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Let
$\mathcal{C}$
be a fusion category over an algebraically closed field
$\mathbb{k}$
of arbitrary characteristic. Two numerical invariants of
$\mathcal{C}$
, that is, the Casimir number and the determinant of
$\mathcal{C}$
are considered in this paper. These two numbers are both positive integers and admit the property that the Grothendieck algebra
$(\mathcal{C})\otimes_{\mathbb{Z}}K$
over any field K is semisimple if and only if any of these numbers is not zero in K. This shows that these two numbers have the same prime factors. If moreover
$\mathcal{C}$
is pivotal, it gives a numerical criterion that
$\mathcal{C}$
is nondegenerate if and only if any of these numbers is not zero in
$\mathbb{k}$
. For the case that
$\mathcal{C}$
is a spherical fusion category over the field
$\mathbb{C}$
of complex numbers, these two numbers and the Frobenius–Schur exponent of
$\mathcal{C}$
share the same prime factors. This may be thought of as another version of the Cauchy theorem for spherical fusion categories.
We show that the only summands of the theta divisor on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the powers of the curve and the Fano surface of lines on the threefold. The proof only uses the decomposition theorem for perverse sheaves, some representation theory and the notion of characteristic cycles.
We introduce and investigate new invariants of pairs of modules
$M$
and
$N$
over quantum affine algebras
$U_{q}^{\prime }(\mathfrak{g})$
by analyzing their associated
$R$
-matrices. Using these new invariants, we provide a criterion for a monoidal category of finite-dimensional integrable
$U_{q}^{\prime }(\mathfrak{g})$
-modules to become a monoidal categorification of a cluster algebra.
For
${\mathcal{C}}$
a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show:
(1)
${\mathcal{C}}$
always contains a simple projective object;
(2) if
${\mathcal{C}}$
is in addition ribbon, the internal characters of projective modules span a submodule for the projective
$\text{SL}(2,\mathbb{Z})$
-action;
(3) the action of the Grothendieck ring of
${\mathcal{C}}$
on the span of internal characters of projective objects can be diagonalised;
(4) the linearised Grothendieck ring of
${\mathcal{C}}$
is semisimple if and only if
${\mathcal{C}}$
is semisimple.
Results (1)–(3) remain true in positive characteristic under an extra assumption. Result (1) implies that the tensor ideal of projective objects in
${\mathcal{C}}$
carries a unique-up-to-scalars modified trace function. We express the modified trace of open Hopf links coloured by projectives in terms of
$S$
-matrix elements. Furthermore, we give a Verlinde-like formula for the decomposition of tensor products of projective objects which uses only the modular
$S$
-transformation restricted to internal characters of projective objects. We compute the modified trace in the example of symplectic fermion categories, and we illustrate how the Verlinde-like formula for projective objects can be applied there.
We completely describe the algebraic part of the rational cohomology of the Torelli groups of the manifolds
$\#^{g}S^{n}\times S^{n}$
relative to a disc in a stable range, for
$2n\geqslant 6$
. Our calculation is also valid for
$2n=2$
assuming that the rational cohomology groups of these Torelli groups are finite-dimensional in a stable range.
We study exact sequences of finite tensor categories of the form Rep G → 𝒞 → 𝒟, where G is a finite group. We show that, under suitable assumptions, there exists a group Γ and mutual actions by permutations ⊳ : Γ × G → G and ⊲ : Γ × G→ Γ that make (G, Γ) into matched pair of groups endowed with a natural crossed action on 𝒟 such that 𝒞 is equivalent to a certain associated crossed extension 𝒟(G,Γ) of 𝒟. Dually, we show that an exact sequence of finite tensor categories VecG → 𝒞 → 𝒟 induces an Aut(G)-grading on 𝒞 whose neutral homogeneous component is a (Z(G), Γ)-crossed extension of a tensor subcategory of 𝒟. As an application we prove that such extensions 𝒞 of 𝒟 are weakly group-theoretical fusion categories if and only if 𝒟 is a weakly group-theoretical fusion category. In particular, we conclude that every semisolvable semisimple Hopf algebra is weakly group-theoretical.
We study Tate motives with integral coefficients through the lens of tensor triangular geometry. For some base fields, including
$\overline{\mathbb{Q}}$
and
$\overline{\mathbb{F}_{p}}$
, we arrive at a complete description of the tensor triangular spectrum and a classification of the thick tensor ideals.
The tube category of a modular tensor category is a variant of the tube algebra, first introduced by Ocneanu. As a category, it can be decomposed in two different, but related, senses. Firstly, via the Yoneda embedding, the Hom spaces decompose into summands factoring through irreducible functors, in a manner analogous to decomposing an algebra as a sum of matrix algebras. We describe these summands. Secondly, under the Yoneda embedding, each object decomposes into irreducibles, which correspond to primitive idempotents in the category itself. We identify these idempotents. We make extensive use of diagram calculus in the description and proof of these decompositions.
We provide explicit and unified formulas for the cocycles of all degrees on the normalized bar resolutions of finite abelian groups. This is achieved by constructing a chain map from the normalized bar resolution to a Koszul-like resolution for any given finite abelian group. With a help of the obtained cocycle formulas, we determine all the braided linear Gr-categories and compute the Dijkgraaf–Witten Invariants of the n-torus for all n.
We introduce the oriented Brauer–Clifford and degenerate affine oriented Brauer–Clifford supercategories. These are diagrammatically defined monoidal supercategories that provide combinatorial models for certain natural monoidal supercategories of supermodules and endosuperfunctors, respectively, for the Lie superalgebras of type Q. Our main results are basis theorems for these diagram supercategories. We also discuss connections and applications to the representation theory of the Lie superalgebra of type Q.
In this paper, we complete the ADE-like classification of simple transitive 2-representations of Soergel bimodules in finite dihedral type, under the assumption of gradeability. In particular, we use bipartite graphs and zigzag algebras of ADE type to give an explicit construction of a graded (non-strict) version of all these 2-representations.
Moreover, we give simple combinatorial criteria for when two such 2-representations are equivalent and for when their Grothendieck groups give rise to isomorphic representations.
Finally, our construction also gives a large class of simple transitive 2-representations in infinite dihedral type for general bipartite graphs.
In this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category (
$\mathscr{C},\otimes$
) endowed with a symmetric 2-trace, i.e., an
$F\in \text{Fun}(\mathscr{C},\text{Vec})$
satisfying some natural trace-like conditions, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the (co)cyclic homology of the (co)algebra “with coefficients in
$F$
”. Furthermore, we observe that if
$\mathscr{M}$
is a
$\mathscr{C}$
-bimodule category and
$(F,M)$
is a stable central pair, i.e.,
$F\in \text{Fun}(\mathscr{M},\text{Vec})$
and
$M\in \mathscr{M}$
satisfy certain conditions, then
$\mathscr{C}$
acquires a symmetric 2-trace. The dual notions of symmetric 2-contratraces and stable central contrapairs are derived as well. As an application we can recover all Hopf cyclic type (co)homology theories.
Fibonacci anyons are attractive for use in topological quantum computation because any unitary transformation of their state space can be approximated arbitrarily accurately by braiding. However, there is no known braid that entangles two qubits without leaving the space spanned by the two qubits. In other words, there is no known ‘leakage-free’ entangling gate made by braiding. In this paper, we provide a remedy to this problem by supplementing braiding with measurement operations in order to produce an exact controlled rotation gate on two qubits.
We prove two results on the tube algebras of rigid C*-tensor categories. The first is that the tube algebra of the representation category of a compact quantum group G is a full corner of the Drinfeld double of G. As an application, we obtain some information on the structure of the tube algebras of the Temperley–Lieb categories 𝒯ℒ(d) for d > 2. The second result is that the tube algebras of weakly Morita equivalent C*-tensor categories are strongly Morita equivalent. The corresponding linking algebra is described as the tube algebra of the 2-category defining the Morita context.
We formulate and study Howe–Moore type properties in the setting of quantum groups and in the setting of rigid
$C^{\ast }$
-tensor categories. We say that a rigid
$C^{\ast }$
-tensor category
${\mathcal{C}}$
has the Howe–Moore property if every completely positive multiplier on
${\mathcal{C}}$
has a limit at infinity. We prove that the representation categories of
$q$
-deformations of connected compact simple Lie groups with trivial center satisfy the Howe–Moore property. As an immediate consequence, we deduce the Howe–Moore property for Temperley–Lieb–Jones standard invariants with principal graph
$A_{\infty }$
. These results form a special case of a more general result on the convergence of completely bounded multipliers on the aforementioned categories. This more general result also holds for the representation categories of the free orthogonal quantum groups and for the Kazhdan–Wenzl categories. Additionally, in the specific case of the quantum groups
$\text{SU}_{q}(N)$
, we are able, using a result of the first-named author, to give an explicit characterization of the central states on the quantum coordinate algebra of
$\text{SU}_{q}(N)$
, which coincide with the completely positive multipliers on the representation category of
$\text{SU}_{q}(N)$
.
This paper deals with the Green ring
$\mathcal{G}(\mathcal{C})$
of a finite tensor category
$\mathcal{C}$
with finitely many isomorphism classes of indecomposable objects over an algebraically closed field. The first part of this paper deals with the question of when the Green ring
$\mathcal{G}(\mathcal{C})$
, or the Green algebra
$\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$
K over a field K, is Jacobson semisimple (namely, has zero Jacobson radical). It turns out that
$\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$
K is Jacobson semisimple if and only if the Casimir number of
$\mathcal{C}$
is not zero in K. For the Green ring
$\mathcal{G}(\mathcal{C})$
itself,
$\mathcal{G}(\mathcal{C})$
is Jacobson semisimple if and only if the Casimir number of
$\mathcal{C}$
is not zero. The second part of this paper focuses on the case where
$\mathcal{C}=\text{Rep}(\mathbb {k}G)$
for a cyclic group G of order p over a field
$\mathbb {k}$
of characteristic p. In this case, the Casimir number of
$\mathcal{C}$
is computable and is shown to be 2p2. This leads to a complete description of the Jacobson radical of the Green algebra
$\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$
K over any field K.
In this paper we introduce and study Miyashita action in the context of monoidal categories aiming by this to provide a common framework of previous studies in the literature. We make a special emphasis of this action on Azumaya monoids. To this end, we develop the theory of invertible bimodules over different monoids (a sort of Morita contexts) in general monoidal categories as well as their corresponding Miyashita action. Roughly speaking, a Miyashita action is a homomorphism of groups from the group of all isomorphic classes of invertible subobjects of a given monoid to its group of automorphisms. In the symmetric case, we show that for certain Azumaya monoids, which are abundant in practice, the corresponding Miyashita action is always an isomorphism of groups. This generalizes Miyashita’s classical result and sheds light on other applications of geometric nature which cannot be treated using the classical theory. In order to illustrate our methods, we give a concrete application to the category of comodules over commutative (flat) Hopf algebroids. This obviously includes the special cases of split Hopf algebroids (action groupoids), which for instance cover the situation of the action of an affine algebraic group on an affine algebraic variety.