Braided Monoidal Category Research Articles

Quantum character varieties and braided module categories

We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants int _S{mathcal {A}} of a surface S, determined by the choice of a braided tensor category {mathcal {A}}, and computed via factorization homology. We identify the algebraic data governing marked points and boundary components with the notion of a braided module category for {mathcal {A}}, and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called quantum moment maps. We then show that the quantum character variety of a decorated surface is obtained from that of the corresponding punctured surface as a quantum Hamiltonian reduction. Characters of braided {mathcal {A}}-modules are objects of the torus category int _{T^2}{mathcal {A}}. We initiate a theory of character sheaves for quantum groups by identifying the torus integral of {mathcal {A}}={text {Rep}}_{q}G with the category {mathcal {D}}_q(G/G)-mod of equivariant quantum {mathcal {D}}-modules. When G=GL_n, we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra {mathbb {SH}}_{q,t}.

Group actions on 2-categories

We study actions of discrete groups on 2-categories. The motivating examples are actions on the 2-category of representations of finite tensor categories and their relation with the extension theory of tensor categories by groups. Associated to a group action on a 2-category, we construct the 2-category of equivariant objects. We also introduce the G-equivariant notions of pseudofunctor, pseudonatural transformation and modification. Our first main result is a coherence theorem for 2-categories with an action of a group. For a 2-category $${\mathcal B}$$ with an action of a group G, we construct a braided G-crossed monoidal category $$\mathcal {Z}_G({\mathcal B})$$ with trivial component the Drinfeld center of $${\mathcal B}$$ . We prove that, in the case of a G-action on the 2-category of representation of a tensor category $${\mathcal C}$$ , the 2-category of equivariant objects is biequivalent to the module categories over an associated G-extension of $${\mathcal C}$$ . Finally, we prove that the center of the equivariant 2-category is monoidally equivalent to the equivariantization of a relative center, generalizing results obtained in Gelaki et al. (Algebra Number Theory 3(8):959–990, 2009).

Classification of symmetry-protected phases for interacting fermions in two dimensions

Recently, it has been shown that two-dimensional bosonic symmetry-protected topological(SPT) phases with on-site unitary symmetry $G$ can be completely classified by the group cohomology class $H^3(G, \mathrm{U}(1))$. Later, group super-cohomology class was proposed as a partial classification for SPT phases of interacting fermions. In this work, we revisit this problem based on the mathematical framework of $G$-extension of unitary braided tensor category(UBTC) theory. We first reproduce the partial classifications given by group super-cohomology, then we show that with an additional $H^1(G, \mathbb{Z}_2)$ structure, a complete classification of SPT phases for two-dimensional interacting fermion systems for a total symmetry group $G\times\mathbb{Z}_2^f$ can be achieved. We also discuss the classification of interacting fermionic SPT phases protected by time-reversal symmetry.

Nichols algebras over weak Hopf algebras

In this paper, we study a Yetter-Drinfeld module V over a weak Hopf algebra ℍ. Although the category of all left ℍ-modules is not a braided tensor category, we can define a Yetter-Drinfeld module. Using this Yetter-Drinfeld modules V, we construct Nichols algebra B(V) over the weak Hopf algebra ℍ, and a series of weak Hopf algebras. Some results of [8] are generalized.

Knots, Music and DNA

Musical gestures connect the symbolic layer of the score to the physical layer of sound. I focus here on the mathematical theory of musical gestures, and I propose its generalization to include braids and knots. In this way, it is possible to extend the formalism to cover more case studies, especially regarding conducting gestures. Moreover, recent developments involving comparisons and similarities between gestures of orchestral musicians can be contextualized in the frame of braided monoidal categories. Because knots and braids can be applied to both music and biology (they apply to knotted proteins, for example), I end the article with a new musical rendition of DNA.

Space in Monoidal Categories

The category of Hilbert modules may be interpreted as a naive quantum field theory over a base space. Open subsets of the base space are recovered as idempotent subunits, which form a meet-semilattice in any firm braided monoidal category. There is an operation of restriction to an idempotent subunit: it is a graded monad on the category, and has the universal property of algebraic localisation. Spacetime structure on the base space induces a closure operator on the idempotent subunits. Restriction is then interpreted as spacetime propagation. This lets us study relativistic quantum information theory using methods entirely internal to monoidal categories. As a proof of concept, we show that quantum teleportation is only successfully supported on the intersection of Alice and Bob's causal future.

A 2-categorical extension of Etingof–Kazhdan quantisation

Let k be a field of characteristic zero. Etingof and Kazhdan constructed a quantisation U_h(b) of any Lie bialgebra b over k, which depends on the choice of an associator Phi. They prove moreover that this quantisation is functorial in b. Remarkably, the quantum group U_h(b) is endowed with a Tannakian equivalence F_b from the braided tensor category of Drinfeld-Yetter modules over b, with deformed associativity constraints given by Phi, to that of Drinfeld-Yetter modules over U_h(b). In this paper, we prove that the equivalence F_b is functorial in b.

Introductory lectures on topological quantum field theory

These notes offer a lightening introduction to topological quantum field theory in its functorial axiomatisation, assuming no or little prior exposure. We lay some emphasis on the connection between the path integral motivation and the definition in terms symmetric monoidal categories, and we highlight the algebraic formulation emerging from a formal generators-and-relations description. This allows one to understand (oriented, closed) 1- and 2-dimensional TQFTs in terms of a finite amount of algebraic data, while already the 3-dimensional case needs an infinite amount of data. We evade these complications by instead discussing some aspects of 3-dimensional extended TQFTs, and their relation to braided monoidal categories.

Logarithmic link invariants of [formula omitted] and asymptotic dimensions of singlet vertex algebras

We study relationships between the restricted unrolled quantum group U‾qH(sl2) at q=eπi/r, and the singlet vertex operator algebra M(r), r≥2. We use deformable families of modules to efficiently compute (1,1)-tangle invariants colored with projective U‾qH(sl2)-modules. These invariants relate to the colored Alexander tangle invariants studied in [6,40]. It follows that the regularized asymptotic dimensions of characters of M(r), studied previously by the first two authors, coincide with the corresponding modified traces of open Hopf link invariants. We also discuss various categorical properties of M(r)-mod in connection to braided tensor categories.

Drinfeld center of enriched monoidal categories

We define the Drinfeld center of a monoidal category enriched over a braided monoidal category, and show that every modular tensor category can be realized in a canonical way as the Drinfeld center of a self-enriched monoidal category. We also give a generalization of this result for important applications in physics.

Monoidal Categories Enriched in Braided Monoidal Categories

AbstractWe introduce the notion of a monoidal category enriched in a braided monoidal category $\mathcal{V}$. We set up the basic theory, and prove a classification result in terms of braided oplax monoidal functors to the Drinfeld centre of some monoidal category $\mathcal{T}$.Even the basic theory is interesting; it shares many characteristics with the theory of monoidal categories enriched in a symmetric monoidal category, but lacks some features. Of particular note, there is no cartesian product of braided-enriched categories, and the natural transformations do not form a 2-category, but rather satisfy a braided interchange relation.Strikingly, our classification is slightly more general than what one might have anticipated in terms of strong monoidal functors $\mathcal{V}\to Z(\mathcal{T})$. We would like to understand this further; in a future article, we show that the functor is strong if and only if the enriched category is ‘complete’ in a certain sense. Nevertheless it remains to understand what non-complete enriched categories may look like.One should think of our construction as a generalization of de-equivariantization, which takes a strong monoidal functor ${\mathsf {Rep}}(G) \to Z(\mathcal{T})$ for some finite group $G$ and a monoidal category $\mathcal{T}$, and produces a new monoidal category $\mathcal{T} _{{/\hspace{-2px}/}G}$. In our setting, given any braided oplax monoidal functor $\mathcal{V} \to Z(\mathcal{T})$, for any braided $\mathcal{V}$, we produce $\mathcal{T} _{{/\hspace{-2px}/}\mathcal{V}}$: this is not usually an ‘honest’ monoidal category, but is instead $\mathcal{V}$-enriched. If $\mathcal{V}$ has a braided lax monoidal functor to ${\mathsf {Vec}}$, we can use this to reduce the enrichment to ${\mathsf {Vec}}$, and this recovers de-equivariantization as a special case. This is the published version of arXiv:1701.00567.

Pointed finite tensor categories over abelian groups

We give a characterization of finite pointed tensor categories obtained as de-equivariantizations of the category of corepresentations of finite-dimensional pointed Hopf algebras with abelian group of group-like elements only in terms of the (cohomology class of the) associator of the pointed part. As an application we prove that every coradically graded pointed finite braided tensor category is a de-equivariantization of the category of corepresentations of a finite-dimensional pointed Hopf algebras with abelian group of group-like elements.

Factorizable R-Matrices for Small Quantum Groups

Representations of small quantum groups $u_q({\mathfrak{g}})$ at a root of unity and their extensions provide interesting tensor categories, that appear in different areas of algebra and mathematical physics. There is an ansatz by Lusztig to endow these categories with the structure of a braided tensor category. In this article we determine all solutions to this ansatz that lead to a non-degenerate braiding. Particularly interesting are cases where the order of $q$ has common divisors with root lengths. In this way we produce familiar and unfamiliar series of (non-semisimple) modular tensor categories. In the degenerate cases we determine the group of so-called transparent objects for further use.

Symmetry and pseudosymmetry of v-Yetter–Drinfeld categories for Hom–Hopf algebras

The purpose of this paper is to introduce the category of [Formula: see text]-Yetter–Drinfeld modules ([Formula: see text]) over a Hom–Hopf algebra. We first prove that every category of [Formula: see text]-Yetter–Drinfeld modules over a Hom–Hopf algebra with a bijective antipode [Formula: see text] is a braided tensor category and that every [Formula: see text]-Yetter–Drinfeld module can provide the solution of the Hom–Yang–Baxter equation. Secondly, we find sufficient and necessary conditions for [Formula: see text] to be symmetric and pseudosymmetric, respectively. Finally, we construct examples of [Formula: see text]-Yetter–Drinfeld modules by a quasitriangular Hom–Hopf algebra and study their relationship.

Meromorphic tensor equivalence for Yangians and quantum loop algebras

Let ${\mathfrak g}$ be a complex semisimple Lie algebra, and $Y_h({\mathfrak g})$, $U_q(L{\mathfrak g})$ the corresponding Yangian and quantum loop algebra, with deformation parameters related by $q=\exp(\pi i h)$. When $h$ is not a rational number, we constructed in arXiv:1310.7318 a faithful functor $\Gamma$ from the category of finite-dimensional representations of $Y_h ({\mathfrak g})$ to those of $U_q(L{\mathfrak g})$. The functor $\Gamma$ is governed by the additive difference equations defined by the commuting fields of the Yangian, and restricts to an equivalence on a subcategory of $Y_h({\mathfrak g})$ defined by choosing a branch of the logarithm. In this paper, we construct a tensor structure on $\Gamma$ and show that, if $|q|\neq 1$, it yields an equivalence of meromorphic braided tensor categories, when $Y_h({\mathfrak g})$ and $U_q(L{\mathfrak g})$ are endowed with the deformed Drinfeld coproducts and the commutative part of the universal $R$-matrix. This proves in particular the Kohno-Drinfeld theorem for the abelian $q$KZ equations defined by $Y_h({\mathfrak g})$. The tensor structure arises from the abelian $q$KZ equations defined by a appropriate regularisation of the commutative $R$-matrix of $Y_h({\mathfrak g})$.

Weak Multiplier Bimonoids

Based on the novel notion of `weakly counital fusion morphism', regular weak multiplier bimonoids in braided monoidal categories are introduced. They generalize weak multiplier bialgebras over fields and multiplier bimonoids in braided monoidal categories. Under some assumptions the so-called base object of a regular weak multiplier bimonoid is shown to carry a coseparable comonoid structure; hence to possess a monoidal category of bicomodules. In this case, appropriately defined modules over a regular weak multiplier bimonoid are proven to constitute a monoidal category with a strict monoidal forgetful type functor to the category of bicomodules over the base object. Braided monoidal categories considered include various categories of modules or graded modules, the category of complete bornological spaces, and the category of complex Hilbert spaces and continuous linear transformations.

Some remarks about monoidal Hom-algebras

In this paper, for 𝒞 a strict braided monoidal category with tensor product ⊗, we improve the definition of Hom-associative algebra by removing the multiplicativity condition of the automorphism α. After that we state the close connection between the classical notions of (co)algebra, (co)module and Hopf algebra and the corresponding ones in the Hom-world. As a consequence we can study the questions about the Hom-world by passing to the classical one, and the main contribution of this paper is to illustrate the suitable procedure to move from one side to the other. Finally, we apply our techniques to get in the Hom setting some classical results about cleft and Galois extensions.

Crossed S-matrices and character sheaves on unipotent groups

Let k be the algebraic closure of a finite field Fq of characteristic p. Let G be a connected unipotent group over k equipped with an Fq-structure given by a Frobenius map F:G⟶G. We will denote the corresponding algebraic group defined over Fq by G0. Character sheaves on G are certain special objects in the triangulated braided monoidal category DG(G) of bounded conjugation equivariant Q‾l-complexes (where l≠p is a prime number) on G. Boyarchenko has proved that the “trace of Frobenius” functions associated with F-stable character sheaves on G form an orthonormal basis of the space of class functions on G0(Fq) and that the matrix relating this basis to the basis formed by the irreducible characters of G0(Fq) is block diagonal with “small” blocks. In particular, there is a partition of the set of character sheaves as well as a partition of the set of irreducible characters of G0(Fq) into “small” families known as L-packets. In this paper we describe these block matrices relating character sheaves and irreducible characters corresponding to each L-packet. We prove that these matrices can be described as certain “crossed S-matrices” associated with each L-packet. We will also derive a formula for the dimensions of the irreducible representations of G0(Fq) in terms of certain modular categorical data associated with the corresponding L-packet. In fact we will formulate and prove more general results which hold for possibly disconnected groups G such that G∘ is unipotent. To prove our results, we will establish a formula (which holds for any algebraic group G) which expresses the inner product of the “trace of Frobenius” function of any F-stable object of DG(G) with any character of G0(Fq) (or of any of its pure inner forms) in terms of certain categorical operations.

Quantum walled Brauer algebra: commuting families, Baxterization, and representations

For the quantum walled Brauer algebra, we construct its Specht modules and (for generic parameters of the algebra) seminormal modules. The latter construction yields the spectrum of a commuting family of Jucys–Murphy elements. We also propose a Baxterization prescription; it involves representing the quantum walled Brauer algebra in terms of morphisms in a braided monoidal category and introducing parameters into these morphisms, which allows constructing a ‘universal transfer matrix’ that generates commuting elements of the algebra.

Symplectic fermions and a quasi-Hopf algebra structure on [formula omitted

We consider the (finite-dimensional) restricted quantum group U‾qsℓ(2) at q=i. We show that U‾isℓ(2) does not allow for a universal R-matrix, even though U⊗V≅V⊗U holds for all finite-dimensional representations U,V of U‾isℓ(2). We then give an explicit coassociator Φ and a universal R-matrix R such that U‾isℓ(2) becomes a quasi-triangular quasi-Hopf algebra.Our construction is motivated by the two-dimensional chiral conformal field theory of symplectic fermions with central charge c=−2. There, a braided monoidal category, SF, has been computed from the factorisation and monodromy properties of conformal blocks, and we prove that Rep(U‾isℓ(2),Φ,R) is braided monoidally equivalent to SF.