QUIVER VARIETIES, AFFINE LIE ALGEBRAS, ALGEBRAS OF BPS STATES, AND SEMICANONICAL BASIS

IGOR FRENKEL, ANTON MALKIN, AND MAXIM VYBORNOV

Abstract. We suggest a (conjectural) construction of a basis in the plus part of the affine Lie algebra of type ADE indexed by irreducible components of certain quiver varieties. This construction is closely related to a stringtheoretic construction of a Lie algebra of BPS states. We then study the new combinatorial questions about the (classical) root systems naturally arising from our constructions and Lusztig’s semicanonical basis.

1. Introduction

1.1. Since the appearance of the first manifestation of the connection between theory of representations of quivers and the structure theory of Lie algebras [Gab72], several authors discovered constructions of Lie algebras arising from the quiver theory. C. M. Ringel constructed the plus part of the simple Lie algebra of type ADE, [Rin90], and a (related) construction of the plus part of any Kac-Moody Lie algebra is implicit in Lusztig’s construction of (quantized) enveloping algebras of Kac-Moody Lie algebras in terms of functions on a class of remarkable affine varieties Λ_{V}, [Lus91, Lus92].

Inspired by Ringel and Lusztig, the present authors came up with a construction of the plus part 𝔫̃_{+} of the affine Lie algebra of type ADE in terms of indecomposable representations of quivers [FMV01].

A similar construction is suggested by string theorists [FM00], who use the stable representations of (double) quivers rather than indecomposable representations of oriented quivers (as do Ringel and the present authors). The concept of stability has been very useful in representation theory: H. Nakajima discovered that modules over Kac-Moody Lie algebras may be described using functions on the varieties Λ_{V}^{s} /G_{V} where Λ_{V}^{s} are the stable points of Lusztig’s varieties Λ_{V} [Nak94, Nak98].

1.2. One of the main goals of this paper is to study the relationship between the Lusztig’s construction of 𝔫̃_{+}, the construction by the present authors [FMV01], and the “stable” construction suggested by physicists. Using the notion of semistable diagonal in Λ_{V} suggested in [HM98], we make a conjecture (Conjecture 4.2.5) directly relating the Lusztig’s construction and the “stable” construction. Moreover, we (conjecturally) obtain a basis in 𝔫̃_{+} parameterized by irreducible components of algebraic varieties. One can look at this conjectural basis as “semicanonical” basis for 𝔫̃_{+}.

1.3. In section 5 we use our methods to obtain some simple results describing new aspects (arising from quiver constructions) of combinatorics of the root systems and Weyl groups, and ask many more questions than we can answer at the moment.

One of the results of the Lusztig construction of the (quantized) universal enveloping algebras in terms of functions on the varieties Λ_{V} is the appearance of the semicanonical basis in the non-quantized enveloping algebras of simply laced Kac-Moody Lie algebras [Lus00]. Semicanonical basis is indexed by the irreducible components of Λ_{V}. It was expected that this basis would coincide with the specialization of the Lusztig’s canonical basis to q = 1, but a counterexample was found by M. Kashiwara and Y. Saito [KS97]. The relationship between these two bases is quite mysterious at the moment, even though they share their combinatorial properties.

Let 𝔫_{+} ⊂ U(𝔫_{+}) be the plus part of a simple Lie algebra of type ADE, and U(𝔫_{+}) be its universal enveloping algebra. For every positive root α ∈ R_{+} we have a canonically (up to a sign) defined element E_{α}^{∗} ∈ 𝔫_{α} ⊂ U(𝔫_{+}), where 𝔫_{α} is the one dimensional root subspace of 𝔫_{+} corresponding to α. We study the decomposition of E_{α}^{∗} with respect to the semicanonical basis. More precisely, let us consider Λ_{V}, dim V = α. Then the irreducible components of Λ_{V} may be indexed by decompositions of α into the sum of positive roots. Let α = α_{1} + α_{2} + ⋯ + α_{l}, α_{i} ∈ R_{+} be such a decomposition and let e_{α_{1} + α_{2} +···+α_{l}} be the element of the semicanonical basis corresponding to this decomposition (and the corresponding irreducible component of Λ_{V}). Then

E_{α}^{∗} = ∑_{α=α_{1} + α_{2} +···+α_{l}} c_{α_{1} + α_{2} +···+α_{l}}e_{α_{1} + α_{2} +···+α_{l}},

where c_{α_{1} + α_{2} +···+α_{l}} ∈ ℤ is the coefficient of e_{α_{1} + α_{2} +···+α_{l}}. In other words, with the help of the semicanonical basis we assign an integer c_{α_{1} + α_{2} +···+α_{l}} to every decomposition α = α_{1} + α_{2} + ⋯ + α_{l} of the root α into a sum of positive roots. It would be very interesting give a purely combinatorial description of these numbers without appealing to the semicanonical basis. We manage to obtain such a description in the A_{n} case in terms of the sign character of the Weyl group W = S_{n+1}.

1.4. Acknowledgment. We are grateful to Naihuan Jing who presented us with the opportunity to give this talk at the CBMS conference he organized at NCSU in June 2001. We tried to preserve a bit of the informal style of a talk in this paper. We are grateful to B. Fiol, J. Humphreys, G. Lusztig, M. Mariño, and C. Ringel for useful conversations. M.V. is grateful to MSRI and Max-Planck-Institut für Mathematik, where parts of this work were done, for their hospitality. The research of I. F. was supported in part by NSF. The research of A. M. and M. V. was supported by NSF Postdoctoral Research Fellowships.

2. Preliminaries

We always assume that our ground field is the field of complex numbers ℂ. Our notation and conventions are mostly lifted from [Lus91, Lus00, FMV01].

2.1. Quivers.

2.1.1. To a graph Q, with no edges joining a vertex with itself, we associate a pair of sets: I (vertices), and H (oriented edges), and two maps from H to I: 2.1.1.a. a map H → I denoted h → h′ (initial vertex), 2.1.1.b. a map H → I denoted h → h′′ (terminal vertex), 2.1.1.c. an involution h → h̄ on H which maps an oriented edge to the same edge

with the opposite orientation.

An orientation of (I, H) is a choice of a subset Ω ⊂ H such that Ω ∪ \overline{Ω} = H and Ω ∩ \overline{Ω} = ∅. Abusing terminology we will call both (I, H) and (I, Ω) quivers.

2.1.2. Let V be the category of I-graded vector spaces V = ⊕_{i∈I}V_{i}. We define:

E_{V} = ⨁_{h∈H} Hom(V_{h′}, V_{h′′}),
E_{V,Ω} = ⨁_{h∈Ω} Hom(V_{h′}, V_{h′′}).

For an element x ∈ E_{V} we denote by x_{h}, h ∈ H its component in Hom(V_{h′}, V_{h′′}). A pair (V, x), x ∈ E_{V} (resp. x ∈ E_{V,Ω}) is called a representation of (I, H) (resp. (I, Ω)). We will also sometimes call (V, x) a module over (I, H), and moreover denote such a module by V if it is clear what x we consider.

Let ℤ[I] be the free abelian group generated by the set I. The dimension of (V, x) with x ∈ E_{V} or x ∈ E_{V,Ω} is an element of ℤ[I] defined as follows:

dim(V, x) = dim V = ∑_{i∈I} dim_{ℂ}(V_{i})i ∈ ℤ[I].

The algebraic group G_{V} = ∏_{i∈I} GL(V_{i}) acts on E_{V} in a natural way.

2.1.3. We say that x ∈ E_{V,Ω} is indecomposable if (V, x) is indecomposable as a representation of (I, Ω). The subset of indecomposable elements in E_{V,Ω} is denoted by E_{V,Ω}^{ind}.

2.1.4. We say that x ∈ E_{V} is nilpotent if there exists an N ≥ 2 such that for any sequence h_{1}, h_{2}, …, h_{N} ∈ H, such that h′_{1} = h′′_{2}, h′_{2} = h′′_{3}, …, h′_{N−1} = h′′_{N}, the composition x_{h_{1}}x_{h_{2}} … x_{h_{N}} = 0. The subset of nilpotent elements in E_{V} is denoted by E_{V}^{nil}.

2.1.5. Following [Lus91] we consider the moment map attached to the G_{V} -action on E_{V}. The i-component m_{i} of this map is given by

m_{i}(x) = ∑_{h∈H:h′′=i} ϵ(h)x_{h}xh̄,

where ϵ : H → ℂ^{∗} is a function such that ϵ(h) + ϵ(h̄) = 0 for all h ∈ H. Following Lusztig we introduce the subvariety Λ_{V} of E_{V} as follows

Definition. Λ_{V} is the closed subvariety of all nilpotent elements x ∈ E_{V} such that m_{i}(x) = 0 for all i ∈ I.

2.2. Convolution product.

2.2.1. Let

L(V) = M_{G_{V}} (Λ_{V})

be the set of constructible ℂ-valued functions on Λ_{V} which are constant on G_{V} - orbits in Λ_{V}. Let L_{ℚ}(V) (resp. L_{ℤ}(V)) be the set of all f ∈ L(V) with rational (resp. integral) values.

2.2.2. Let V, V′, V′′ ∈ V be such that dim V′ + dim V′′ = dim V. Let f′ ∈ L(V′), f′′ ∈ L(V′′). We lift the convolution construction from [Lus91, 12.10]. Consider the diagram

Λ_{V′} × Λ_{V′′} ⟵^{p_{1}} F′ ⟶^{p_{2}} F′′ ⟶^{p_{3}} Λ_{V} ,

where F′′ is the variety of all pairs (x, W) where x ∈ Λ_{V} and W ⊂ V is an x-stable subspace, dim W = dim V′′.

F′ is the variety of all quadruples (x, W, R′′, R′) where (x, W) ∈ F′′, and R′, R′′ are V-isomorphisms R′ : V′ ≃ V/W, R′′ : V′′ ≃ W.

We have p_{1}(x, W, R′′, R′) = (x′, x′′) where x_{h}R′_{h′} = R′_{h′′}x′_{h}, and x_{h}R′′_{h′} = R′′_{h′′}x′′_{h} for all h ∈ H.

We have p_{2}(x, W, R′′, R′) = (x, W), p_{3}(x, W) = x.

Let f_{1} ∈ L(Λ_{V′} × Λ_{V′′}) be given by f_{1}(x′, x′′) = f′(x′)f′′(x′′). Then there is a unique function f_{3} ∈ L(F′′) such that p_{1}^{∗}f_{1} = p_{2}^{∗}f_{3}. Then by definition

f′ ∗ f′′ = (p_{3})_{!}(f_{3}).

There exists an analogous construction for the oriented quiver [Lus91, 10.19]. We will denote the corresponding convolution product by ∗_{Ω} for an orientation Ω.

2.2.3. If dim V = i ∈ I, then Λ_{V} is a point, and we denote by E_{i}(Λ_{V}) ≡ 1 the function which is identically 1 on this point. Let L_{0} be the associative algebra with ∗-product generated by {E_{i}}_{i∈I}. (The associative algebra of functions on E_{V,Ω} with ∗_{Ω}-product generated by {E_{i}}_{i∈I} will be denoted by L_{0,Ω}.) One can consider L_{0} as a ℤ_{+}[I]-graded Lie algebra over ℚ, with the following Lie bracket:

[f, g] = f ∗ g − g ∗ f.

We denote by 𝔫^{∗} the Lie subalgebra of L_{0} generated by {E_{i}}_{i∈I}.

2.3. Geometric realization of the enveloping algebra.

2.3.1. Let (I, H) be a quiver. For i, j ∈ I define i · j = − Card{h ∈ H | h′ = i, h′′ = j}, if i ≠ j and i · j = 2 if i = j. Let U^{+} be the ℂ-algebra defined by generators e_{i}, i ∈ I and the Serre relations:

∑_{\begin{matrix}p,q∈ℤ_{≥0} \\ p+q=−i·j+1 \end{matrix}}(−1)^{p} \frac{e_{i}^{p}}{p!} e_{j} \frac{e_{i}^{q}}{q!} = 0

for any i ≠ j in I. U^{+} = U(𝔫) is the enveloping algebra of the Lie algebra 𝔫 defined by generators e_{i}, i ∈ I and the Serre relations:

(ad e_{i})^{−i·j+1}(e_{j}) = 0

for any i ≠ j in I.

Let (V, x) be a representation of (I, H). Let U_{dim V}^{+} denote the subspace of U^{+} generated by e_{i_{1}}e_{i_{2}} … e_{i_{n}} for sequences i_{1}, i_{2}, …, i_{n} in which i appears dim V_{i} times for any i ∈ I.

Let U_{ℤ}^{+} be the subring of U^{+} generated by the elements e_{i}^{p}/p! for all i ∈ I, p ∈ ℤ_{≥0}. Then U_{ℤ,dim V}^{+} = U_{ℤ}^{+} ∩ U_{dim V}^{+} , see [Lus00, 1.1].

2.3.2. In [Lus91, 12.12] Lusztig defines a ℂ-linear map ψ_{V} : U_{dim V}^{+} → L(V), such that 2.3.2.a. ψ = ⨁_{dim V} ψ_{V} : U^{+} ≃ L_{0} is an isomorphism of algebras, 2.3.2.b. ψ(e_{i}) = E_{i}, for any i ∈ I, 2.3.2.c. ψ_{V} (U_{ℤ,dim V}^{+} ) ⊂ L_{ℤ}(V). The isomorphism ψ also restricts to an isomorphism of Lie algebras ψ : 𝔫 → 𝔫^{∗} such that ψ(e_{i}) = E_{i} for any i ∈ I.

In [Lus00] Lusztig defines a ℚ version of the map ψ_{V} : U_{dim V}^{+} → L(V), which restricts to a map ψ_{V} : U_{ℤ,dim V}^{+} → L_{ℤ}(V).

2.4. Lie algebra based on the Euler cocycle: classical ADE case. Let the graph underlying the quiver (I, H) be the Dynkin diagram of type ADE. In this case we can identify ℤ[I] with the root lattice of type ADE. The elements i ∈ I are considered simple roots, and we have the root system R ⊂ ℤ[I] and its positive part R_{+} ⊂ ℤ_{≥0}[I].

Let us fix a cocylcle ϵ : ℤ[I] × ℤ[I] → ℤ /2 ℤ. In the simply laced Dynkin diagram case, the Lie algebra 𝔫 = 𝔫^{∗} defined above is isomorphic to the Lie algebra n^{ϵ} spanned by the elements ẽ_{α}, α ∈ R_{+} with the bracket defined by ϵ as in [FMV01, 4.2.1.a].

Now let us define the functions E_{α}^{∗} ∈ L(V), α ∈ R_{+}, dim V = α as follows:

E_{α}^{∗} = ψ(ẽ_{α}). (2.4.0.a)

Notice that if α = i ∈ I, then Λ_{V} is a point and E_{i}^{∗} = E_{i}.

2.5. Lie algebra based on the Euler cocycle: affine ADE case. Let the graph underlying the quiver (I, H) be the extended Dynkin diagram of type ADE. In this case we can identify ℤ[I] with the affine root lattice of type ADE. The elements i ∈ I are considered simple roots, and we have the affine root system R ⊂ ℤ[I] and its positive part R_{+} ⊂ ℤ_{≥0}[I].

Let us choose an extending vertex p ∈ I, and let I′ = I − {p}.

Again, let us fix a cocylcle ϵ : ℤ[I] × ℤ[I] → ℤ/2 ℤ. Let 𝔫_{α} be the root space of the algebra 𝔫 corresponding to α ∈ R_{+}. Let δ be the indivisible imaginary root. The Lie algebra 𝔫 is isomorphic to the Lie algebra 𝔫^{ϵ} spanned by the elements ẽ_{α} ∈ 𝔫_{α}^{ϵ}, α ∈ R_{+}^{re}, and α_{k}(n) ∈ 𝔫_{nδ}^{ϵ}, n ≥ 1, where we identify 𝔫_{nδ}^{ϵ} = ℂ[I′] = the vector space spanned by I′, and α_{k} = k ∈ I′ ⊂ ℤ[I′] are simple roots. The bracket is defined by ϵ as in [FMV01, 5.2.1.a]. In particular,

[ẽ_{α}, ẽ_{β}] = \begin{cases}ϵ(α, β)ẽ_{α+β} & if α + β ∈ R_{+}^{re}, \\ ϵ(α, β)α(n) & if α + β = nδ, \\ 0 & if α + β ∉ R_{+}. \end{cases} (2.5.0.b)

Now let us define the functions E_{α}^{∗} ∈ L(V), dim V = α ∈ R_{+}^{re} as follows:

E_{α}^{∗} = ψ(ẽ_{α}), (2.5.0.c)

and the functions E_{k}^{∗}(n) ∈ L(V), dim V = nδ as follows:

E_{k}^{∗}(n) = ψ(α_{k}(n)). (2.5.0.d)

Again, if α = i ∈ I, then Λ_{V} is a point and E_{i}^{∗} = E_{i}.

3. Stability and simple Lie algebras of type ADE

3.1. Stability after King and Rudakov.

3.1.1. Stability after King. Let us consider the abelian category of nilpotent representations of a quiver (I, H) and its Grothendieck group K_{0}. A character on K_{0} is an additive function θ : K_{0} → ℝ.

Definition. [King94] A point x ∈ E_{V} is called θ-stable (resp. θ-semistable) if θ(V) = 0, and for any x-stable nonzero proper subspace V′ ⊂ V we have θ(V′) > 0 (resp. θ(V′) ≥ 0).

Definition. Two θ-semistable points x′, x′′ ∈ E_{V} are S-equivalent (notation: x′ ∼^{S} x′′) if the orbit closures \overline{G_{V} · x′} ∩ \overline{G_{V} · x′′} intersect in the set of θ-semistable points E_{V}^{ss}.

3.1.2. Stability after Rudakov. One can also define stable points using a “slope” stability condition: μ = c/r where c and r are additive functions K_{0} → ℝ and r(V) > 0 for any V ∈ V, see [Rud97, 3].

Definition. [Rud97] A point x ∈ E_{V} is called μ-stable (resp. μ-semistable) if for any x-stable nonzero proper subspace V′ ⊂ V we have μ(V′) < μ(V) (resp. μ(V′) ≤ μ(V)).

If we fix such slope stability condition μ and V ∈ V we can define a character θ_{μ} : K_{0} → ℝ as follows:

θ_{μ}(V′) = −c(V′) + \frac{c(V)}{r(V)}r(V′). (3.1.2.a)

According to [Rud97, Proposition 3.4] a point x ∈ E_{V} is μ-stable if and only if it is θ_{μ}-stable.

3.1.3. Let us fix an orientation Ω ⊂ H until the end of this section. Following [King94] we construct a character θ = Θ_{V,Ω} associated to Ω. Let V, V′ ∈ V. We define

Θ_{V,Ω}(V′) = ∑_{h∈Ω}(dim V_{h′} dim V′_{h′′} − dim V′_{h′} dim V_{h′′}).

3.2. Stability Lemma. In this section we will consider quivers of finite ADE type, i.e. the underlying non-oriented graph Q is a Dynkin graph of (finite) ADE type. A fact similar to the lemma below was independently conjectured by M. Reineke [Rei02, Conjecture 7.1]

Lemma. Let x ∈ Λ_{V}, dim V = α ∈ R_{+}. Then x is Θ_{V,Ω}-stable if and only if x ∈ E_{V,Ω}^{ind}.

Proof. Since in the Dynkin quiver case every root is a Schur root [King94], a generic point in E_{V,Ω} is Θ_{V,Ω}-stable according to [King94, Scho92]. Since E_{V,Ω}^{ind} is an open dense G_{V} -orbit in E_{V,Ω}, any x ∈ E_{V,Ω}^{ind} is Θ_{V,Ω}-stable.

Let us prove the “only if” part. If x ∈ Λ_{V} −E_{V,Ω}^{ind}, then we have x = y +z, where y ∈ E_{V,Ω}, z ∈ E_{V,\overline{Ω}}. Following [Lus91, 14] we see that there exists a decomposition:

(V, y) = ⊕_{p=1}^{ν}(V^{p}, y)

such that Hom(V^{p}, V^{p′}) = 0 whenever p′ < p. There is a direct sum decomposition:

E_{V,Ω} = ⨁_{1≤p,p′≤ν} E_{V,Ω}^{p,p′}  where  E_{V,Ω}^{p,p′} = Hom_{h∈Ω}(V_{h′}^{p}, V_{h′′}^{p′}).

Now we need the following claim: there exists y′ ∈ ⨁_{p>p′} E_{V,Ω}^{p,p′} such that y+y′ =: x′ ∈ E_{V,Ω}^{ind}. It is not very hard to prove this claim using the methods of [Lus90, 4].

Following [Lus91, 14] we denote V^{(q)} = ⊕_{p>q}V^{p}. Then we have the x-invariant filtration:

V = V^{(0)} ⊃ V^{(1)} ⊃ ⋯ ⊃ V^{(ν)} = 0.

Denote V_{(q)} = ⊕_{p<q}V^{p}. Then we have the x′-invariant filtration:

0 = V_{(0)} ⊂ V_{(1)} ⊂ ⋯ ⊂ V_{(ν)} = V.

The vector space V_{(q)} is a submodule in (V, x′). Since x′ is Θ_{V,Ω}-stable we have Θ_{V,Ω}(V_{(q)}) > 0. However V_{(q)} is a quotient module in (V, x). Therefore, (V, x) is unstable.

3.3. Stable construction.

3.3.1. Let us consider the cocycle ϵ_{Ω} associated to our fixed orientation Ω, see [FK81], [FMV01, 1.3.4]. By construction, the functions E_{α}^{∗}, α ∈ R_{+} on Λ_{V} defined as in 2.4 with the cocycle ϵ = ϵ_{Ω} form a basis of the Lie algebra n^{∗}, and the ∗-bracket is given by:

[E_{α}^{∗}, E_{β}^{∗}] = \begin{cases}ϵ(α, β)E_{α+β}^{∗}, & if α + β ∈ R_{+}, \\ 0 & if α + β ∉ R_{+}. \end{cases} (3.3.1.a)

3.3.2. It is clear that the affine variety E_{V,Ω} ⊂ Λ_{V} is one of the irreducible components of Λ_{V}. We can define a constructible function E_{α}, α ∈ R_{+} on E_{V,Ω}, dim V = α as follows, see [FMV01, 4.3.1.b]:

(3.3.2.a) E_{α} = \begin{cases}1, & if x ∈ E_{V,Ω}^{ind}, \\ 0 & otherwise . \end{cases}

According to [FMV01, 4.3.4], the space spanned by the functions E_{α}, α ∈ R_{+} is a Lie algebra (which we will denote here by 𝔫^{Ω}) with the bracket [f, g]_{Ω} = f ∗_{Ω} g − g ∗_{Ω} f. The map 𝔫^{ϵ} → 𝔫^{Ω} given by ẽ_{α} → E_{α} is an isomorphism. In other words, the bracket can be explicitly described as follows:

[E_{α}, E_{β}]_{Ω} = \begin{cases}ϵ(α, β)E_{α+β}, & if α + β ∈ R_{+}, \\ 0 & if α + β ∉ R_{+}. \end{cases} (3.3.2.b)

3.3.3. Let us take the character Θ_{V,Ω}, and denote the identity function on Λ_{V}^{s}, dim V = α, by Ẽ_{α}, Ẽ_{α}(Λ_{V}^{s}) ≡ 1, where Λ_{V}^{s} is the set of points in Λ_{V} stable with respect to Θ_{V,Ω}. Notice that by Lemma 3.2, Λ_{V}^{s} = E_{V,Ω}^{ind}, and so the function Ẽ_{α} is constant on a single G_{V} -orbit in Λ_{V}^{s}.

Lemma. Up to a sign: 3.3.3.a. E_{α}^{∗}|_{E_{V,Ω}} = E_{α}, 3.3.3.b. E_{α}^{∗}|_{Λ_{V}^{s}} = Ẽ_{α}.

Proof. Follows immediately from definitions, the proof of [Lus91, Theorem 12.13], and the formulas 3.3.1.a, 3.3.2.b.

3.3.4. Let us denote the space spanned by the functions Ẽ_{α}, α ∈ R_{+} by 𝔫^{stable}. Since Λ_{V}^{s} = E_{V,Ω}^{ind}, by lemma 3.2, the restrictions from Λ_{V} to E_{V,Ω} to Λ_{V}^{s} provide us with the based isomorphisms of vector spaces (cf. Lemma 3.3.3)

𝔫^{∗} ⟶_{restriction}^{≃} 𝔫^{Ω} ⟶_{restriction}^{≃} 𝔫^{stable},

defined by E_{α}^{∗} ↦ E_{α} ↦ Ẽ_{α} for α ∈ R_{+}. It is clear that the first restriction is an isomorphism of Lie algebras, and the second restriction equips the space 𝔫^{stable} of functions on Λ_{V}^{s} constant on G_{V} -orbits (or equivalently, functions on Λ_{V}^{s} /G_{V} = pt_{V}) with the structure of a Lie algebra isomorphic to 𝔫^{∗} ≃ 𝔫^{Ω} ≃ 𝔫^{ϵ}.

4. Stability, affine Lie algebras of type ADE, and a conjectural construction of the algebras of BPS states

4.1. Physics.

4.1.1. One of the current models of string theory defines D-branes as objects in the derived category D^{b}(Coh(Z)) of the coherent sheaves on the algebraic (Calabi-Yau) variety Z. Physicists consider the moduli space M_{ζ}(n) (where ζ = (ζ_{1}, ζ_{2}, …, ζ_{N}) are the Fayet-Iliopulos terms) of semistable D-branes of charge n, cf. [HM98, DFR00, FM00]. In the semiclassical approximation, the space of BPS states is given by the cohomology of the moduli space:

H_{BPS} = H^{∗}(M_{ζ}(n)).

4.1.2. Dictionary. We will consider the D-branes on the resolution { \atop ℂ^{2} /Γ} of a simple singularity ℂ^{2} /Γ. Let Q be the extended Dynkin diagram corresponding to Γ via the McKay correspondence. We suggest a mathematical model of the physical situation in this case with which we will work. In the next subsection 4.1.3 we will justify our model. For now, we provide the physics/mathematics dictionary:

the charge lattice : the positive part ℤ_{≥0}[I] of the affine root lattice associated to Q;

places in the the charge lattice occupied by the single particle BPS states : positive roots R_{+} ⊂ ℤ_{≥0}[I];

a D-brane of charge α ∈ ℤ[I] on { \atop ℂ^{2} /Γ} : a representation x ∈ Λ_{V}, dim V = α of the quiver (I,H) (with the underlying non-oriented graph Q);

Fayet-Iliopulos terms ζ : an additive function c which defines Rudakov’s slope stability together with r(V) = ∑_{i∈I} dim V_{i};

ζ-stable D-branes : stable elements of Λ_{V} with respect to a stability condition; moduli spaces of D-branes of charge α ∈ R_{+}^{re} : Λ_{V}^{s} /G_{V} = pt_{α}, dim V = α ∈ R_{+}^{re}; moduli spaces of D-branes of charge δ = indivisible imaginary root : Λ_{V}^{s} /G_{V} =

𝔏 = 𝔏_{1}, where 𝔏 is the exceptional fiber of the resolution of simple singularity;

moduli spaces of D-branes of charge mδ, m > 1 : Λ_{V}^{ssd}/(S − equivalence) = 𝔏 = 𝔏_{m}, where Λ_{V}^{ssd} is the semistable diagonal (see 4.2.4.a for the definition, cf. [HM98]), and where 𝔏 is the exceptional fiber of the resolution of simple singularity;

BPS states : certain constructible functions on our moduli spaces, see 4.2.8 for details.

4.1.3. Justification. If we consider the D-branes on { \atop ℂ^{2} /Γ} then due to M. Kapranov and E. Vasserot [KV00]:

D^{b}(Coh({ \atop ℂ^{2} /Γ})) ≃ D^{b}(Rep(Q))

where Q is the extended Dynkin diagram associated to the Γ ⊂ SL(2, ℂ) via the McKay correspondence, and Rep(Q) is the category of finite-dimensional double representations of Q in the sense of [KV00, 3.4].

We are only interested in D-branes of charge α ∈ R_{+} which are 0-complexes on the right hand side, i.e. double representations of Q rather than complexes of such representations. Such representations are identified with elements x ∈ E_{V}, dim V = α satisfying m_{i}(x) = 0 for all i ∈ I (see section 2.1.5 for the definition of m_{i}). The notion of stability becomes the usual GIT stability adapted to the quiver situation by King and Rudakov (see section 3). We choose a “non-degenerate” stability condition (cf. 4.2.6).

If dim V = α ∈ R_{+}^{re}, then the moduli space is a point: M_{ζ}(α) = pt_{α}.

If dim V = δ, where δ is the indivisible imaginary root, then the moduli space is the resolution of a simple singularity:

M_{ζ}(δ) = { \atop ℂ^{2} /Γ}.

In the two cases above one may replace the (middle) cohomology of the moduli spaces with the constructible functions on the exceptional fiber see [Nak94, 10.16]. Following this logic, we suggest the interpretation of BPS states as in the dictionary above and in 4.2.8 below if dim V = mδ, m > 1 where the situation is more complicated and a clear connection between physical and mathematical results was not available before, as far as we know.

4.1.4. Physics implies that the space of BPS states on { \atop ℂ^{2} /Γ} of all possible charges should form a Lie algebra isomorphic to the plus part of the affine Lie algebra corresponding to Γ via the McKay correspondence [FM00]. The main purpose of this section is to offer a conjectural, but mathematically rigorous, validation of this claim in 4.2.8.

4.2. The conjecture. In this section we consider quivers of affine ADE type, i.e. the underlying non-oriented graph Q is an extended Dynkin graph of ADE type.

4.2.1. Let us fix an extending vertex p of the graph Q. For every α = dim V let us fix the Nakajima’s character: θ(V) = −1, if dim V = k ∈ I′ = I − {p}, and θ(V) = ∑_{i∈I′} dim_{ℂ}(V_{i}) if dim V = p ∈ I (cf. [Nak98]). (Semi)stable points in this section are considered with respect to this stability condition. If V_{p} = 0 we can take a King’s character Θ_{V,Ω} (see 3.1.3) associated to any orientation that “flows to the extending vertex” i.e. the extending vertex is a sink and one can get to the extending vertex from any other point in the quiver going along the oriented edges. In the DE case there is only one such orientation (once the extending vertex is fixed). In the A_{n} case there are n such orientations.

4.2.2. For dim V = α ∈ R_{+}^{re} we have (cf. [CB01]):

Λ_{V}^{s} /G_{V} = pt_{α} . (4.2.2.a)

Let us denote by Ẽ_{α} the function identically 1 on the point pt_{α}, Ẽ_{α}(pt_{α}) ≡ 1.

4.2.3. For dim V = δ = indivisible imaginary root we have ([Kron89], [Nak94]):

Λ_{V}^{s} /G_{V} = 𝔏, (4.2.3.a)

where 𝔏 = 𝔏_{1} is the exceptional fiber of the resolution of simple singularity

𝔏 ↪ { \atop ℂ^{2} /Γ} → ℂ^{2} /Γ.

Here Γ is the finite subgroup of SL(2, ℂ) corresponding to the diagram Q via the McKay correspondence.

It is well known that 𝔏 is a configuration of lines ℙ^{1} which are its irreducible components and which may be indexed by the vertices k ∈ I′ = I − {p}. We will denote the k^{th} irreducible component of 𝔏 by Y_{k}, k ∈ I′. Let us denote the characteristic function of Y_{k} by Ẽ_{k}(1), k ∈ I′, Ẽ_{k}(1)(Y_{k}) ≡ 1, Ẽ_{k}(1)(𝔏 − Y_{k}) ≡ 0.

4.2.4. Finally, for dim V = mδ, m > 1 we introduce the semistable diagonal Λ_{V}^{ssd} ⊂ Λ_{V} as follows (cf. [HM98]): x is S-equivalent to the direct sum of m isomorphic representations in Λ_{δ}^{s}. Here Λ_{δ}^{s} = Λ_{V′}^{s}, dim V′ = δ. Formally:

Λ_{V}^{ssd} = {x ∈ Λ_{V} | x ∼^{S} x_{1} ⊕ x_{2} ⊕ ⋯ ⊕ x_{m}, x_{1} ≃ x_{2} ≃ ⋯ ≃ x_{m}, x_{i} ∈ Λ_{δ}^{s}} (4.2.4.a)

It is clear that:

Λ_{V}^{ssd}/(S-equivalence) = 𝔏

where 𝔏 = 𝔏_{m} is the same exceptional variety as above. Let us denote the characteristic function of the k^{th} irreducible component Y_{k} of 𝔏 = 𝔏_{m} by Ẽ_{k}(m), k ∈ I′, Ẽ_{k}(m)(Y_{k}) ≡ 1, Ẽ_{k}(m)(𝔏 − Y_{k}) ≡ 0.

4.2.5. We have defined the functions E_{α}^{∗} on Λ_{V}, dim V = α ∈ R_{+}^{re}, and the functions E_{k}^{∗}(m), k ∈ I′ = I − {p} on Λ_{V}, dim V = mδ, m ≥ 1 in section 2.5. We need one more definition (cf. 2.5.0.b):

Ê_{k}^{∗}(m) := ϵ(α_{k}, β_{k})[E_{α_{k}}^{∗}|_{Λ_{V_{1}}^{s}} , E_{β_{k}}^{∗}|_{Λ_{V_{2}}^{s}}],

where dim V_{1} = α_{k} = k ∈ I′ ⊂ R_{+}^{re}, dim V_{2} = β_{k} = (mδ − α_{k}) ∈ R_{+}^{re} (here we treat elements of I′ as simple roots), and the bracket is in the sense of 2.2.3.

Since the functions E_{α}^{∗}, Ê_{k}^{∗}(m) are constant on G_{V} -orbits, we can consider them as functions on Λ_{V} /G_{V}. The equalities in the following conjecture are understood in this sense.

Conjecture. 4.2.5.a. Up to a sign, E_{α}^{∗}|_{Λ_{V}^{s}} = Ẽ_{α}, dim V = α ∈ R_{+}^{re}. 4.2.5.b. Up to a sign, Ê_{k}^{∗}(1)|_{Λ_{V}^{s}} = Ẽ_{k}(1), dim V = δ. 4.2.5.c. Let dim V = mδ, m > 1. We conjecture that if x′, x′′ ∈ Λ^{ssd} are two ele-

ments in the same S-equivalence class and Ê_{k}^{∗}(m)(x′) ≠ 0, Ê_{k}^{∗}(m)(x′′) ≠ 0, then Ê_{k}^{∗}(m)(x′) = Ê_{k}^{∗}(m)(x′′). Thus we can consider Ê_{k}^{∗}(m)|_{Λ_{V}^{ssd}} as a function on Λ^{ssd}/S-equivalence by setting for an S-equivalence class X:

Ê_{k}^{∗}(m)(X) = \begin{cases}Ê_{k}^{∗}(m)(x), & if there exists x ∈ X with Ê_{k}^{∗}(m)(x) ≠ 0, \\ 0, & otherwise. \end{cases}

4.2.5.d. Up to a sign, Ê_{k}^{∗}(m)|_{Λ_{V}^{ssd}} = Ẽ_{k}(m), dim V = mδ.

The conjecture is verified in the Â_{1} case.

4.2.6. Remark. We would like to explain here that there is nothing special about the Nakajima’s choice of character. If dim V = mδ is an imaginary root, then the space of all characters defining stability may be naturally identified with a (classical) Cartan subalgebra 𝔥 = ℂ-vector space spanned by i ∈ I′. A character θ is called non-degenerate if it does not lie on a wall of a Weyl chamber. We expect the conjecture to be true for any non-degenerate character in the same Weyl chamber as the Nakajima’s character (this is the fundamental chamber). Moreover, we could take a non-degenerate character in any other Weyl chamber, but that would change our choice of simple roots which we identify with the set I′ = I − {p}. One could perhaps reformulate our conjecture in this case modifying the functions E_{k}^{∗}(m) using the action of the Weyl group.

4.2.7. Remark. The functions Ẽ_{k}(m) parameterize the set Irr 𝔏 = Irr 𝔏_{m} of the irreducible components of the exceptional variety 𝔏. The components of 𝔏 are a

basis in the space H^{2}({ \atop ℂ^{2} /Γ}), see [Nak94].

4.2.8. Denote the Lie algebra spanned by the functions E_{α}^{∗} on Λ_{V}, dim V = α ∈ R_{+}^{re}, and the functions E_{k}^{∗}(m) k ∈ I′ = I − {p} on Λ_{V}, dim V = mδ, m ≥ 1 by n^{∗}(Q), cf. section 2.5.

Let us denote the vector space spanned by the functions Ẽ_{α}, α ∈ R_{+}^{re}, and the functions Ẽ_{k}(m) on 𝔏 = 𝔏_{m} = Λ_{mδ}^{ssd}/(S − equivalence) by 𝔫^{stable}(Q). This is the vector space of constructible functions on moduli spaces of (semi)stable points in Λ, which are linear combinations of characteristic functions of irreducible components. The space 𝔫^{stable}(Q) is our model of the space of BPS states at the orbifold ℂ^{2} /Γ, with Q corresponding to Γ via the McKay correspondence.

The conjecture, if true, would imply that the restriction from Λ to Λ^{s} (or Λ^{ssd}) gives us the based isomorphism of vector spaces

𝔫^{∗}(Q) ⟶^{≃} 𝔫^{stable}(Q)

such that E_{α}^{∗} ↦ Ẽ_{α} and E_{k}^{∗}(m) ↦ Ê_{k}(m) ↦ Ẽ_{k}(m) for α ∈ R_{+}^{re}, k ∈ I′, m ≥ 1. This map equips the space of BPS states 𝔫^{stable}(Q) with the structure of a Lie algebra isomorphic to the plus part of the affine Lie algebra corresponding to Q.

On the other hand the above isomorphism provides the plus part of the affine Lie algebra corresponding to Q, with a basis {Ẽ_{α}, α ∈ R_{+}^{re}; Ẽ_{k}(m), k ∈ I′, m ∈ ℤ_{>0}} indexed by irreducible components of algebraic varieties pt_{α} = Λ_{α}^{s}/G_{α}, and 𝔏_{m} = Λ_{mδ}^{ssd}/(S − equivalence), m ∈ ℤ_{>0}.

5. Remarks on semicanonical basis for simple Lie algebras

5.1. Semicanonical basis. This subsection is lifted from [Lus00, 2.4-5]. Recall the setup of 2.3. Let V = ⊕_{i∈I}V_{i} be a I-graded vector space. If Y ∈ Irr Λ_{V} is an irreducible component of Λ_{V} and f ∈ L_{ℤ}(V) then there is a unique c ∈ ℤ such that f^{−1}(c) ∩ Y contains an open dense subset of Y. Note that f ↦ c is a linear function ρ_{Y} : L_{ℤ}(V) → ℤ.

Lemma. Let Y ∈ Irr Λ_{V}. There exists f = e_{Y} ∈ ψ_{V} (U_{ℤ,V}^{+} ) such that ρ_{Y} (f) = 1 and ρ_{Y′}(f) = 0 for any Y′ ∈ Irr Λ_{V} − {Y }.

Lusztig proves that the functions e_{Y}, Y ∈ Irr Λ_{V} form a basis in ψ_{V} (U_{V}^{+}), and therefore, the collection of e_{Y}, Y ∈ Irr Λ_{V} for all possible dimensions of V forms a basis in the algebra ψ(U^{+}) = L_{0} isomorphic to U^{+}. This basis is called semicanonical.

In the remainder of this section we consider quivers of finite ADE type.

5.1.1. Since our quiver is of finite ADE type we know [Lus91, 14.2] that the irreducible components of Λ_{V} are the closures of conormal bundles of the various G_{V} -orbits in E_{V,Ω}, where Ω is some orientation. Such G_{V} -orbits are indexed by the decomposition of (V, x), x ∈ E_{V,Ω} into a direct sum of indecomposable submodules (V, x) = (V_{1}, x_{1}) ⊕ ⋯ ⊕ (V_{l}, x_{l}) with x_{i} ∈ E_{V_{i},Ω}^{ind}. (The decompositions are considered up to the order of the summands.) Thus, there is a one-to one correspondence between Irr Λ_{V} and the decompositions dim V = dim V_{1} + ⋯ + dim V_{l} of dim V ∈ ℤ[I] into a sum of positive roots dim V_{i} ∈ R_{+}. (Recall that E_{V,Ω}^{ind} ≠ ∅ if and only if dim V ∈ R_{+}.) Thus the elements of the semicanonical basis may be indexed by decompositions of dim V into a sum of positive roots.

5.2. Open problem. Recall that we have defined the functions E_{α}^{∗} on Λ_{V}, dim V = α ∈ R_{+} in 2.4. By construction these functions are defined canonically up to a sign which depends on the choice of the cocycle ϵ. Since E_{α}^{∗} ∈ ψ_{V} (U_{ℤ,dim V}^{+} ), we can decompose E_{α}^{∗} with respect to the semicanonical basis:

E_{α}^{∗} = ∑_{Y ∈Irr Λ_{V}} c_{Y} e_{Y} . (5.2.0.a)

The problem is to calculate the integer coefficients c_{Y} for all irreducible components Y of Λ_{V}. These coefficients are independent of any choices up to simultaneous multiplication of all of them by −1. Suppose that an irreducible component Y ∈ Irr Λ_{V}, dim V = α corresponds to a decomposition

α = α_{1} + ⋯ + α_{l} (5.2.0.b)

of α into a sum of positive roots α_{1}, …, α_{l} . Then the formula 5.2.0.a may be regarded as an assignment {α = α_{1} + ⋯ + α_{l}} ↦ c_{Y} of the integer c_{Y} to every decomposition 5.2.0.b. It would be very interesting to describe this assignment in terms of combinatorics of the root system without appealing to the semicanonical basis. We obtain such a description in the A_{n} case using the Weyl group W = S_{n+1}. We also succeed in calculating c_{Y} when Y = E_{V,Ω} are irreducible components “arising from orientations”, and we calculate the examples of D_{4} and D_{5}.

5.3. Irreducible components arising from orientations.

5.3.1. Let us define the subvariety O_{V} ⊆ Λ_{V} as follows:

O_{V} = ⋃_{Ω} E_{V,Ω}

where the union is taken over all orientations Ω of our quiver. Since each Y = E_{V,Ω} is an irreducible component of Λ_{V} (we will call such Y irreducible components arising from orientations), we have Irr O_{V} ⊆ Irr Λ_{V}. We will split the sum 5.2.0.a into two parts:

E_{α}^{∗} = E′_{α} + E′′_{α}

where

(5.3.1.a)  E′_{α} = ∑_{Y ∈Irr O_{V}} c_{Y} e_{Y}  and  E′′_{α} = ∑_{Y ∈Irr Λ_{V} −Irr O_{V}} c_{Y} e_{Y}

5.3.2. If a root α ∈ R_{+} is not simple, represent it as a sum of simple roots α_{k_{1}}, α_{k_{2}}, …, α_{k_{h}}, k_{l} ∈ I, where h = ht α

α = α_{k_{1}} + α_{k_{2}} + ⋯ + α_{k_{h}} (5.3.2.a)

in such a way that α_{k_{1}} + α_{k_{2}} + ⋯ + α_{k_{j}} is a root for 1 ≤ j ≤ h.

It follows from our construction and definitions that up to a sign

E_{α}^{∗} = [… [E_{α_{k_{1}}}^{∗} , E_{α_{k_{2}}}^{∗}] … E_{α_{k_{h}}}^{∗}] (5.3.2.b)

Now we can calculate the coefficients of E′_{α} (cf. section 3 for a similar discussion).

Proposition. Up to a sign,

E′_{α} = ∑_{Y =E_{V,Ω}∈Irr O_{V}} (∏_{i<j} ϵ_{Ω}(α_{k_{i}}, α_{k_{j}}))e_{Y} ,

where ϵ_{Ω} is the Frenkel-Kac cocycle corresponding to Ω, [FMV01, 1.3.4]. This presentation does not depend on the decomposition 5.3.2.a.

Proof. Let us fix some orientation Ω, and let us assume that Y = E_{V,Ω}. According to Lusztig [Lus92, 12.13] we have a homomorphism of algebras L_{0} → L_{0,Ω} induced by the restriction on functions from Λ_{V} to E_{V,Ω} (see 2.2.3 for the definitions of L_{0} and L_{0,Ω}). Then

E_{α}^{∗}|_{Y} = E′_{α}|_{Y} = [… [E_{k_{1}}, E_{k_{2}}]_{Ω} … E_{k_{h}}]_{Ω} = ∏_{i<j} ϵ_{Ω}(α_{k_{i}}, α_{k_{j}})E_{α}, (5.3.2.c)

where E_{α} is defined by 3.3.2.a, and the last equality follows from [FMV01, 4.3.4.b]. Now it is clear that ρ_{Y} (E_{α}^{∗}) = ρ_{Y} (E_{α}^{∗}|_{Y}). Then

c_{Y} = ρ_{Y} (E_{α}^{∗}) = ρ_{Y} (E_{α}^{∗}|_{Y}) = ∏_{i<j} ϵ_{Ω}(α_{k_{i}}, α_{k_{j}})ρ_{Y} (E_{α,Ω}) = ∏_{i<j} ϵ_{Ω}(α_{k_{i}}, α_{k_{j}}).

Notice that due to the formula 5.3.2.c we know explicitly E_{α}^{∗}|_{O} i.e., the value E_{α}^{∗}(x) at any point x ∈ O ⊆ Λ.

5.3.3. Let Q be the Dynkin graph of type A_{n}. In this case it is clear that for V ∈ V, dim V = α ∈ R_{+} any irreducible component of Λ_{V} is of the form E_{V,Ω} for some orientation Ω. Thus Proposition 5.3.2 gives a complete answer to our question 5.2 in this case.

5.4. Conjugacy classes of the Weyl group.

5.4.1. Recall the isomorphism ψ : 𝔫 → 𝔫^{∗} (see 2.3.2), and let us take ẽ_{α} ∈ 𝔫^{ϵ} ≃ 𝔫 such that ψ(ẽ_{α}) = E_{α}^{∗}, α ∈ R_{+}, as in 2.4. Let us fix Y ∈ Irr Λ_{V} and let α = β_{1} + ⋯ + β_{l} be the corresponding decomposition of α into the sum of positive roots. Now consider the element ẽ_{β_{1}} + ⋯ + ẽ_{β_{l}} ∈ 𝔫^{ϵ} ≃ 𝔫, and consider its adjoint orbit. In such a way we obtain a map from Irr Λ_{V} to the set N of nilpotent orbits of the Lie algebra corresponding to the Dynkin diagram Q. Moreover, following Kazhdan-Lusztig [KL88] and Spaltenstein [Spal88], we can construct a map from N to the set Cl(W) of conjugacy classes of the Weyl group W. Thus we obtain a map cl : Irr Λ_{V} → Cl(W) for any V ∈ V. We will use this map in the following subsection.

5.4.2. Example: A_{n}.

Proposition. Up to a sign,

E_{α}^{∗} = ∑_{Y ∈Irr Λ_{V}} sgn(cl Y) e_{Y} ,

where sgn(cl Y) is the value of the one-dimensional sign character of W = S_{n+1} on the conjugacy class cl Y.

Proof. The proof is straightforward and is left to the reader.

5.4.3. Open question. We have the map cl : Irr Λ_{V} → Cl(W) for all ADE quivers, see 5.4.1. Looking at the A_{n} case (Proposition 5.4.2), one may ask if in the DE case the coefficients c_{Y} are also governed by characters of the Weyl group. Unfortunately, the authors’ naive attempt to replicate the A_{n} result fails already in the D_{4} case (however the explicit calculation for the maximal root of D_{4} is given below).

5.5. Example: D_{4}.

5.5.1. We index the vertices of the Dynkin diagram as follows:

[figure: 1; 0; 2; 3]

The simple roots are denoted by α_{0}, α_{1}, α_{2}, α_{3}. Let α = 2α_{0} + α_{1} + α_{2} + α_{3} be the maximal root. Indexing the irreducible components of Λ_{V}, dim V = α by the decompositions of α we obtain by a straighforward computation using the properties of the semicanonical basis:

E_{α}^{∗} = E′_{α} + E′′_{α}

= E′_{α} − e_{(α_{0}+α_{1})+(α_{0}+α_{2}+α_{3})} − e_{(α_{0}+α_{2})+(α_{0}+α_{1}+α_{3})} − e_{(α_{0}+α_{3})+(α_{0}+α_{1}+α_{2})} − e_{α_{0}+α_{1}+(α_{0}+α_{2}+α_{3})} − e_{α_{0}+α_{2}+(α_{0}+α_{1}+α_{3})} − e_{α_{0}+α_{3}+(α_{0}+α_{1}+α_{2})} + 2e_{α_{0}+(α_{0}+α_{1}+α_{2}+α_{3})}.

5.6. Example: D_{5}.

5.6.1. We index the vertices of the Dynkin diagram as follows:

[figure: 1; 0; 2; 3; 4]

The simple roots are denoted by α_{0}, α_{1}, α_{2}, α_{3}, α_{4}. Let α = 2α_{0} + α_{1} + α_{2} + 2α_{3} + α_{4} be the maximal root. There are 55 root partitions of α, e.g. α = α_{0} + α_{3} +(α_{0} + α_{1} + α_{2} + α_{3} + α_{4}). Indexing the irreducible components of Λ_{V}, dim V = α by the decompositions of α we obtain by a tedious but straightforward computation using the properties of the semicanonical basis:

E_{α}^{∗} =  E′_{α} + E′′_{α} = E′_{α}+
+ e_{(2 α_{0} + α_{1} + α_{2} + α_{3})+(α_{3} + α_{4})} + e_{α_{2} +(α_{0} + α_{1})+(α_{0} + α_{3})+(α_{3} + α_{4})}
+ e_{α_{1} +(α_{0} + α_{2})+(α_{0} + α_{3})+(α_{3} + α_{4})} + e_{α_{1} + α_{2} +(α_{0} + α_{3})+α_{0} +(α_{3} + α_{4})}
+ e_{(2 α_{0} + α_{1} + α_{2} + α_{3})+α_{3} + α_{4}} + e_{α_{2} +(α_{0} + α_{1})+α_{3} + α_{4} +(α_{0} + α_{3})}
+ e_{α_{1} +(α_{0} + α_{2})+(α_{0} + α_{3})+α_{3} + α_{4}} +  e_{(α_{0} + α_{3})+α_{0} + α_{2} + α_{3} + α_{4}}
+ e_{(α_{2} + α_{0} + α_{3})+(α_{1} + α_{0} + α_{3} + α_{4})} + e_{(α_{1} + α_{2} + α_{0} + α_{3})+(α_{0} + α_{3})+α_{4}}
− e_{(α_{2} + α_{0} + α_{1})+(α_{0} + α_{3})+(α_{3} + α_{4})} + e_{(α_{2} + α_{0} + α_{1})+α_{0} + α_{3} +(α_{3} + α_{4})}
+ e_{(α_{2} + α_{0} + α_{1})+α_{0} +2 α_{3} + α_{4}} − e_{(α_{0} + α_{1} + α_{2})+(α_{0} + α_{3})+α_{4}}
− e_{(α_{0} + α_{2})+(α_{0} + α_{1} + α_{3})+(α_{3} + α_{4})} + e_{(α_{0} + α_{1} + α_{3})+(α_{0} + α_{2} + α_{3} + α_{4})}
− e_{α_{2} +(α_{0} + α_{1} + α_{3})+α_{0} +(α_{3} + α_{4})} + e_{α_{2} +(α_{0} + α_{1} + α_{3})+(α_{0} + α_{3} + α_{4})}
− e_{(α_{0} + α_{1})+(α_{2} + α_{0} + α_{3})+(α_{3} + α_{4})} + e_{(α_{2} + α_{0} + α_{3})+(α_{0} + α_{1} + α_{3} + α_{4})}
− e_{α_{0} + α_{1} +(α_{2} + α_{0} + α_{3})+(α_{3} + α_{4})} + e_{α_{1} +(α_{2} + α_{0} + α_{3})+(α_{0} + α_{3} + α_{4})}
+ e_{(α_{0} + α_{2})+α_{3} +(α_{0} + α_{1} + α_{3} + α_{4})} − e_{(α_{0} + α_{2})+α_{3} +(α_{0} + α_{1} + α_{3})+α_{4}}
+ e_{α_{0} + α_{2} + α_{3} +(α_{0} + α_{1} + α_{3} + α_{4})} − e_{α_{0} + α_{2} + α_{3} +(α_{0} + α_{1} + α_{3})+α_{4}}
+ e_{α_{3} +(α_{0} + α_{1})+(α_{0} + α_{2} + α_{3} + α_{4})} − e_{α_{3} +(α_{0} + α_{1})+(α_{0} + α_{2} + α_{3})+α_{4}}
+ e_{α_{3} + α_{0} + α_{1} +(α_{0} + α_{2} + α_{3} + α_{4})} − e_{α_{3} + α_{0} + α_{1} +(α_{0} + α_{2} + α_{3})+α_{4}}
+ 2e_{α_{0} +(α_{3} + α_{4})+(α_{0} + α_{1} + α_{2} + α_{3})} − 2e_{(α_{0} + α_{3} + α_{4})+(α_{0} + α_{1} + α_{2} + α_{3})}
− 2e_{α_{0} + α_{3} +(α_{0} + α_{1} + α_{2} + α_{3} + α_{4})} + 2e_{α_{0} + α_{3} + α_{4} +(α_{0} + α_{1} + α_{2} + α_{3})}
− 2e_{(2 α_{0} + α_{1} + α_{2} α_{3} + α_{4})+α_{3})} − 2e_{α_{2} +(α_{0} + α_{1})+(α_{0} + α_{3} + α_{4})+α_{3}}
− 2e_{α_{1} +(α_{0} + α_{2})+(α_{0} + α_{3} + α_{4})+α_{3}} − 2e_{(α_{0} + α_{3} + α_{4})+α_{0} + α_{1} + α_{2} + α_{3}}
+ 2e_{(α_{0} + α_{3} + α_{4})+(α_{0} + α_{1} + α_{2})+α_{3}}

5.6.2. While in the D_{5} case the coefficients in the decomposition for the maximal root are ±1, ±2, in the D_{6} case the coefficients are ±1, ±2, ±4. We expect that in the D_{n} case all the coefficients are the powers of 2.

5.7. More open questions. The questions of decomposing “canonically” defined basis elements of affine Lie algebra (such as E_{α}^{∗}, E_{k}^{∗}(n), see 2.5) with respect to semicanonical basis make sense for any affine Lie algebra of type ADE. It would be very interesting to develop a purely combinatorial approach to these questions in the affine case as well.

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Department of Mathematics, Yale University, 10 Hillhouse Ave, New Haven, CT 06520

   E-mail address: frenkel@math.yale.edu

Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139

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

Department of Mathematics, University of Massachusetts, Lederle Graduate Research Tower, Amherst, MA 01003

   E-mail address: vybornov@math.umass.edu