Duality Functor Research Articles

Dg-Hecke Duality and Tensor Products

Abstract We continue our study of the monoidal category $D(G)$ begun in [ 12]. At the level of cohomology we transfer the duality functor $R\underline{\operatorname{Hom}}(-,k)$ to the derived category of dg-modules $D(H_{U}^{\bullet })$. In the process we develop a more general and streamlined approach to the anti-involution $\mathscr J$ from [ 8]. We also verify that the tensor product on $D(G)$ corresponds to an operadic tensor product on the dg-side (cf [ 5]). This uses a result of Schnürer on dg-categories with a model structure.

Twisted Drinfeld centers and framed string-nets

We discuss a string-net construction on 2 -framed surfaces, taking as algebraic input a finite, rigid tensor category, which is assumed to be neither pivotal nor semi-simple. It is shown that circle categories of our framed string-net construction essentially compute Drinfeld centers twisted by powers of the double dual functor.

Standard $\lambda$-lattices, rigid $\mathrm{C}^{*}$ tensor categories, and (bi)modules

In this article, we construct a 2 -shaded rigid \mathrm{C}^{*} multitensor category with canonical unitary dual functor directly from a standard \lambda -lattice. We use the notions of traceless Markov towers and lattices to define the notion of module and bimodule over standard \lambda -lattice(s), and we explicitly construct the associated module category and bimodule category over the corresponding 2 -shaded rigid \mathrm{C}^{*} multitensor category. As an example, we compute the modules and bimodules for Temperley–Lieb–Jones standard \lambda -lattices in terms of traceless Markov towers and lattices. Translating into the unitary 2-category of bigraded Hilbert spaces, we recover De Commer–Yamashita’s classification of \mathcal{T LJ} module categories in terms of edge weighted graphs, and a classification of \mathcal{T LJ} bimodule categories in terms of biunitary connections on square-partite weighted graphs. As an application, we show that every (infinite depth) subfactor planar algebra embeds into the bipartite graph planar algebra of its principal graph.

A Lie conformal superalgebra and duality of representations for E(4,4)

We construct a duality functor in the category of continuous representations of the Lie superalgebra E(4,4), the only exceptional simple linearly compact Lie superalgebra, for which it wasn't known. This is achieved by constructing a Lie conformal superalgebra of type (4,4) for which E(4,4) is the annihilation algebra. Along the way we obtain an explicit realization of E(4,4) by vector fields on a (4|4)-dimensional supermanifold.

A geometric interpretation of Ranicki duality

Consider a commutative ring $R$ and a simplicial map, $X\mathop {\longrightarrow }\limits ^{\pi }K,$ of finite simplicial complexes. The simplicial cochain complex of $X$ with $R$ coefficients, $\Delta ^*X,$ then has the structure of an $(R,K)$ chain complex, in the sense of Ranicki . Therefore it has a Ranicki-dual $(R,K)$ chain complex, $T \Delta ^*X$ . This (contravariant) duality functor $T:\mathcal {B} R_K\to \mathcal {B} R_K$ was defined algebraically on the category of $(R,K)$ chain complexes and $(R,K)$ chain maps. Our main theorem, 8.1, provides a natural $(R,K)$ chain isomorphism: \[ T\Delta^*X\cong C(X_K) \] where $C(X_K)$ is the cellular chain complex of a CW complex $X_K$ . The complex $X_K$ is a (nonsimplicial) subdivision of the complex $X$ . The $(R,K)$ structure on $C(X_K)$ arises geometrically.

External Spanier–Whitehead duality and homology representation theorems for diagram spaces

We construct a Spanier-Whitehead type duality functor relating finite $\mathcal{C}$-spectra to finite $\mathcal{C}^{\mathrm{op}}$-spectra and prove that every $\mathcal{C}$-homology theory is given by taking the homotopy groups of a balanced smash product with a fixed $\mathcal{C}^{\mathrm{op}}$-spectrum. We use this to construct Chern characters for certain rational $\mathcal{C}$-homology theories.

PBW theory for quantum affine algebras

Let U_q'(\mathfrak{g}) be a quantum affine algebra of arbitrary type and let \mathscr{C}_\mathfrak{g}^0 be Hernandez-Leclerc’s category. We can associate the quantum affine Schur–Weyl duality functor \mathcal{F}_\mathcal{D} to a duality datum \mathcal{D} in \mathscr{C}_\mathfrak{g}^0 . In this paper, we introduce the notion of a strong (complete) duality datum \mathcal{D} and prove that, when \mathcal{D} is strong, the induced duality functor \mathcal{F}_\mathcal{D} sends simple modules to simple modules and preserves the invariants \Lambda , \tilde\Lambda and \Lambda^\infty introduced by the authors. We next define the reflections \mathscr{S}_k and \mathscr{S}^{-1}_k acting on strong duality data \mathcal{D} . We prove that if \mathcal{D} is a strong (resp.\ complete) duality datum, then \mathscr{S}_k(\mathcal{D}) and \mathscr{S}_k^{-1}(\mathcal{D}) are also strong (resp. complete) duality data. This allows us to make new strong (resp. complete) duality data by applying the reflections \mathscr{S}_k and \mathscr{S}^{-1}_k from known strong (resp. complete) duality data. We finally introduce the notion of affine cuspidal modules in \mathscr{C}_\mathfrak{g}^0 by using the duality functor \mathcal{F}_\mathcal{D} , and develop the cuspidal module theory for quantum affine algebras similar to the quiver Hecke algebra case. When \mathcal{D} is complete, we show that all simple modules in \mathscr{C}_\mathfrak{g}^0 can be constructed as the heads of ordered tensor products of affine cuspidal modules. We further prove that the ordered tensor products of affine cuspidal modules have the unitriangularity property. This generalizes the classical simple module construction using ordered tensor products of fundamental modules.

Affine highest weight categories and quantum affine Schur-Weyl duality of Dynkin quiver types

For a Dynkin quiver Q Q (of type A D E \mathrm {ADE} ), we consider a central completion of the convolution algebra of the equivariant K K -group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc’s monoidal category C Q \mathcal {C}_{Q} of modules over the quantum loop algebra U q ( L g ) U_{q}(L\mathfrak {g}) via Nakajima’s homomorphism. As an application, we show that Kang-Kashiwara-Kim’s generalized quantum affine Schur-Weyl duality functor gives an equivalence between the category of finite-dimensional modules over the quiver Hecke algebra associated with Q Q and Hernandez-Leclerc’s category C Q \mathcal {C}_{Q} , assuming the simpleness of some poles of normalized R R -matrices for type E \mathrm {E} .

Cluster algebra structures on module categories over quantum affine algebras

We study monoidal categorifications of certain monoidal subcategories C J $\mathcal {C}_J$ of finite-dimensional modules over quantum affine algebras, whose cluster algebra structures on their Grothendieck rings K ( C J ) $K(\mathcal {C}_J)$ are closely related to the category of finite-dimensional modules over quiver Hecke algebra of type A ∞ $A_\infty$ via the generalized quantum Schur–Weyl duality functors. In particular, when the quantum affine algebra is of type A $A$ or B $B$ , the subcategory coincides with the monoidal category C g 0 $\mathcal {C}_{\mathfrak {g}}^0$ introduced by Hernandez–Leclerc. As a consequence, the modules corresponding to cluster monomials are real simple modules over quantum affine algebras.

Homological duality for covering groups of reductive $p$-adic groups

In this largely expository paper we extend properties of the homological duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is the Hecke algebra of a reductive $p$-adic group, to the case where it is the Hecke algebra of a finite central extension of a reductive $p$-adic group. The most important properties being that $RHom_{\mathcal H}(-,{\mathcal H})$ is concentrated in a single degree for irreducible representations and that it gives rise to Schneider--Stuhler duality for Ext groups (a Serre functor like property). Along the way we also study Grothendieck--Serre duality with respect to the Bernstein center and provide a proof of the folklore result that on admissible modules this functor is nothing but the contragredient duality. We single out a necessary and sufficient condition for when these three dualities agree on finite length modules in a given block. In particular, we show this is the case for all cuspidal blocks as well as, due to a result of Roche, on all blocks with trivial stabilizer in the relative Weyl group.

Dual spaces of operator systems

The aim of this article is to give an infinite dimensional analogue of a result of Choi and Effros concerning dual spaces of finite dimensional unital operator systems. An (not necessarily unital) operator system is a self-adjoint subspace of L(H), equipped with the induced matrix norm and the induced matrix cone. We say that an operator system T is dualizable if one can find an equivalent dual matrix norm on the dual space T⁎ such that under this dual matrix norm and the canonical dual matrix cone, T⁎ becomes a dual operator system. We show that an operator system T is dualizable if and only if the ordered normed space M∞(T)sa satisfies a form of bounded decomposition property. In this case,‖f‖d:=sup⁡{‖[fi,j(xk,l)]‖:x∈Mn(T)+;‖x‖≤1;n∈N}(f∈Mm(T⁎);m∈N), is the largest dual matrix norm that is equivalent to and dominated by the original dual matrix norm on T⁎ that turns it into a dual operator system, denoted by Td. It can be shown that Td is again dualizable. For every completely positive completely bounded map ϕ:S→T between dualizable operator systems, there is a unique weak-⁎-continuous completely positive completely bounded map ϕd:Td→Sd which is compatible with the dual map ϕ⁎. From this, we obtain a full and faithful functor from the category of dualizable operator systems to that of dualizable dual operator systems. Moreover, we will verify that if S is either a C⁎-algebra or a unital operator system, then S is dualizable and the canonical weak-⁎-homeomorphism from the unital operator system S⁎⁎ to the operator system (Sd)d is a completely isometric complete order isomorphism. Furthermore, via the duality functor above, the category of C⁎-algebras and that of unital operator systems (both equipped with completely positive complete contractions as their morphisms) can be regarded as full subcategories of the category of dual operator systems (with weak-⁎-continuous completely positive complete contractions as its morphisms). Consequently, a nice duality framework for operator systems is obtained, which includes all C⁎-algebras and all unital operator systems.

Cosheaf representations of relations and Dowker complexes

The Dowker complex is an abstract simplicial complex that is constructed from a binary relation in a straightforward way. Although there are two ways to perform this construction—vertices for the complex are either the rows or the columns of the matrix representing the relation—the two constructions are homotopy equivalent. This article shows that the construction of a Dowker complex from a relation is a non-faithful covariant functor. Furthermore, we show that this functor can be made faithful by enriching the construction into a cosheaf on the Dowker complex. The cosheaf can be summarized by an integer weight function on the Dowker complex that is a complete isomorphism invariant for the relation. The cosheaf representation of a relation actually embodies both Dowker complexes, and we construct a duality functor that exchanges the two complexes.

-modules on rigid analytic spaces III: weak holonomicity and operations

AbstractWe develop a dimension theory for coadmissible$\widehat {\mathcal {D}}$-modules on rigid analytic spaces and study those which are of minimal dimension, in analogy to the theory of holonomic$\mathcal {D}$-modules in the algebraic setting. We discuss a number of pathologies contained in this subcategory (modules of infinite length, infinite-dimensional fibres). We prove stability results for closed immersions and the duality functor, and show that all higher direct images of integrable connections restricted to a Zariski open subspace are coadmissible of minimal dimension. It follows that the local cohomology sheaves$\underline {H}^{i}_Z(\mathcal {M})$with support in a closed analytic subset$Z$of$X$are also coadmissible of minimal dimension for any integrable connection$\mathcal {M}$on$X$.

Equivalence between module categories over quiver Hecke algebras and Hernandez–Leclerc's categories in general types

We prove in full generality that the generalized quantum affine Schur–Weyl duality functor, introduced by Kang–Kashiwara–Kim, gives an equivalence between the category of finite-dimensional modules over a quiver Hecke algebra and a certain full subcategory of finite-dimensional modules over a quantum affine algebra which is a generalization of the Hernandez–Leclerc's category CQ. This was previously proved in untwisted ADE types by Fujita using the geometry of quiver varieties, which is not applicable in general. Our proof is purely algebraic, and so can be extended uniformly to general types.

PBW theoretic approach to the module category of quantum affine algebras

Let $U_{q}'(\mathfrak{g})$ be a quantum affine algebra of untwisted affine ADE type and let $\mathcal{C}_{\mathfrak{g}}^{0}$ be Hernandez-Leclerc’s category. For a duality datum $\mathcal{D}$ in $\mathcal{C}_{\mathfrak{g}}^{0}$, we denote by $\mathcal{F}_{\mathcal{D}}$ the quantum affine Weyl-Schur duality functor. We give a sufficient condition for a duality datum $\mathcal{D}$ to provide the functor $\mathcal{F}_{\mathcal{D}}$ sending simple modules to simple modules. Moreover, under the same condition, the functor $\mathcal{F}_{\mathcal{D}}$ has compatibility with the new invariants introduced by the authors. Then we introduce the notion of cuspidal modules in $\mathcal{C}_{\mathfrak{g}}^{0}$, and show that all simple modules in $\mathcal{C}_{\mathfrak{g}}^{0}$ can be constructed as the heads of ordered tensor products of cuspidal modules. We next state that the ordered tensor products of cuspidal modules have the unitriangularity property.

A Functorial Presentation of Units of Burnside Rings

Let \(B^\times \) be the biset functor over \(\mathbb {F}_2\) sending a finite group G to the group \(B^\times (G)\) of units of its Burnside ring B(G), and let \(\widehat{B^\times }\) be its dual functor. The main theorem of this paper gives a characterization of the cokernel of the natural injection from \(B^\times \) in the dual Burnside functor \(\widehat{\mathbb {F}_2B}\), or equivalently, an explicit set of generators \(\mathcal {G}_\mathcal {S}\) of the kernel L of the natural surjection \(\mathbb {F}_2B\rightarrow \widehat{B^\times }\). This yields a two terms projective resolution of \(\widehat{B^\times }\), leading to some information on the extension functors \(\mathrm{Ext}^1(-,B^\times )\). For a finite group G, this also allows for a description of \(B^\times (G)\) as a limit of groups \(B^\times (T/S)\) over sections (T, S) of G such that T/S is cyclic of odd prime order, Klein four, dihedral of order 8, or a Roquette 2-group. Another consequence is that the biset functor \(B^\times \) is not finitely generated, and that its dual \(\widehat{B^\times }\) is finitely generated, but not finitely presented. The last result of the paper shows in addition that \(\mathcal {G}_\mathcal {S}\) is a minimal set of generators of L, and it follows that the lattice of subfunctors of L is uncountable.

Lie conformal superalgebras and duality of modules over linearly compact Lie superalgebras

We construct a duality functor on the category of continuous representations of linearly compact Lie superalgebras, using representation theory of Lie conformal superalgebras. We compute the dual representations of the generalized Verma modules.

English

We classify which dual functors on a unitary multitensor category are compatible with the dagger structure in terms of groupoid homomorphisms from the universal grading groupoid to $\mathbb{R}_{>0}$ where the latter is considered as a groupoid with one object. We then prove that all unitary dual functors induce unitarily equivalent bi-involutive structures. As an application, we provide the unitary version of the folklore correspondence between shaded planar ${\rm C^*}$ algebras with finite dimensional box spaces and unitary multitensor categories with a chosen unitary dual functor and chosen generator. We make connection with the recent work of Giorgetti-Longo to determine when the loop parameters in these planar algebras are scalars. Finally, we show that we can correct for many non-spherical choices of dual functor by adding the data of a spherical state on $\operatorname{End}_{\mathcal{C}}(1_{\mathcal{C}})$, similar to the spherical state for a graph planar algebra.

Schur algebras for the alternating group and Koszul duality

We introduce the alternating Schur algebra $AS_F(n,d)$ as the commutant of the action of the alternating group $A_d$ on the $d$-fold tensor power of an $n$-dimensional $F$-vector space. When $F$ has characteristic different from $2$, we give a basis of $AS_F(n,d)$ in terms of bipartite graphs, and a graphical interpretation of the structure constants. We introduce the abstract Koszul duality functor on modules for the even part of any $\mathbf Z/2\mathbf Z$-graded algebra. The algebra $AS_F(n,d)$ is $\mathbf Z/2\mathbf Z$-graded, having the classical Schur algebra $S_F(n,d)$ as its even part. This leads to an approach to Koszul duality for $S_F(n,d)$-modules that is amenable to combinatorial methods. We characterize the category of $AS_F(n,d)$-modules in terms of $S_F(n,d)$-modules and their Koszul duals. We use the graphical basis of $AS_F(n,d)$ to study the dependence of the behavior of derived Koszul duality on $n$ and $d$.

Structure and representation of semimodules over inclines

An incline S is a commutative semiring where r+1=1 for any r∈S. We note that the ideal lattice of an S-semimodule is naturally an S-semimodule and so is its congruence lattice when S is transitive. We prove that the categories of complete S-semimodules, together with dual functor, internal hom and tensor product, is a ⋆-autonomous category. We define the locally and globally maximal congruences which are related to Birkhoff subdirect product decomposition. We show that the categories of S-semimodules, algebraic S-semimodules and topological S-semimodules are equivalent. Finally, we get a sheaf representation of any S-semimodule.