjectures and open questions in Chapter 4.

Most of the results here appeared elsewhere; however, several remarks are new. Throughout this survey the base field is ℂ.

1. Basic Definitions and Examples

1.1. We begin with the formal definition of a conformal algebra.

Definition 1.1. [19] A conformal algebra C is a ℂ[∂]-module endowed with bilinear operations ◯^{n} : C ⊗ C → C, n ∈ ℤ_{≥0}, such that for any a, b ∈ C

(1) (locality axiom) a ◯^{n} b = 0 for n > N(a, b) (N(a, b) is called the order of locality of a and b);

(2) (Leibniz rule) ∂(a ◯^{n} b) = (∂a) ◯^{n} b + a ◯^{n} (∂b);

(3) (∂a) ◯^{n} b = −na ^{n−1} b.

A more succinct way to present operations in a conformal algebra C is via the so-called λ-product. Let λ be a formal variable. Define the map C ⊗ C → ℂ[λ] ⊗ C as

(1.1)  a_{λ}b = ∑_{n} \frac{λ^{n}}{n!} a ◯^{n} b, a, b ∈ C, n ∈ ℤ_{≥0}.

We then arrive at an alternative definition of a conformal algebra: this is a ℂ[∂]- module with a bilinear λ-product satisfying analogs of axioms (2) and (3) (a quick exercise is to deduce them explicitly!). Clearly locality is automatic here; after all, the λ-product produces a polynomial in λ. The λ-product works extremely well in calculations; we will see some evidence of this below.

1.2. A set of mutually local formal distributions closed with respect to product ◯^{n} and ∂_{z} is a conformal algebra. Thus conformal algebras provide an algebraic formalism for algebras of local formal distributions. Conversely, every conformal algebra can be made into an algebra of formal distributions. This is the procedure:

Let C be a conformal algebra. For each integer n, consider the linear space Â(n) isomorphic to C. The element corresponding to a ∈ C is denoted â(n). Let Â = ⨁_{n∈ℤ} Â(n) and let E be the subspace of Â spanned by the elements of the form (∂a)(n) + na(n − 1) for all a ∈ C, n ∈ ℤ. The quotient space Coeff C = Â/E is the coefficient algebra of C. The image â(n) in Coeff C is denoted a(n).

It remains to introduce the operation on Coeff C. The following formula holds for any two formal distributions a and b:

a(m)b(n) = ∑_{j≥0} \binom{m}{j}(a ^{j} b)(m + n − j).

Now we take it as the definition of the product in Coeff C. It follows that C is isomorphic to an algebra of formal distributions over Coeff C (and ∂ acts properly because we factored out E).

Remark 1.2. The above construction is taken from [31]. [19] follows a somewhat different approach by introducing a conformal structure on Â. (This is more similar to the original construction for vertex algebras in [6].)

Coeff C is universal among all algebras of possible coefficients. Namely, let B be an algebra such that there exists a homomorphism C → B[[z, z^{−1}]]. Then there exists a unique homomorphism ϕ : Coeff C → B such that the diagram

Coeff C[[z, z^{−1}]] ⟶^{ϕ} B[[z, z^{−1}]]

↖ ↗

C

commutes.

1.3. Now we can define Lie and associative conformal algebras.

Let X be a variety of algebras (Lie, associative, commutative, etc). Then we say that C is X conformal if Coeff C is X (or more rigorously lies in X).

This definition is in a sense unsatisfactory: it refers to another object and a nonconformal one at that. An improvement would be to define a variety of conformal algebras directly. So, assume that X is defined by identities {f_{α}} (e.g. the Jacobi identity and anti-commutativity for Lie algebras). Then one can produce identities {g_{α}} such that a conformal algebra C satisfies them if and only if C is X conformal. The algorithm for the construction of such identities g_{α} can be found in [19]; a more detailed discussion appears in [24].

Conformal identities look much better in their λ-form. To provide a few examples, the conformal law of associativity is (for any a, b, c ∈ C)

(1.2)  a_{λ}(b_{μ}c) = (a_{λ}b)_{λ+μ}c,

and anti-commutativity and the Jacobi identity are, respectively,

[a_{λ}b] = −[b_{−∂−λ}a], (1.3)

[a_{λ}[b_{μ}c]] = [[a_{λ}b]_{λ+μ}c] + [b_{μ}[a_{λ}c]] (1.4)

(we denote the λ-product in a Lie conformal algebra as [ _{λ} ] to emphasize its relation to the ordinary Lie bracket. That is, for a Lie conformal algebra C we have [ _{λ} ] : C ⊗ C → C[λ]).

Remark 1.3. An associative conformal algebra can always be turned into a Lie conformal one. The λ-bracket is defined as

[a_{λ}b] = a_{λ}b − b_{−λ−∂}a.

This allows us to abuse the language sometimes and speak of Lie subalgebras of an associative conformal algebra C without mentioning that we first turn C into a Lie conformal algebra as above. The same goes for embeddings of Lie conformal algebras into associative ones etc.

At this point a reasonable question would be, If we have something like “conformal varieties”, can we also get free conformal algebras? In such generality, the answer is negative. Indeed, a free object must map onto any other, and for conformal algebras this means that the orders of locality for generators in a free conformal algebra would be unbounded. However, if we restrict the orders of locality for generators beforehand, we can construct a free conformal algebra. Namely, let S be a set of letters with a given function N : S × S → ℤ_{≥0}. Consider the category of X conformal algebras generated by S such that for any a, b ∈ S and n > N(a, b), a ◯^{n} b=0. This category does possess the universal object called the free conformal algebra corresponding to the function N. For details, see [31] or [5].

The above discussion shows that a conformal algebra can be described as universal in some category only if all objects in this category satisfy given locality conditions. With this in mind, one can ask if there exist universal enveloping algebras for Lie conformal algebras. This turns out to be true for some algebras (e.g. finite) but in general there exist Lie conformal algebras that can not be embedded into associative conformal algebras (in the sense of Remark 1.3), see [32].

1.4. The paragraphs above dealt with “universal algebra”; we continue with the definitions from the structure theory.

A subalgebra of a conformal algebra C is a ℂ[∂]-submodule of C closed with respect to all the operations ◯^{n} . An ideal of C is a ℂ[∂]-submodule I such that for all n, C ◯^{n} I ⊂ C and I ◯^{n} C ⊂ C.

A conformal algebra C is simple if its only ideals are C and 0.

A conformal algebra C is nilpotent of order d if for any n_{1}, n_{2}, …, n_{d−1} ∈ ℤ_{≥0}, C ^{n_{1}} C … ^{n_{d−1}} C = 0. (There is an abuse of notation going on: the product of d elements of C is not defined unless we say how the brackets are inserted. Here we mean that the product is 0 for any insertion of brackets.)

An associative conformal algebra C is semisimple if it has no nonzero nilpotent ideals.

The derived conformal algebra of a Lie conformal algebra C is C′ = ∑_{n} C ◯^{n} C. As usual we set C^{(1)} = C′, C^{(m+1)} = (C^{(m)})′, m ≥ 1, and say that C is solvable if C^{(m)} = 0 for some m. A Lie conformal algebra is semisimple if it has no nonzero solvable ideals.

One can go on and define, e.g., prime conformal algebras, various conformal radicals, and so on, but we do not need them in this survey.

1.5. Now we turn to examples of conformal algebras.

The first example shows that every “ordinary” algebra can be made conformal.

Example 1.4. Let B be an algebra. Consider the affinization B[t^{±1}] of B and the collection F ⊂ B[t^{±1}][[z, z^{−1}]] of formal distributions of the form

(1.5)  b̃ = ∑_{m∈ℤ} bt^{m}z^{−m−1}, b ∈ B.

We claim that the module Cur B = ℂ[∂] ⊗ F is a conformal algebra. For this we have only to show that all elements of Cur B are pairwise mutually local.

It is easy to see that for any a, b ∈ B,

ã ◯^{0} b̃ = ãb,  ã ◯^{n} b̃ = 0, n > 0

(where the products ◯^{n} are understood in the sense of (0.1)), and as for the rest of the elements of Cur B, their mutual locality follows from axioms (2) and (3).

Cur B is called the current algebra over B.

Remark 1.5. It was easy to check mutual locality for all elements of Cur B; however, this might not be so for an arbitrary collection of formal distributions. Fortunately, there is a way out for Lie and associative algebras.

Lemma 1.6 (Dong’s Lemma). Let a, b, and c be pairwise mutually local formal distributions over a Lie or associative algebra. Then for all n ∈ ℤ_{≥0} the formal distributions a ◯^{n} b and c are mutually local.

In particular, Dong’s lemma implies that for Lie and associative conformal algebras we have only to check that the generators are local.

For associative algebras, there exists a generalization of current algebras:

Example 1.7. Let B be an associative algebra with a locally nilpotent derivation δ. Consider its (localized) Ore extension B[t^{±1}; δ] and the collection F ⊂ B[t^{±1}; δ][[z, z^{−1}]] of formal distributions of the type (1.5). For a, b ∈ B, we have

ã ◯^{n} b̃ = (−1)^{n}{ \atop aδ^{n}(b)}

(or ã_{λ}b̃ = { \atop ae^{−λδ}b} in the λ-notation).

The conformal algebra Diff B = ℂ[∂] ⊗ F is called the differential algebra over B.

In the Lie case, the smallest non-current conformal algebra is the Virasoro conformal algebra:

Example 1.8. Consider the Lie algebra Vect ℂ^{×} of regular vector fields on ℂ^{×}. It is well known that the fields L_{n} = −t^{n+1}∂_{t}, n ∈ ℤ, form a basis of Vect ℂ^{×}. The formal distribution L(z) = ∑_{n} L_{n}z^{−n−2} is local with itself:

[L_{λ}L] = (∂ + 2λ)L.

The conformal algebra Vir = ℂ[∂] ⊗ L is called the Virasoro conformal algebra.

Remark 1.9. Vect ℂ^{×} is the algebra of infinitesimal conformal transformations of ℂ^{×}. This explains the choice of the term “conformal” since Vir is the smallest non-trivial (i.e. non-current) conformal algebra.

Of course, the “ordinary” Virasoro algebra is not Vect ℂ^{×} but its central extension. However, one can also construct the central extension V̂ir of Vir such that Coeff V̂ir is the Virasoro algebra:

Example 1.10. As a vector space V̂ir = Vir ⊕ℂc (here ∂c = 0) and the λ-brackets are

[L_{λ}L] = (∂ + 2λ)L + \frac{λ^{3}}{12} c, [L_{λ}c] = 0, [c_{λ}c] = 0.

In the same vein, for a simple finite-dimensional Lie algebra 𝔤, one can construct the central extension { \atop Cur 𝔤} of Cur 𝔤 such that Coeff { \atop Cur 𝔤} = 𝔤̂. Namely, { \atop Cur 𝔤} is generated by elements g̃, g ∈ 𝔤, and c (again, ∂c = 0) such that

[g̃_{λ}h̃] = { \atop [g, h]} + λ(g|h)c, [g̃_{λ}c] = 0, [c_{λ}c] = 0.

1.6. For brevity we say that a conformal algebra is finite if it is finite as a ℂ[∂]- module.

A current algebra Cur B is simple (respectively, finite) if and only if B is simple (respectively, finite). Thus, we already know a number of (admittedly not very interesting) examples of finite simple conformal algebras. Are there others?

In the Lie case, the Virasoro conformal algebra is also simple and finite but this is it:

Theorem 1.11. (1) [11] Let C be a finite simple Lie conformal algebra. Then C is isomorphic to either Vir or a current algebra Cur 𝔤 over a simple finite-dimensional Lie algebra 𝔤.

(2) [20] Let C be a finite simple associative conformal algebra. Then C is isomorphic to a current algebra Cur Mat_{n}(ℂ).

The first part of Theorem 1.11 is proved by the careful study of the Lie algebra of non-negative coefficients of C. When completed with respect to its natural topology, this algebra becomes linearly compact and then, after some additional work, one applies the Cartan–Guillemin theorem to obtain the complete classification.

The second part follows from the first via standard algebraic techniques.

Remark 1.12. As we have just seen, in the conformal setting the (centerless) Virasoro algebra and the Laurent extensions of finite-dimensional simple Lie algebras appear as coefficient algebras of finite simple Lie conformal algebras. In the nonconformal universe, however, these belong to two very distinct worlds of Cartan and affine Kac–Moody algebras.

We can extend Theorem 1.11 to the semisimple case but it is not as straightforward as one may think:

Example 1.13. For a finite-dimensional Lie algebra 𝔤, the ℂ[∂]-module Vir ⊕ Cur 𝔤 carries the following conformal structure:

[L_{λ}L] = (∂ + 2λ)L, [g̃_{λ}h̃] = { \atop [g, h]}, [L_{λ}g̃] = (∂ + λ)g̃.

So, we obtain the semidirect product of Vir and Cur 𝔤. When 𝔤 is semisimple, this semidirect product is semisimple too.

This, however, is the only surprise in the classification of finite semisimple Lie conformal algebras:

Theorem 1.14. (1)[11] Let C be a finite semisimple conformal Lie algebra. Then C is isomorphic to a direct sum of copies of finite simple Lie conformal algebras and semidirect products of Vir and Cur 𝔤 for semisimple finite-dimensional 𝔤’s.

(2)[20] A finite semisimple associative conformal algebra is isomorphic to a direct sum of copies of simple associative conformal algebras.

The second part of this theorem seems to be a weak version of the Artin– Wedderburn theorem: in the end, all we get is matrices. However, from the representation-theoretic point of view, current algebras over matrices and, in general, finite conformal algebras are not the right analogs of ordinary matrices. We discuss this in the next chapter.

2. Representation Theory of Conformal Algebras

Defining a module M over a conformal algebra C is easy and we now have a choice of two approaches: either modify Definition 1.1 by considering the products C ⊗ M → M or define a module of formal distributions over an algebra of formal distributions imitating (0.1). However, it is even better to take a more general approach and start with the definition of a conformal linear map. In particular, this will help us to construct certain representations later on.

2.1. Let M and N be two ℂ[∂]-modules. A conformal linear map from M to N is a ℂ-linear map ϕ : M → ℂ[λ] ⊗ N, denoted ϕ_{λ} : M → N, such that ∂ϕ_{λ} − ϕ_{λ}∂ = −λϕ_{λ}. As above, we can define the operators ϕ ◯^{n} via ϕ_{λ}m = ∑_{n}(λ^{n}/n!)ϕ ◯^{n} m.

We can “differentiate” conformal linear maps by putting (∂ϕ)_{λ} = −λϕ_{λ}.

The space of all conformal linear maps from M to N is denoted Chom(M, N). It carries a natural structure of a ℂ[∂]-module. When M = N, we can also define the products ψ ◯^{n} ϕ of conformal linear maps by applying the conformal law of associativity (1.2). However, in general this does not make Chom(M, M) into a conformal algebra as the locality condition fails. For instance, let M be of infinite rank and ϕ ∈ Chom(M, M) be such that for any n there exists u_{n} ∈ M whose order of locality with ϕ is n. Put v_{n} = ϕ ◯^{n} u_{n}. Let ψ ∈ Chom(M, M) be such that ψ ◯^{0} v_{n} ≠ 0 for all n. Then for any n, (ψ ◯^{n} ϕ) ◯^{n} u_{n} ≠ 0 and thus ψ and ϕ are not local.

On the other hand, if M is of finite rank, any two elements of Chom(M, M) are local with each other. In this case we denote the associative conformal algebra Chom(M, M) as Cend M.

Cend M with a Lie conformal bracket (see Remark 1.3) is denoted gc M.

To simplify notations, we also denote Cend ℂ[∂]^{n} and gc ℂ[∂]^{n} as Cend_{n} and gc_{n}, respectively.

2.2. Now we will define a module over a conformal algebra.

Definition 2.1. A module M over a Lie or associative conformal algebra C is a ℂ[∂]-module endowed with an operation C ⊗ M → ℂ[λ] ⊗ M such that for any a, b ∈ C and v ∈ M,

(1) (∂a)_{λ}v = −λa_{λ}v, a_{λ}∂v = (∂ + λ)(a_{λ}v);

(2) a_{λ}(b_{μ}v) = [a_{λ}b]_{λ+μ}v + b_{μ}(a_{λ}v) if C is Lie;

a_{λ}(b_{μ}v) = (a_{λ}b)_{λ+μ}v if C is associative.

Simply put, M is a C-module if there exists a map C → Chom(M, M) of conformal algebras that satisfies a version of the Jacobi identity or associativity.

As usual we can define a submodule, an irreducible module (contains no nontrivial submodules), an indecomposable module (does not split into a direct sum of non-trivial submodules), etc. We call a module finite if it is finite over ℂ[∂].

Remark 2.2. If C is a Lie conformal algebra and M and N are modules over C, Chom(M, N) also carries a natural structure of a C-module. Namely, we set

(a_{λ}ϕ)_{μ}m = a_{λ}(ϕ_{μ−λ}m) − ϕ_{μ−λ}(a_{λ}m), a ∈ C, ϕ ∈ Chom(M, N), m ∈ M.

Then we can define the contragradient C-module U^{∗} = Chom(M, ℂ), where ℂ stands for the trivial C-module (with the trivial action of ∂).

For a finite M, we also can define M ⊗ N = Chom(M^{∗}, N).

As in Chapter 1, one can show that a module over a conformal algebra C can be always viewed as a module of formal distributions over a “coefficient module” which, in turn, is a module over Coeff C.

2.3. Below we construct modules for finite simple conformal algebras (cf. Theorem 1.11).

Example 2.3. For any Δ, α ∈ ℂ consider the space of densities F (Δ, α) = ℂ[t, t^{−1}]e^{−αt}(dt)^{1−Δ}. This is naturally a module over Vect ℂ^{×}; it is irreducible whenever Δ ≠ 0.

The formal distribution m(z) = ∑_{n}(t^{n}e^{−αt}(dt)^{1−Δ})z^{−n−1} spans the module M(Δ, α) over the Virasoro conformal algebra Vir with the action induced from that of Vect ℂ^{×} on F (Δ, α). Explicitly, L_{λ}m = (∂_{z} + α + Δλ)m. Again, whenever Δ ≠ 0, M(Δ, α) is irreducible.

In fact, the following is true:

Theorem 2.4. [9] Any non-trivial irreducible finite module over the Virasoro conformal algebra is isomorphic to M(Δ, α) with Δ ≠ 0.

Finite indecomposable modules over Vir were studied in [10]. Complete reducibility does not hold here. In fact, by classifying finite central extensions of M(Δ, α), one can arrive directly at the classification of central extensions of certain modules over regular vector fields on ℂ. This is an example of a connection between the cohomology of conformal algebras [3] and the cohomology of infinite-dimensional Lie algebras [14, 13, 18].

Example 2.5. It is easy to construct a module over a current algebra. Let A be an algebra and U an A-module. Then Cur A acts on the module Ũ = ℂ[∂] ⊗ U with the natural action ã_{λ}(1 ⊗ u) = 1 ⊗ au, a ∈ A, u ∈ U.

A companion result to Theorem 2.4, also proved in [9], states that a non-trivial irreducible finite module over Cur 𝔤 for a finite-dimensional semisimple Lie algebra 𝔤 is of the form Ũ for a non-trivial irreducible finite-dimensional 𝔤-module U. We will see below (Theorem 3.10) that an even more general result holds for associative unital algebras: every module over such an algebra is of the form Ũ (and Ũ is irreducible if and only if U is).

3. Cend_{n} and gc_{n}

3.1. Here we present the explicit constructions of the conformal algebras Cend_{n} and gc_{n}.

Every conformal endomorphism of the module ℂ[∂] is determined by the image of ∂, thus roughly speaking, Cend_{1} is isomorphic to ℂ[∂] ⊗_{ℂ} ℂ[∂] (where the second component is responsible for the image and the first, for the ℂ[∂]-module structure of this conformal algebra). In the case of ℂ[∂]^{n} = ℂ[∂]⊗ℂ^{n} we have also to account for the action of Mat_{n}(ℂ) on ℂ^{n}.

However, to get more explicit expressions, it is perhaps better to go down to the level of coefficients.

The coefficient algebra of Cend_{n} or gc_{n} is the algebra Mat_{n}(D(ℂ^{×})) viewed as either an associative or Lie algebra. (Here D(M) denotes the algebra of differential operators on M.) The formal distributions that span Cend_{n} as a ℂ[∂]-module (here and further, we simply write ∂ for ∂_{z}) are

J_{A}^{m} = ∑_{n∈ℤ} At^{n}(−∂_{t})^{m}z^{−n−1}, A ∈ Mat_{n}(ℂ), m ∈ ℤ_{≥0}.

The action on ℂ[∂]^{n} arises from the standard action of Mat_{n}(D(ℂ^{×})) on the space ℂ^{n}[t^{±1}]. Namely, for v ∈ ℂ^{n} let ṽ = ∑_{n} vt^{n}z^{−n−1}. Then

(3.1)  J_{A}^{m}_{λ}ṽ = (∂ + λ)^{m}Av.

We call this action canonical.

Remark 3.1. We can tweak the above construction by putting

J_{A}^{m}_{λ}ṽ = (∂ + λ + α)^{m}Av, α ∈ ℂ.

More explicitly, we consider the action of Mat_{n}(D(ℂ^{×})) on the space ℂ^{n}[t^{±1}]e^{−αt} and then pass to the formal distributions. Notice, though, that we still get an action of Cend_{n} on ℂ[∂]^{n}. We denote this representation E_{n}^{α}.

In fact, this action comes from the automorphism of Cend_{n} sending the generator J_{A}^{1} to J_{A}^{1} + αJ_{Id}^{0}, where Id is the identity matrix.

Another way to obtain the explicit presentation of Cend_{n} is to use the language of differential conformal algebras (see Example 1.7). (This is more in tune with our statement that Cend_{1} = ℂ[∂]⊗ℂ[∂].) Here we have Cend_{n} = Diff(ℂ[∂_{t}]⊗Mat_{n}(ℂ)) for the derivation ad t = [t, ·].

Actually, an easy computation shows that both explicit presentations above produce the same algebra of formal distributions; we just get two different bases, {J_{A}^{m}} and \{{ \atop f(∂_{t})}\}. Though the formula for the action of the second basis on ℂ[∂]^{n} does not look as good as (3.1), it still has merits (see below).

Remark 3.2. The coefficient algebra of gc_{n} can be embedded into the algebra ĝl_{∞} (a central extension of the algebra of infinite-dimensional matrices with finitely many nonzero diagonals). This relation is used in the study of finite modules of gc_{n} but its precise nature–as well as its connection to vertex algebras–is outside the scope of this survey.

3.2. It is clear from the construction that Cend_{n} contains Cur Mat_{n}(ℂ), i.e. that every simple finite conformal algebra is contained in Cend_{n} for some n. And, for every simple finite-dimensional Lie algebra 𝔤, Cur 𝔤 embeds into gc_{n} for some n.

The construction of Cend_{n} (and thus gc_{n}) also implies that gc_{n} contains the Virasoro conformal algebra Vir. Moreover, for every α ∈ ℂ, the conformal subalgebra of gc_{n} generated by L_{α} = J_{Id}^{1} + αJ_{Id}^{0} is isomorphic to Vir (cf. Remark 3.1).

It is more useful here to pass to the basis \{{ \atop f(∂_{t})}\} of gc_{n}. In the above notation, L_{α} = ˜∂_{t}+∂˜1+α˜1. We can go further and consider the element L_{α,β} = ˜∂_{t}+β∂˜1+α˜1 for any β ∈ ℂ. The subalgebra generated by L_{α,β} is also isomorphic to Vir.

This gives us a family of embeddings θ_{α,β} : Vir ↪ gc_{1}. It is not difficult to show that every embedding of Vir into gc_{1} is of this form. Moreover, a map θ_{α,β} and the canonical gc_{1}-action on ℂ[∂] establish the isomorphism ℂ[∂] ≃ M(1−β, α) (see Example 2.3). Thus we have just classified all Vir modules of rank one.

Remark 3.3. A direct classification of all embeddings of Vir into gc_{n} for an arbitrary n would be of great interest. Among other things it would imply the classification of Virasoro elements in gc_{n} and the complete description of finite Vir modules.

3.3. We essentially view conformal algebras Cend_{n} and gc_{n} as analogs of matrix algebras. Since the theory of finite associative algebras is much simpler than its Lie counterpart, we should first focus on the study of Cend_{n}. There are three directions here: the representation theory, a purely algebraic description of Cend_{n} (i.e. the first cornerstone for the analog of Artin–Wedderburn), and a more detailed study of its subalgebras. The latter approach, of course, is a dead end when ordinary matrix algebras are concerned but we already saw that Cend_{n} possesses some interesting subalgebras.

3.4. We begin by focusing on a possible analog of the Artin–Wedderburn theorem.

The biggest problem is that unlike Mat_{n}(ℂ), Cend_{n} is not of finite rank. This suggests to develop a concept of growth for conformal algebras. In analogy with the ordinary theory [25], we define the Gelfand–Kirillov dimension of a finitelygenerated conformal algebra C as (3.2)

GKdim C = lim sup_{r→∞} \frac{log rk_{ℂ[∂]}(ℂ[∂]-span of products of ≤ r generators of C)}{log r}.

By locality all the ranks in (3.2) are finite and, since the function we consider here is monotone, GKdim C is well defined. It possesses the standard properties: GKdim is does not depend on the choice of the generating set, GKdim of a subalgebra or a quotient algebra does not exceed that of the algebra, etc.

For any differential conformal algebra Diff B, GKdim Diff B = GKdim B. In particular, GKdim Cend_{n} = 1.

Remark 3.4. In general, for an associative conformal algebra C, GKdim Coeff C ≤ GKdim C +1 [28] and the inequality is sometimes strict (e.g. when C is torsion). It is still an open question if the equality is always reached for a torsion-free conformal algebra.

So far we can say that Cend_{n} is simple, of GKdimension 1, and differential. The latter property, however, refers to the coefficient algebra and we wish to obtain a description of Cend_{n} in purely conformal terms.

Mimicking the ordinary algebra, we have to start by defining unital conformal algebras. An ordinary unital algebra contains the field (i.e. a subalgebra of dimension one) whose action is nonzero. The only non-trivial conformal algebra of rank one is Cur ℂ and, moreover, every Cur ℂ module M splits as M = M^{0} ⊕ M^{1}, where ˜1 ◯^{0} = Id_{M^{1}} and the action on M^{0} is zero (both facts are not hard to show). Hence, we will call an associative conformal algebra C unital if it contains Cur ℂ and if for the resulting action of Cur ℂ, C = C^{1}.

The element ˜1 is called a conformal identity.

Remark 3.5. A more rigorous name would be a left conformal identity as we have only the left action of Cur ℂ on C. And, unlike, an ordinary (two-sided) identity, a conformal identity is not unique.

A differential algebra Diff B can be always made unital by adjoining identity to B. (It is still unknown if one can adjoin a conformal identity to a torsion-free conformal algebra.) The converse is almost true.

Let the left annihilator of C be the set {a ∋ C | a_{λ}C = 0}.

Theorem 3.6. [28] An associative conformal algebra C with the zero left annihilator is differential, C = Diff B. Also, if C is finitely generated, then so is B.

(The main part of the proof is to show that the conformal structure of C is encoded by the zeroth coefficients and ˜1(1). It immediately follows that the coefficient algebra is an Ore extension.)

Thus a simple unital conformal algebra is always differential. Utilizing the classification of algebras of GKdimension 1 [33], we can finally obtain the algebraic description of Cend_{n}:

Theorem 3.7. [28] A simple unital associative conformal algebra of Gelfand– Kirillov dimension 1 is isomorphic to Cend_{n}.

Remark 3.8. The proof of Theorem 3.7 can be extended and yield the classification of unital semisimple conformal algebras of GKdimension 1. Namely, such an algebra always embeds into a direct sum of Cend_{n} and a current algebra over a semiprime algebra of zero or linear growth [30]. It would be very difficult to classify all such embeddings: for instance, there exists a prime non-current subalgebra of a current algebra of linear growth [30].

3.5. Now we turn to the representation theory of Cend_{n}.

Just as in the case of Cur ℂ-modules, a module M over a unital conformal algebra C always splits, M = M^{0} ⊕ M^{1}. We need only to study the structure of M^{1}. As in the proof of Theorem 3.6, the action of ˜1 gives a certain rigidity to M^{1}. Namely, M^{1} is filtered by the submodules annihilated by ˜1 ◯^{n} . If C is differential, i.e. C = Diff B (with a derivation δ), the lowest non-trivial component of this filtration can be made into a B-module that completely determines the structure of M^{1}.

Conversely, a (unitary) B-module V gives rise to a C-module Ṽ = ℂ[∂]⊗V with the action

(3.3)  ã_{λ}ṽ = ∑_{j} ∂^{j}({ \atop δ^{j}(a)v}), where ṽ = 1 ⊗ v, v ∈ V, and a ∈ A.

Example 3.9. A finite irreducible module E_{n}^{α} over Cend_{n} constructed in Remark 3.1 has the form E_{n}^{α} = ℂ̃^{n}, where ∂_{t} acts on ℂ^{n} as α.

This example can be generalized by considering modules U ⊗ ℂ^{n} over ℂ[∂_{t}] ⊗ Mat_{n}(ℂ) with ∂_{t} acting on U as α ∈ End(U). Thus we obtain a Cend_{n}-module

E_{n}^{α}(U) = { \atop U ⊗ ℂ^{n}}.

Call a C-module M unitary if M = M^{1}. The discussion above implies that there is a bijection between unitary C-modules and unitary B-modules. We can make this statement more precise: consider the category Rep C whose objects are unitary C-modules and morphisms are homomorphisms that commute with a ◯^{n} for every a ∈ C, n ∈ ℤ_{≥0}. Then

Theorem 3.10. [29] Rep B ≃ Rep C.

As the above equivalence is constructed explicitly, we easily deduce that irreducibles correspond to irreducibles, indecomposables to indecomposables, and that an extension of C-modules arises from an extension of corresponding B-modules.

The only concept from representation theory that does not automatically survive is faithfulness: if Ṽ is faithful, we can only conclude that Ann V does not contain any nonzero δ-stable ideals.

Remark 3.11. It would be more useful to define a sort of “conformal category,” i.e. to define Rep C for any conformal algebra with the morphisms also carrying a conformal structure. So far the attempts at such definition have been unsuccessful.

Corollary 3.12. Finite irreducible Cend_{n}-modules are of the form E_{n}^{α}. Finite indecomposable Cend_{n}-modules are of the form E_{n}^{α}(U) for an indecomposable α ∈ End(U).

The above result was first stated in [20]. Another proof was given in [8]. Though more calculation-heavy, it also works for certain non-unital conformal algebras (see below).

3.6. Finite modules over gc_{n} look similar to Cend_{n}-modules; however, the classification methods here are entirely different.

We begin by constructing such modules. Since every representation of Cend_{n} gives rise to a representation of gc_{n}, we already have the family E_{n}^{α}(U) of gc_{n}- modules (see Example 3.9).

Recall that we can also construct contragradient modules E_{n}^{α}(U)^{∗} (see Remark 2.2).

Theorem 3.13. [20, 7] A finite irreducible gc_{n}-module has the form E_{n}^{α} or (E_{n}^{α})^{∗}.

The first step of the proof is to look at certain representations of the coefficient algebra instead. In fact, for a conformal algebra C it suffices to consider modules over the annihilation algebra spanned by ∂ and coefficients a(n), n ≥ 0. Indeed, a C-module M can be viewed as a module over the annihilation algebra. Conversely, a module V over the annihilation algebra such that for each v ∈ V, a(n)v = 0 for n ≫ 0, gives rise to a C-module.

The annihilation algebra of gc_{n} is the direct sum of a one-dimensional Lie algebra and the algebra of matrices of regular differential operators on the line. Since we are interested in finite gc_{n}-modules here, we need to consider only modules with finite-dimensional graded components (i.e. quasifinite). Therefore, the proof of Theorem 3.13 comes down to classifying quasifinite modules over the Lie algebra of regular differential operators on the line. This was achieved in [7] via the technique developed in [21].

It can be shown that modules over the annihilation algebra of gc_{n} are completely reducible [22]. Thus all finite indecomposable modules over gc_{n} are of the form E_{n}^{α}(U) or E_{n}^{α}(U)^{∗}.

3.7. We have already discussed the finite subalgebras of Cend_{n} and gc_{n}. Here we present a continuous family of infinite subalgebras of Cend_{n} acting irreducibly on ℂ[∂]^{n}.

Let P (∂_{t}) be a matrix in Mat_{n}(D(ℂ^{×})). The formal distributions from Cend_{n} whose coefficients lie in Mat_{n}(D(ℂ^{×}))P (∂_{t}) form a subalgebra (in fact, a left ideal) of Cend_{n} denoted Cend_{n,P}. When P (∂_{t}) is non-degenerate, the conformal algebra Cend_{n,P} acts irreducibly on ℂ[∂]^{n}.

By applying first elementary transformations to P and then conjugating Cend_{n,P}, we arrive at an isomorphic subalgebra for P = diag(p_{1}(∂_{t}), …, p_{n}(∂_{t})), where p_{i}(∂_{t}) ≠ 0 are monic and p_{i}|p_{i+1}. Such p_{i}’s are called elementary divisors. Therefore, we obtain a family of non-isomorphic subalgebras of Cend_{n} that act irreducibly on ℂ[∂] and is parametrized by sequences of elementary divisors.

Conjecture 3.14. [20]^{1} A subalgebra of Cend_{n} that acts irreducibly on the standard module ℂ[∂]^{n} is conjugate to either Cur Mat_{n}(ℂ) or Cend_{n,P} for a non-degenerate P.

That all such finite subalgebras are conjugate to Cur Mat_{n}(ℂ) can be deduced from the conformal Cartan–Jacobson theorem [11].

Another particular case of the conjecture is its restriction to unital subalgebras; here the result follows from Theorem 3.7.

1Added in proof. Recently P. Kolesnikov has announced a proof of this conjecture; details are forthcoming.

It is possible to relax the definition of unitality and consider associative conformal algebras that contain Cur ℂ (in this case ˜1 is called a conformal idempotent). For such algebras, the conjecture also holds [36].

However, for arbitrary subalgebras, the only settled case is n = 1. Moreover, here one can classify all subalgebras of Cend_{1} by carefully investigating subalgebra elements of minimal degree (with respect to both bases \{J^{m}\} and \{{ \atop f(∂_{t})}\}), similar to the classical proof that the algebra ℂ[x] is principal [8].

3.8. One of the crucial observations of [8] is that for an associative algebra C and an irreducible C-module M, M ≃ C_{−∂−λ}m|_{λ=α} for some m ∈ M, α ∈ ℂ. Thus we obtain a surjective map from the set of maximal left ideals of C to the set of non-trivial irreducible C-modules (taken up to isomorphism). Hence,

Theorem 3.15. [8] A finite irreducible Cend_{n,P} -module is isomorphic to E_{n}^{α}.

However, the category of representations of Cend_{n,P} is very different from that of Cend_{n}; it is actually wild for P of high degree [16]. This can be seen by constructing the ext-quiver (adapted to conformal algebras) for the finite-dimensional extensions of irreducible Cend_{n,P} -modules classified in [8].

Remark 3.16. If one could define a conformal category (i.e. a category of conformal objects, where morphism carry a conformal structure as well, see Remark 3.11), then perhaps representations of Cend_{n,P} could be described as a deformation of Rep Cend_{n}. Moreover, an analog of the density theorem for endomorphisms in such a category might help in solving Conjecture 3.14.

3.9. Above we passed from Cend_{n} to its subalgebras Cend_{n,P}. In the Lie case, we go from gc_{n} to gc_{n,P} (either by introducing the Lie bracket on Cend_{n,P} or by also considering the subalgebra of formal distributions of gc_{n} with coefficients divisible on the right by P). Thus, we obtain subalgebras of gc_{n} acting irreducibly on ℂ[∂]^{n}.

But there is more. Since gc_{n} is simple, it can be viewed as both the analog of gl_{n} and sl_{n}. What about analogs of other simple Lie algebras? In particular, what are orthogonal and symplectic conformal Lie algebras?

Consider an anti-involution ^{∗} on D(ℂ^{×}): ∂_{t}^{∗} = −∂_{t}, t^{∗} = t. It can be extended to Mat_{n}(ℂ) ⊗ D(ℂ^{×}) by applying ^{∗} to the second component and a matrix antiinvolution to the first. This gives us an anti-involution of Cend_{n}.

Theorem 3.17. [8] Up to conjugation, all anti-involutions of Cend_{n} are of this form.

We continue mimicking the constructions of orthogonal and symplectic Lie algebras. Let σ be an anti-involution on Cend_{n} that arises from a symmetric (resp. skew-symmetric) involution of Mat_{n}(ℂ). The fixed points of −σ form the orthogonal conformal algebra oc_{n} (resp. symplectic conformal algebra spc_{n}). As gc_{n}, both oc_{n} and spc_{n} are simple.

We can go further and define orthogonal and symplectic subalgebras oc_{n,P} and spc_{n,P} of gc_{n,P}. However, this can be done only for hermitian and anti-hermitian P ’s respectively.

Remark 3.18. There exists another construction of oc_{n} and spc_{n} that is more representation-theoretic in spirit. Here we begin by defining a conformal form on a ℂ[∂]-module V: ⟨ , ⟩_{λ} : V ⊗ V → ℂ[λ]. The form is bilinear if

⟨∂v, w⟩_{λ} = −λ⟨v, w⟩_{λ} = −⟨v, ∂w⟩_{λ}, v, w ∈ V

and symmetric (resp. skew-symmetric) if ⟨v, w⟩_{λ} = ⟨w, v⟩_{−λ} (resp. −⟨w, v⟩_{−λ}).

If V is free over ℂ[∂], the form is completely determined by its action on the basis {e_{i}}, i.e. by a matrix P (λ) with entries ⟨e_{i}, e_{j}⟩_{λ}.

A conformal bilinear form gives rise to a map V → V^{∗}, which establishes an isomorphism between V and P (−∂)V^{∗} (for a non-degenerate P).

Thus, though we can not define adjoints with respect to a given form for all operators in Cend_{n}, there is a well-defined adjoint for elements of Cend_{n,P}. (The details here are rather involved, see [8].) Elements that are skew-symmetric with respect to taking the adjoint form up the Lie conformal algebra oc_{n,P} (if the corresponding form is symmetric) or spc_{n,P} (resp. skew-symmetric).

As a companion to Conjecture 3.14, we have

Conjecture 3.19. [35, 8] An infinite subalgebra of gc_{n} that acts irreducibly on the standard module ℂ[∂]^{n} is conjugate to either gc_{n,P}, oc_{n,P}, or spc_{n,P} for a nondegenerate P (if defined).

Remark 3.20. The classification of finite subalgebras of gc_{n} acting irreducibly on ℂ[∂]^{n} is contained in [11].

Remark 3.21. It is worth to emphasize that here we get only the analogs of classical series A_{n}, B_{n}, C_{n}, and D_{n} (though the series are not discrete). There is no place for exceptional Lie conformal algebras in this conjecture.

And indeed, so far all attempts to construct such by the analogs of ordinary methods (Tits–Kantor–Koecher construction, sums of representations of algebras of small rank, etc.) have failed.

The existing evidence for this conjecture covers several important cases. In all cases we assume a presence of an important finite subalgebra:

• [36] If C is a simple Lie conformal algebra that has a faithful finite representation and such that C ⊃ Cur sl_{2}, then C is one of the subalgebras from Conjecture 3.19.

• [12] If C is a subalgebra of gc_{n} that acts irreducibly on ℂ[∂]^{n} and is fixed by the action of L ◯^{n} , n = 0, 1, 2, for the Virasoro element L = ˜∂_{t} + α∂˜1, then C is one of the subalgebras from Conjecture 3.19.

Remark 3.22. The second statement above can be reformulated: the (centerless) Virasoro Lie algebra contains an sl_{2} spanned by L_{−1}, L_{0}, and L_{1}. A subalgebra fixed by this sl_{2} and acting irreducibly on ℂ[∂] is from Conjecture 3.19.

Finally, as far as the study of representations of infinite subalgebras of gc_{n} goes, only few inroads have been made: [7] has the classification of finite representations of subalgebras of gc_{1} containing a Virasoro subalgebra but this is all.

4. Future Developments

4.1. We restate here Conjectures 3.14 and 3.19:

Conjecture 4.1. Let C be an infinite conformal algebra that acts faithfully and irreducibly on ℂ[∂]. Then

• if C is associative, it is isomorphic to Cend_{n,P};

• if C is Lie, it is isomorphic to either gc_{n,P}, oc_{n,P}, or spc_{n,P}.

It is well known that any study of infinite-dimensional Lie algebras of linear growth is in general hopeless; fortunately, there is a natural subclass of affine algebras that gives us a controllable rich theory. Since these algebras are finite from the conformal point of view, the subalgebras of gc_{n} (or, rather, their coefficients) form a good class of algebras of quadratic growth.

More generally, one should study finite subalgebras of gc_{n} as well: this may lead, for instance, to the classification of Virasoro elements (see Remark 3.3) and other important results.

In view of Theorem 3.7 and its generalization in [36], we also propose a generalization of Conjecture 3.14:

Conjecture 4.2. Let C be a simple associative algebra of Gelfand–Kirillov dimension 1. Then C is isomorphic to Cend_{n,P} for some n and P.

4.2. A closely related issue is the study of representations of Cend_{n,P}, gc_{n,P}, oc_{n,P} and spc_{n,P}.

We have already mentioned in Remarks 3.11 and 3.16 that the best strategy here might be to define a conformal category with the space of morphisms carrying a conformal structure and then describe this category in terms of some equivalent data. Though successful for unital conformal algebras (see Theorem 3.10), this task is very hard in general. This is not the only obstacle for such a project. For instance, as mentioned above, there is no bijection between the set of maximal left ideals and irreducible modules–i.e. even if one could define a “conformal kernel,” it would yield less information than its ordinary analog.

In the Lie case, these questions are closely related to the study of representations of ĝl_{∞} and its subalgebras. This is a very important subject for infinite-dimensional representation theory and hopefully, the use of conformal language can move it further.

4.3. Representations of Lie algebras are a source for a lot of combinatorics. What about their conformal counterparts?

The study of subalgebras of gc_{n} that are normalized by the sl_{2} part of a Virasoro element (see Remark 3.22) produced a surprising connection with classical Jacobi polynomials [12]. This seems to be the only deep combinatorial result in the conformal algebra field but we can hope for more: for instance, character formulas for gc_{n} (and, for that matter a good definition of “conformal” characters) and its subalgebras should turn out very interesting from the combinatorial point of view.

4.4. Finally, we should mention a generalization of conformal algebras.

It is clear that instead of ℂ[∂]-modules in Definition 1.1 we can consider modules over ℂ[∂_{1}, …, ∂_{k}] (and take n in ◯^{n} to be a multiindex). A more involved procedure allow us to endow modules over any cocommutative Hopf algebra with a conformallike structure. Such objects are called pseudoalgebras.

So far we have the classification of finite pseudoalgebras, the beginnings of representation theory (both in [1]), and the theory of unital pseudoalgebras [29]. It should be mentioned that simple finite Lie pseudoalgebras arise from either affine algebras or algebras of Cartan type (of any GK dimension), so in a sense this theory unifies the Kac–Moody and the Cartan type sides of infinite-dimensional Lie algebras, cf. Remark 1.12. As far as infinite pseudoalgebras are concerned, some results in [29] (for the associative case only) suggest that their theory is rich and manageable. These objects should be given more attention.

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Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139

   E-mail address: retakh@math.mit.edu