POLARIZATIONS OF TRIANGULATED CATEGORIES.

WILL TURNER

Abstract. We introduce the notion of a polarization of triangulated categories.

1. Introduction

The term noncommutative geometry is somewhat provocative [8]. In a weak sense, its meaning is obvious: geometry in which functions are allowed to be noncommutative. Or, to phrase it another way, the study of noncommutative rings. But in what sense is this geometry ? If the term is to have a serious meaning, one must interpret geometric notions in the noncommutative setting.

According to one formulation, a noncommutative manifold is defined to be a triangulated category, as introduced by J-L. Verdier circa 1970 [14]. Thus, the character of a noncommutative space is encoded in the homological algebra of the corresponding triangulated category. Geometric notions should thus be defined in homological terms, using objects, morphisms, and functors.

Which classical geometric notions have a setting in this context ? One is dimension. Indeed, R. Rouquier has defined a dimension for triangulated categories, leading to solutions of problems concerning finite dimensional algebras [16]. A second is Serre duality, which has a natural formulation for triangulated categories. A third example is stratification: A. Beilinson, J. Bernstein and P. Deligne have defined a recollement of triangulated categories [2] which, in a noncommutative setting, has lead to the theory of highest weight categories [7].

In Riemannian geometry, the sequence of inclusions of holonomy groups,

SO(n) > U(\frac{n}{2}) > SU(\frac{n}{2}) > Sp(\frac{n}{4})

defines a hierarchy of manifold types,

(Riemannian) ⊃ (Kähler) ⊃ (Calabi-Yau) ⊃ (Holomorphic Symplectic). M. Kontsevich has defined a Calabi-Yau triangulated category to be a triangu-

lated category with Serre duality, whose Serre functor is some power of the shift functor (see [9]). Such categories have generated a large amount of interest for various reasons, not least the grandiose claims made by string theorists, concerning the significance of Calabi-Yau manifolds for the structure of the universe. However, there are interesting mathematical reasons for contemplating Calabi-Yau categories. To give one, a conjecture of Kontsevich predicts equivalences between triangulated categories associated to mirror Calabi-Yau manifolds [12]. A second, which makes best sense in the noncommutative setting, is the interpretation of algebras whose derived categories are Calabi-Yau of dimension three via quivers with superpotentials [9].

An obvious question is whether one can understand the Holomorphic Symplectic restriction in homological terms. I do not know a general answer to this. However, in this paper, we describe a homological condition which appears to be somewhat symplectic in nature. The condition is the existence of a polarization on a triangulated category.

A basic example of a triangulated category which admits a polarization is the derived category of the ring of functions on a symplectic vector space. We also expect polarizations to arise in certain situations in Lie theory. A rich class of such examples is given by derived categories of Cubist algebras.

2. Polarizations

Let k be a field. We denote by V^{∗} the k-linear dual of a k-vector space V.

Let T be a k-linear triangulated category.

We denote the opposite of T by T^{op}. The category T^{op} is triangulated; a shift by i in T^{op} is identified with a shift by −i in T . For any exact functor ψ : R → T between triangulated categories, we have a corresponding exact functor ψ^{∨} : R^{op} → T^{op}.

Let M be a collection of objects in T . We say M generates T , if Hom(X, M[i]) = 0, for all i ∈ ℤ, and all M ∈ M, implies X = 0.

Let M be an object in T . We say M is compact if the functor Hom(M, −) commutes with infinite direct sums. We denote by T^{c} the thick subcategory of T of compact objects.

T is said to be Calabi-Yau of dimension d, if

Hom(M, N) ≅ Hom(N, M[d])^{∗},

naturally in M ∈ T^{c}, N ∈ T .

Let Q be a quiver. Let A = k[[Q]]/I be a quotient of the completed path algebra of Q by an ideal of relations I ⊂ kQ_{≥1} generated in degrees ≥ 1. Then

A = lim_{n}(kQ/(kQ_{≥n} + I))

is an inverse limit of finite dimensional algebras. We write K = k[[Q]]/k[[Q]]_{≥1}, a semisimple A-module.

We write A -mod for the category of finitely generated A-modules, and D^{b}(A -mod) for the corresponding bounded derived category.

Definition 1. The algebra A is Calabi-Yau of dimension d, if the derived categories D^{b}(A -mod) and D^{b}(A^{op} -mod) are Calabi-Yau of dimension d.

Example 2 Let A = k[[V]] be the power series ring on a vector space of dimension d. Then A is Calabi-Yau of dimension d.

Theorem 3. Suppose A is Calabi-Yau of dimension d. Then we have an equivalence of triangulated categories,

RHom(−, A) : D^{b}(A -mod) → D^{b}(A^{op} -mod)^{op},

sending an irreducible A-module S to S^{∗}[d].

Proof. Suppose d = 0. Then A ≅ RHom(A, A) ≅ RHom(A, A)^{∗} ≅ A^{∗}, as A-A-bimodules, and so A is a symmetric algebra [15]. Therefore _{A}A and A_{A} have injective dimension zero.

Suppose d > 0. Then Ext^{i}(M, A) ≅ RHom(A, M[d−i])^{∗} ≅ 0, for all A-modules M, and i > d, and therefore _{A}A has injective dimension ≤ d. Working with A^{op}- modules, we find that A_{A} has injective dimension ≤ d.

In conclusion, whether d is zero or not, the injective dimensions of _{A}A and A_{A} are finite. Therefore, the bimodule_{A}A_{A} defines a dualizing complex for A, implying the aforementioned equivalence [18].

Finally, thanks to the Calabi-Yau condition, we have

RHom(S, A) ≅ RHom(A, S[d])^{∗} ≅ S^{∗}[d].
□

We would like to know what it means for a noncommutative variety to be holomorphic symplectic. We do not know a general answer to this. However, the following definition carries the flavour of symplecticism, and of the Calbi-Yau property, by Theorem 3.

Definition 4. A polarization of T is a quadruple (ϕ, X^{+}, X^{−}, d), where d ∈ ℤ_{≥0}, where X^{+}, X^{−} are generating collections of objects in T , where

Hom(X^{+}, X^{−}[i]) = Hom(X^{−}, X^{+}[i]) = 0, for i ≠ d,

and where ϕ : T → T^{op} is an equivalence, such that

ϕ(X^{+}) = X^{−}, ϕ(X^{−}) = X^{+}, ϕ^{∨}.ϕ = 1.

The pair (X^{+}, X^{−}) defining a polarization are analogous to a complementary pair of Lagrangian subspaces of a symplectic vector spaces, as illustrated by the following proposition.

Let W be a symplectic vector space of dimension 2w, with alternating form ⟨−, −⟩. Let us fix a basis of W, such that ⟨v, w⟩ = v^{T} Jw, with respect to this basis, where J = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ).

Proposition 5. Let G < Sp(W) be a finite group, such that G = G^{T}. Let A = k[[W]] ⋊ G. Then the triangulated category D^{b}(A -mod) admits a polarization (ϕ, Δ^{+}, Δ^{−}, w).

Before proving Proposition 5, let us record a lemma.

Lemma 6. Let A be a Calabi-Yau algebra of dimension d, and let S be a finite dimensional irreducible A-module. Then Hom_{A}(S, M[d]) ≅ S^{∗} ⊗_{A} M, for all irreducible A-modules S, and all free A-modules M.

Proof. We have S^{∗} ⊗_{A} A ≅ Hom_{k}(S^{∗} ⊗_{A} A, k)^{∗} ≅ Hom_{A}(A, Hom_{k}(S^{∗}, k))^{∗} ≅

Hom_{A}(A, S)^{∗} ≅ Hom_{A}(S, A[d]).

Consequently the functors S^{∗} ⊗_{A} − and Hom_{A}(S, −[d]) are isomorphic on the category of free A-modules. □

We now prove Proposition 5.

Proof. Let W = V^{+} ⊕ V^{−} be a decomposition into Lagrangian subspaces, where J is an endomorphism of W which exchanges V^{+}, V^{−}. The group Sp(W) is the set of matrices g, such that gJg^{T} = J.

The automorphism J of W extends to an automorphism of k[[W]]. Let σ(g.x) = x^{J}.g^{T} ∈ A, for x ∈ k[[W]], g ∈ G. The condition G = G^{T} implies that σ sends A to A. The condition that G is symplectic implies that σ is an anti-automorphism of A:

σ((g′x′).(gx)) = σ(g′gx^{′g}x) = (x^{′g}x)^{J}(g′g)^{T} = x^{′gJ}x^{J}g^{T} g^{′T} ,
σ(gx).σ(g′x′) = x^{J}g^{T} x^{′J}g^{′T} = x^{J}x^{′Jg^{−T}} g^{T} g^{′T} = x^{′gJ}x^{J}g^{T} g^{′T} .

The functor ϕ = RHom(−, A[w])^{σ} gives an equivalence between D^{b}(A -mod) and D^{b}(A -mod)^{op}, by Theorem 3.

Let

Δ^{+} = A ⊗_{k[[V^{+}]]} k,
Δ^{−} = A ⊗_{k[[V^{−}]]} k.

We have isomorphisms of vector spaces,

Δ^{+} ≅ kG ⊗ k[[V^{−}]], Δ^{−} ≅ kG ⊗ k[[V^{+}]].

The free module _{A}A is the limit of a filtration, with sections Δ^{+}. Consequently Δ^{+} generates D^{b}(A -mod). Similarly Δ^{−} generates D^{b}(A -mod).

We wish to show that Δ^{+} and Δ^{−} are exchanged by ϕ. Inducing the Koszul complex

k[[V^{+}]] ⊗ ⋀(V^{+})^{∗} ↠ k

for k[[V^{+}]] up to A, we obtain a projective resolution,

A ⊗ ⋀(V^{+})^{∗} ↠ Δ^{+}.

Applying Hom(−, A[w]) to our resolution of Δ^{+}, we obtain a complex Hom_{A}(A ⊗ ⋀(V^{+})^{∗}, A[w]) ≅

Hom_{k}(⋀(V^{+})^{∗}, A[w]) ≅
Hom_{k}(⋀(V^{+})^{∗}, Hom_{k[[V^{+}]]}(k[[V^{+}]], A[w])) ≅
Hom_{k[[V^{+}]]}(k[[V^{+}]] ⊗ ⋀(V^{+})^{∗}, A[w]),

whose cohomology is RHom_{k[[V^{+}]]}(k, A[w]). However,

RHom_{k[[V^{+}]]}(k, A[w]) ≅ k ⊗_{k[[V^{+}]]} A,

by Lemma 6, since A is a free k[[V^{+}]]-module, and k[[V^{+}]] is Calabi-Yau of dimension w. Applying σ, we obtain A ⊗_{k[[V^{−}]]} k = Δ^{−}. Thus, ϕ(Δ^{+}) = Δ^{−}.

An identical calculation shows that ϕ(Δ^{−}) = Δ^{+}.

We finally show that Δ^{+} and Δ^{−} satisfy the required homological orthogonality properties. Indeed, applying Hom_{A}(−, Δ^{−}[w]) to our resolution of Δ^{+}, we obtain a complex

Hom_{A}(A ⊗ ⋀(V^{+})^{∗}, Δ^{−}[w]) ≅
Hom_{k}(⋀(V^{+})^{∗}, kG ⊗ k[[V^{+}]][w]),

whose cohomology computes RHom(Δ^{+}, Δ^{−}[w]). This complex is isomorphic to a complex

kG ⊗ Hom_{k[[V^{+}]]}(k[[V^{+}]] ⊗ ⋀(V^{+})^{∗}, k[[V^{+}]][w]),

whose cohomology is

kG ⊗ RHom_{k[[V^{+}]]}(k, k[[V^{+}]][w]) ≅ kG ⊗ k.

This proves the required Ext-orthogonality. □

Remark 7 Let W be a complex symplectic vector space, on which a finite group G < Sp(W) acts. Let { \atop W/G} denote a resolution of the singular space W/G, such that the symplectic form induced on the smooth part of W/G has an extension to a nondegenerate closed 2-form on { \atop W/G}. D. Kaledin and R. Bezrukavnikov have proved the McKay correspondence

D^{b}({ \atop W/G} -coh) ≅ D^{b}(k[W] ⋊ G -mod)

holds, in this case [4] (cf. the article of T. Bridgeland, A. King, and M. Reid [5]).

Example 8 Let A be a Calabi-Yau algebra of dimension d. The triangulated category T = D^{b}(Aˆ⊗A^{op} -mod) admits a polarization

X^{+} = A ⊗ K^{op}, X^{−} = K ⊗ A^{op}, ϕ = RHom(−, Aˆ⊗A^{op}[d])^{σ},

where σ denotes a twist by the anti-automorphism of Aˆ⊗A^{op} which swaps A, A^{op}.

Remark 9 Let us comment on V. Ginzburg’s approach to the Calabi-Yau property from the polarized perspective [9]. According to Ginzburg, a Calabi-Yau algebra of dimension d is an algebra A, with an isomorphism

f : A → RHom_{A⊗A^{op}}(A, A ⊗ A^{op}[d])

of A-A-bimodules, such that

f = RHom(A ⊗ A^{op})(f, A ⊗ A^{op}[d]).

Suppose that A is a Calabi-Yau algebra, according to definition 1, as well as Ginzburg’s definition. Then the triangulated category T = D^{b}(Aˆ⊗A^{op} -mod) admits a polarization, by example 8. We also have an isomorphism f : A → ϕ(A), such that ϕ(f) = f. In other words, _{A}A_{A} is a fixed point under ϕ.

If T is a triangulated category admitting a polarization (ϕ, X^{+}, X^{−}, d), we define a CY object to be a fixed point under ϕ.

3. Lie theory

We consider polarizations on triangulated categories arising in Lie theory.

Throughout this chapter, we work with ℤ_{+}-graded k-algebras A = ⊕_{i∈ℤ_{≥0}}A^{i}, where A^{0} has a k-basis {e_{s} | s ∈ R}, consisting of idempotents e_{s}. We assume the finiteness condition dim e_{s}A_{i} < ∞ for all s ∈ R and i ∈ ℤ. Let A be the category of all graded A-modules, where the space of morphisms between graded modules M and N, which we denote Hom_{A}(M, N), consists of A-module homomophisms preserving degree. We denote by A -god the full subcategory consisting of modules M such that dim e_{s}M_{i} < ∞ for all s ∈ R and i ∈ ℤ, and that M_{i} = 0 for i << 0.

If M = ⊕_{i∈ℤ}M^{i} is a ℤ-graded module, let M⟨i⟩ denote the i^{th} degree shift of M. Thus (M⟨j⟩)^{n} = M^{j−n}. Let M̃ = ⊕_{i∈ℤ}M⟨i⟩.

For definitions, and further technicalities concerning highest weight categories [7], cellularity [10], and standard Koszulity [1] in this setting, the reader is invited to consult the article ”Cubist algebras” [6].

Theorem 10. (see [6]) Let A, B be Koszul dual algebras. Then D^{b}(A -god) is Calabi-Yau if, and only if, B is super-symmetric.

So the Calabi-Yau condition has a natural formulation for Koszul algebras, in terms of its Koszul dual. Beneath, we present a sufficient condition for the derived category of a Koszul algebra of global dimension 2w to possess a polarization, in terms of its Koszul dual. Such conditions are rather intricate, but are satisfied by certain algebras arising in Lie theory.

Theorem 11. Let A, B be Koszul dual algebras. Suppose that B is self-injective, and that B is standard Koszul, with respect to partial orderings ⪯_{i} of a set Λ, for i ∈ ℤ/2. Suppose that the standard modules, taken with respect to ⪯_{i}, are also standard modules for a graded cellular structure on B, for i ∈ ℤ/2. Suppose that all standard B-modules have Loewy length w+1, and there exists a bijection θ : Λ → Λ, such that

Δ_{i}(λ) ≅ ∇_{i+1}(λ^{θ}),
Hom(Δ_{i}(λ), ∇_{i}(μ)) = 0,

for λ ≠ μ ∈ Λ, i ∈ ℤ/2. Then D^{b}(A -god) admits a polarization (ϕ, Δ_{0}, Δ_{1}, w).

Proof. We first construct our antiautomorphism ϕ. An algebra A is Koszul if, and only if, its opposite algebra A^{op} is Koszul. Let us note that Koszul duality, on the left and right, exhanges the regular A-A-bimodule with the regular B-B-bimodule. Consequently, the diagram

\xymatrix{D^{b}(A -god) \ar[r]^{RHom(−,Ã)} \ar[d]^{K} & D^{b}(A^{op} -god)^{op} \ar[d]^{K^{op}} \\ D^{b}(B -god) \ar[r]^{RHom(−,B̃)} & D^{b}(B^{op} -god)^{op}}

commutes, where K denotes the Koszul duality equivalence [3]. The algebra B is graded cellular, and and therefore admits a graded involutory antiautomorphism ω_{B}, such that S^{∗ω} ≅ S, for all simple modules S, by a result of S. König and C. Xi ([11], theorem 1.1). We thus have a commuting diagram of equivalences

\xymatrix{D^{b}(A^{op} -god) \ar[r]^{θ} \ar[d]^{K} & D^{b}(A -god) \ar[d]^{K} \\ D^{b}(B^{op} -god) \ar[r]^{ω_{B}} & D^{b}(B -god).}

The Koszul duality functor K takes a projective A^{op}-module to a corresponding simple B^{op}-module, and a projective A-module to a corresponding B-module, whilst a twist by ω_{B} takes a simple module to a simple module. It follows that the top horizontal functor θ takes the regular A^{op}-module to the regular A-module, thus defining an antihomomorphism A^{op} → A. By symmetry, this antihomomorphism has an inverse, and therefore have an involutory antiautomorphism ω_{A} of A, such that the following diagram commutes:

\xymatrix{D^{b}(A^{op} -god) \ar[r]^{ω_{A}} \ar[d]^{K} & D^{b}(A -god) \ar[d]^{K} \\ D^{b}(B^{op} -god) \ar[r]^{ω_{B}} & D^{b}(B -god).}

Gluing diagrams, we have a commutative diagram

\xymatrix{D^{b}(A -god) \ar[r]^{RHom(−,Ã)} \ar[d]^{K} & D^{b}(A^{op} -god)^{op} \ar[r]^{ω_{A}} \ar[d]^{K^{op}} & D^{b}(A -god)^{op} \ar[d]^{K^{op}} \\ D^{b}(B -god) \ar[r]^{RHom(−,B̃)} & D^{b}(B^{op} -god)^{op} \ar[r]^{ω_{B}} & D^{b}(B -god)^{op}.}

We define ϕ to be RHom(−, Ã)^{ω_{A}}[w].

Thus, B is a positively graded self-injective algebra. By a theorem of Martinez-Villa, its projective modules all have the same Loewy length ([13], proposition 3.2). Because standard modules have Loewy length w + 1, they are concentrated in degrees 0, 1, ..., w. Since B is a graded cellular algebra, with a filtration by twosided ideals of the form Δ⊗Δ^{ω}, the algebra B is concentrated in degrees 0, 1, ..., 2w. Consequently, projective B-modules all have Loewy length 2w + 1.

Let

Δ_{i}^{A} = {Δ_{i}^{A}(λ)⟨j⟩|λ ∈ Λ, j ∈ ℤ}, Δ_{i}^{B} = {Δ_{i}^{B}(λ)⟨j⟩|λ ∈ Λ, j ∈ ℤ}

for i ∈ ℤ.

The collection Δ_{i}^{A} generates D^{b}(A -god), for i ∈ ℤ/2, since projective A-modules are filtered by standard modules.

Since B is a self-injective algebra, RHom_{B}(−, B̃) ≅ Hom_{B}(−, B̃) is isomorphic to the duality functor, twisted by the Nakayama automorphism. Therefore Hom_{B}(−, B̃)^{ω_{B}} sends a standard module Δ_{i}^{B}(λ) to a costandard module ∇_{i}^{B}(h(λ)), shifted in degree. Since S^{∗ω} ≅ S, for all simple modules S, we have h(λ) = λ. Since B is concentrated in degrees 0, 1, ..., 2w, and ∇(λ) has Loewy length w, the degree shift is w. Therefore, Δ_{i}^{B} ≅ Hom(Δ_{i+1}^{B}, B̃)^{ω_{B}}⟨w⟩.

We have K(Δ_{i}^{A}) = Δ_{i}^{B}, by standard Koszulity ([1], theorem 2.7). Our commutative square shows us we have

ϕ(Δ_{i}^{A}) = RHom(Δ_{i}^{A}, Ã[w])^{ω} ≅
K^{−1}(RHom(Δ_{i}^{B}, B̃[w])^{ω}) ≅ K^{−1}(Δ_{i+1}^{B}⟨w⟩[w]) ≅
Δ_{i+1}^{A}⟨−w⟩[−w][w] ≅ Δ_{i+1}^{A}.

We finally prove the required Ext-orthogonality relation between Δ_{1}^{A} and Δ_{2}^{A}. Computing in the derived category of graded modules, we have

Hom(Δ_{i}^{A}(λ), Δ_{i+1}^{A}(μ)⟨l⟩[j]) ≅
Hom(Δ_{i}^{B}(λ), Δ_{i+1}^{B}(μ)⟨−l⟩[j − l]) ≅
Hom(Δ_{i}^{B}(λ), ∇_{i}^{B}(μ)⟨−l⟩[j − l]),

which is isomorphic to zero unless j = l, since B is quasi-hereditary. But our assumptions decree that Hom(Δ_{i}^{B}(λ), ∇_{i+1}^{B}(μ)⟨−j⟩) is isomorphic to k, if λ = μ and j = w, and and zero otherwise. This completes the proof of the theorem. □

Let T be a tiling of Euclidean space by parallelohedra. We have associated to such a tiling a Koszul dual pair of algebras U_{T}, and V_{T}, the Cubist algebras associated to T [17].

Theorem 12. The derived category D^{b}(V_{T} -god) admits a polarization.

Proof. This is an application of Theorem 11 (see [17]). □

Remark 13 The above theorem applies more generally to parallelohedral tilings of Euclidean space of any dimension which are ordered, in a suitable sense [17].

We expect that polarizations can be found in other situations in Lie theory. Let us give an example.

Let G be a semimple algebraic group, defined over a field whose characteristic is greater than the Coxeter number of G. Let T be a maximal torus in G, and u its restricted enveloping algebra. Let û denote the basic associative algebra, whose category of finite dimensional modules is equivalent to the category of finite dimensional u-T -modules.

Question 14. Does B = û satisfy the hypotheses of theorem 11 ?

If Question 14 has an affirmative answer for G, then the derived category D^{b}(B -god) of B admits a polarization, by Theorem 11. The question has a positive answer, for G = GL_{2}. A positive answer also seems plausible, in case G = GL_{3}.

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