results for au:Galindo_C in:math

- We study novel invariants of modular categories that are beyond the modular data, with an eye towards a simple set of complete invariants for modular categories. Our focus is on the $W$-matrix $-$the quantum invariant of a colored framed Whitehead link from the associated TQFT of a modular category. We prove that the $W$-matrix and the set of punctured $S$-matrices are strictly beyond the modular data $(S,T)$. Whether or not the triple $(S,T,W)$ constitutes a complete invariant of modular categories remains an open question.
- Mar 16 2018 math.QA arXiv:1803.05568v2We introduce the notion of a $\textit{reflection fusion category}$, which is a type of a $G$-crossed category generated by objects of Frobenius-Perron dimension $1$ and $\sqrt{p}$, where $p$ is an odd prime. We show that such categories correspond to orthogonal reflection groups over $\mathbb{F}_p$. This allows us to use the known classification of irreducible reflection groups over finite fields to classify irreducible reflection fusion categories.
- Jan 16 2018 math.QA arXiv:1801.04296v1Acyclic anyon models are non-abelian anyon models for which thermal anyon errors can be corrected. In this note, we characterize acyclic anyon models and raise the question if the restriction to acyclic anyon models is a deficiency of the current protocol or could it be intrinsically related to the computational power of non-abelian anyons. We also obtain general results on acyclic anyon models and find new acyclic anyon models such as $SO(8)_2$ and untwisted Dijkgraaf-Witten theories of nilpotent finite groups.
- Dec 20 2017 math.QA arXiv:1712.07097v1The purpose of this paper is to study minimal modular extensions of braided fusion categories doing emphases in minimal modular extensions of super-Tannakian fusion categories. We define actions of finite super-groups on fermionic fusion categories and spin-braided fusion categories. Similar to the case of groups, our motivation came from the study of fusion categories containing the representation category of a super-group. We develop many analog results to the Tannakian case, including cohomological obstructions, relation with braided $G$-crossed categories and minimal modular extensions. We apply the general results to the construction and classification of minimal modular extensions of super-groups and braided fusion categories. In particular, we exhibit some examples of braided fusion categories without minimal modular extensions.
- We give sufficient conditions for self-orthogonality with respect to symplectic, Euclidean and Hermitian inner products of a wide family of quasi-cyclic codes of index two. We provide lower bounds for the symplectic weight and the minimum distance of the involved codes. Supported in the previous results, we show algebraic constructions of good quantum codes and determine their parameters.
- We introduce a new class of evaluation linear codes by evaluating polynomials at the roots of a suitable trace function. We give conditions for self-orthogonality of these codes and their subfield-subcodes with respect to the Hermitian inner product. They allow us to construct stabilizer quantum codes over several finite fields which substantially improve the codes in the literature and that are records at [http://www.codetables.de] for the binary case. Moreover, we obtain several classical linear codes over the field $\mathbb{F}_4$ which are records at [http://www.codetables.de].
- LCD codes are linear codes with important cryptographic applications. Recently, a method has been presented to transform any linear code into an LCD code with the same parameters when it is supported on a finite field with cardinality larger than 3. Hence, the study of LCD codes is mainly open for binary and ternary fields. Subfield-subcodes of $J$-affine variety codes are a generalization of BCH codes which have been successfully used for constructing good quantum codes. We describe binary and ternary LCD codes constructed as subfield-subcodes of $J$-affine variety codes and provide some new and good LCD codes coming from this construction.
- We give a characterization of finite pointed tensor categories obtained as de-equivariantizations of finite-dimensional pointed Hopf algebras over abelian groups 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 a finite dimensional pointed Hopf algebras over an abelian group.
- Jul 14 2017 math.QA arXiv:1707.03884v1We study representations of the braid groups from braiding gapped boundaries of Dijkgraaf-Witten theories and their twisted generalizations, which are (twisted) quantum doubled topological orders in two spatial dimensions. We show that the resulting braid (pure braid) representations are all monomial with respect to some specific bases, hence all such representation images of the braid groups are finite groups. We give explicit formulas for the monomial matrices and the ground state degeneracy of the Kitaev models that are Hamiltonian realizations of Dijkgraaf-Witten theories. Our results imply that braiding gapped boundaries alone cannot provide universal gate sets for topological quantum computing with gapped boundaries.
- May 16 2017 math.QA arXiv:1705.05293v2We pursue a classification of low-rank super-modular categories parallel to that of modular categories. We classify all super-modular categories up to rank=$6$, and spin modular categories up to rank=$11$. In particular, we show that, up to fusion rules, there is exactly one non-split super-modular category of rank $2,4$ and $6$, namely $PSU(2)_{4k+2}$ for $k=0,1$ and $2$. This classification is facilitated by adapting and extending well-known constraints from modular categories to super-modular categories, such as Verlinde and Frobenius-Schur indicator formulae.
- We prove that the Newton-Okounkov body of the flag $E_{\bullet}:= \left\{ X=X_r \supset E_r \supset \{q\} \right\}$, defined by the surface $X$ and the exceptional divisor $E_r$ given by any divisorial valuation of the complex projective plane $\mathbb{P}^2$, with respect to the pull-back of the line-bundle $\mathcal{O}_{\mathbb{P}^2} (1)$ is either a triangle or a quadrilateral, characterizing when it is a triangle or a quadrilateral. We also describe the vertices of that figure. Finally, we introduce a large family of flags for which we determine explicitly their Newton-Okounkov bodies which turn out to be triangular.
- 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 B with an action of a group G, we construct a braided G-crossed monoidal category Z_G(B) with trivial component the Drinfeld center of B. We prove that, in the case of a G-action on the 2-category of representation of a tensor category C, the 2-category of equivariant objects is biequivalent to the module categories over an associated G-extension of 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 [S. Gelaki, D. Naidu and D. Nikshych, Centers of graded fusion categories, Algebra Number Theory 3, No. 8 (2009), 959--990.]
- Self-orthogonal $J$-affine variety codes have been successfully used to obtain quantum stabilizer codes with excellent parameters. In a previous paper we gave formulae for the dimension of this family of quantum codes, but no bound for the minimum distance was given. In this work, we show how to derive quantum stabilizer codes with designed minimum distance from $J$-affine variety codes and their subfield-subcodes. Moreover, this allows us to obtain new quantum codes, some of them either, with better parameters, or with larger distances than the previously known codes.
- Two new constructions of linear code pairs $C_2 \subset C_1$ are given for which the codimension and the relative minimum distances $M_1(C_1,C_2)$, $M_1(C_2^\perp,C_1^\perp)$ are good. By this we mean that for any two out of the three parameters the third parameter of the constructed code pair is large. Such pairs of nested codes are indispensable for the determination of good linear ramp secret sharing schemes [35]. They can also be used to ensure reliable communication over asymmetric quantum channels [47]. The new constructions result from carefully applying the Feng-Rao bounds [18,27] to a family of codes defined from multivariate polynomials and Cartesian product point sets.
- We consider the last value $\hat{\mu} (\nu)$ of the vanishing sequence of $H^0(L)$ along a divisorial or irrational valuation $\nu$ centered at $\mathcal{O}_{\mathbb{P}^2,p}$, where $L$ resp. $p$ is a line resp. a point of the projective plane $\mathbb{P}^2$ over an algebraically closed field. This value contains, for valuations, similar information as that given by Seshadri constants for points. It is always true that $\hat{\mu} (\nu) \geq \sqrt{1 / \mathrm{vol}(\nu)}$ and minimal valuations are those satisfying the equality. In this paper, we prove that the Greuel-Lossen-Shustin Conjecture implies a variation of the Nagata Conjecture involving minimal valuations (that extends the one stated in the paper "Very general monomial valuations of $\mathbb{P}^2$ and a Nagata type conjecture" by Dumnicki et al. to the whole set of divisorial and irrational valuations of the projective plane) which also implies the original one. We also provide infinitely many families of very general minimal valuations with an arbitrary number of Puiseux exponents, and an asymptotic result that can be considered as an evidence in the direction of the mentioned conjecture by Dumnicki et al.
- Jun 07 2016 math.QA arXiv:1606.01414v3We address the problem of determining the obstruction to existence of solutions of the hexagon equation for abelian fusion rules and the classification of prime abelian anyons.
- This paper introduces Hopf braces, a new algebraic structure related to the Yang-Baxter equation which include Rump's braces and their non-commutative generalizations as particular cases. Several results of classical braces are still valid in our context. Furthermore, Hopf braces provide the right setting for considering left symmetric algebras as Lie-theoretical analogs of braces.
- Apr 07 2016 math.QA arXiv:1604.01679v2We prove a coherence theorem for actions of groups on monoidal categories. As an application we prove coherence for arbitrary braided $G$-crossed categories.
- Mar 31 2016 math.QA arXiv:1603.09294v3We study spin and super-modular categories systematically as inspired by fermionic topological phases of matter, which are always fermion parity enriched and modelled by spin TQFTs at low energy. We formulate a $16$-fold way conjecture for the minimal modular extensions of super-modular categories to spin modular categories, which is a categorical formulation of gauging the fermion parity. We investigate general properties of super-modular categories such as fermions in twisted Drinfeld doubles, Verlinde formulas for naive quotients, and explicit extensions of $PSU(2)_{4m+2}$ with an eye towards a classification of the low-rank cases.
- Oct 14 2015 math.QA arXiv:1510.03475v3Topological order of a topological phase of matter in two spacial dimensions is encoded by a unitary modular (tensor) category (UMC). A group symmetry of the topological phase induces a group symmetry of its corresponding UMC. Gauging is a well-known theoretical tool to promote a global symmetry to a local gauge symmetry. We give a mathematical formulation of gauging in terms of higher category formalism. Roughly, given a UMC with a symmetry group $G$, gauging is a 2-step process: first extend the UMC to a $G$-crossed braided fusion category and then take the equivariantization of the resulting category. Gauging can tell whether or not two enriched topological phases of matter are different, and also provides a way to construct new UMCs out of old ones. We derive a formula for the $H^4$-obstruction, prove some properties of gauging, and carry out gauging for two concrete examples.
- Sep 07 2015 math.QA arXiv:1509.01548v2We present new Hopf algebras with the dual Chevalley property by determining all semisimple Hopf algebras Morita-equivalent to a group algebra over a finite group, for a list of groups supporting a non-trivial finite-dimensional Nichols algebra.
- New stabilizer codes with parameters better than the ones available in the literature are provided in this work, in particular quantum codes with parameters $[[127,63, \geq 12]]_2$ and $[[63,45, \geq 6]]_4$ that are records. These codes are constructed with a new generalization of the Steane's enlargement procedure and by considering orthogonal subfield-subcodes --with respect to the Euclidean and Hermitian inner product-- of a new family of linear codes, the $J$-affine variety codes.
- We classify all modular categories of dimension $4m$, where $m$ is an odd square-free integer, and all ranks $6$ and $7$ weakly integral modular categories. This completes the classification of weakly integral modular categories through rank $7$. Our results imply that all integral modular categories of rank at most $7$ are pointed (that is, every simple object has dimension $1$). All strictly weakly integral (weakly integral but non-integral) modular categories of ranks $6$ and $7$ have dimension $4m$, with $m$ an odd square free integer, so their classification is an application of our main result. The classification of rank $7$ integral modular categories is facilitated by an analysis of two actions on modular categories: the Galois group of the field generated by the entries of the $S$-matrix and the group of isomorphism classes of invertible simple objects. The interplay of these two actions is of independent interest, and we derive some valuable arithmetic consequences from their actions.
- We give an algorithm for deciding whether a planar polynomial differential system has a first integral which factorizes as a product of defining polynomials of curves with only one place at infinity. In the affirmative case, our algorithm computes a minimal first integral. In addition, we solve the Poincaré problem for the class of systems which admit a polynomial first integral as above in the sense that the degree of the minimal first integral can be computed from the reduction of singularities of the corresponding vector field.
- Two groups are called isocategorical over a field $k$ if their respective categories of $k$-linear representations are monoidally equivalent. We classify isocategorical groups over arbitrary fields, extending the earlier classification of Etingof-Gelaki and Davydov for algebraically closed fields. In order to construct concrete examples of isocategorical groups a new variant of the Weil representation associated to isocategorical groups is defined. We construct examples of non-isomorphic isocategorical groups over any field of characteristic different from two and rational Weil representations associated to symplectic spaces over finite fields of characteristic two.
- Given a fusion category $\mathcal{C}$ and an indecomposable $\mathcal{C}$-module category $\mathcal{M}$, the fusion category $\mathcal{C}^*_\mathcal{M}$ of $\mathcal{C}$-module endofunctors of $\mathcal{M}$ is called the (Morita) dual fusion category of $\mathcal{C}$ with respect to $\mathcal{M}$. We describe tensor functors between two arbitrary duals $\mathcal{C}^*_\mathcal{M}$ and $\mathcal{D}^*_\mathcal{N}$ in terms of data associated to $\mathcal{C}$ and $\mathcal{D}$. We apply the results to $G$-equivariantizations of fusion categories and group-theoretical fusion categories. We describe the orbits of the action of the Brauer-Picard group on the set of module categories and we propose a categorification of the Rosenberg-Zelinsky sequence for fusion categories.
- Stabilizer codes obtained via CSS code construction and Steane's enlargement of subfield-subcodes and matrix-product codes coming from generalized Reed-Muller, hyperbolic and affine variety codes are studied. Stabilizer codes with good quantum parameters are supplied, in particular, some binary codes of lengths 127 and 128 improve the parameters of the codes in http://www.codetables.de. Moreover, non-binary codes are presented either with parameters better than or equal to the quantum codes obtained from BCH codes by La Guardia or with lengths that can not be reached by them.
- We use affine variety codes and their subfield-subcodes for obtaining quantum stabilizer codes via the CSS code construction. With this procedure, we get codes with good parameters and a code whose parameters exceed the CSS quantum Gilbert-Varshamov bound given by Feng and Ma.
- Feb 19 2014 math.AG arXiv:1402.4257v3We consider surfaces $X$ defined by plane divisorial valuations $\nu$ of the quotient field of the local ring $R$ at a closed point $p$ of the projective plane $\mathbb{P}^2$ over an arbitrary algebraically closed field $k$ and centered at $R$. We prove that the regularity of the cone of curves of $X$ is equivalent to the fact that $\nu$ is non positive on ${\mathcal O}_{\mathbb{P}^2}(\mathbb{P}^2\setminus L)$, where $L$ is a certain line containing $p$. Under these conditions, we characterize when the characteristic cone of $X$ is closed and its Cox ring finitely generated. Equivalent conditions to the fact that $\nu$ is negative on ${\mathcal O}_{\mathbb{P}^2}(\mathbb{P}^2\setminus L) \setminus k$ are also given.
- Dec 20 2013 math.QA arXiv:1312.5557v1We investigate braid group representations associated with unitary braided vector spaces, focusing on a conjecture that such representations should have virtually abelian images in general and finite image provided the braiding has finite order. We verify this conjecture for the two infinite families of Gaussian and group-type braided vector spaces, as well as the generalization to quasi-braided vector spaces of group-type.
- Let $K$ be a finite group and let $G$ be a finite group acting on $K$ by automorphisms. In this paper we study two different but intimately related subjects: on the one side we classify all possible multiplicative and associative structures with which one can endow the twisted $G$-equivariant K-theory of $K$, and on the other, we classify all possible monoidal structures with which one can endow the category of twisted and $G$-equivariant bundles over $K$. We achieve this classification by encoding the relevant information in the cochains of a sub double complex of the double bar resolution associated to the semi-direct product $K \rtimes G$; we use known calculations of the cohomology of $K$, $G$ and $K \rtimes G$ to produce concrete examples of our classification. In the case in which $K=G$ and $G$ acts by conjugation, the multiplication map $G \rtimes G \to G$ is a homomorphism of groups and we define a shuffle homomorphism which realizes this map at the homological level. We show that the categorical information that defines the Twisted Drinfeld Double can be realized as the dual of the shuffle homomorphism applied to any 3-cocycle of $G$. We use the pullback of the multiplication map in cohomology to classify the possible ring structures that the Grothendieck ring of representations of the Twisted Drinfeld Double may have, and we include concrete examples of this procedure.
- We classify integral modular categories of dimension pq^4 and p^2q^2 where p and q are distinct primes. We show that such categories are always group-theoretical except for categories of dimension 4q^2. In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara-Yamagami categories and quantum groups. We show that a non-group-theoretical integral modular category of dimension 4q^2 is equivalent to either one of these well-known examples or is of dimension 36 and is twist-equivalent to fusion categories arising from a certain quantum group.
- Nov 28 2012 math.AG arXiv:1211.6274v2We give an explicit formula for the log-canonical threshold of a reduced germ of plane curve. The formula depends only on the first two maximal contact values of the branches and their intersection multiplicities. We also improve the two branches formula given in a paper by Kuwata in Amer. J. Math. 121.
- Sep 11 2012 math.QA arXiv:1209.2022v2We prove that every braiding over a unitary fusion category is unitary and every unitary braided fusion category admits a unique unitary ribbon structure.
- We study the de-equivariantization of a Hopf algebra by an affine group scheme and we apply Tannakian techniques in order to realize it as the tensor category of comodules over a coquasi-bialgebra. As an application we construct a family of coquasi-Hopf algebras $A(H,G,\Phi)$ attached to a coradically-graded pointed Hopf algebra $H$ and some extra data.
- Oct 14 2011 math.DS arXiv:1110.2973v1We solve the Poincaré problem for plane foliations with only one dicritical divisor. Moreover, in this case, we give an algorithm that decides whether a foliation has a rational first integral and computes it in the affirmative case. We also provide an algorithm to compute a rational first integral of prefixed genus $g\neq 1$ of any type of plane foliation $\cf$. When the number of dicritical divisors dic$(\cf)$ is larger than two, this algorithm depends on suitable families of invariant curves. When dic$(\cf) = 2$, it proves that the degree of the rational first integral can be bounded only in terms of $g$, the degree of $\cf$ and the local analytic type of the dicritical singularities of $\cf$.
- We develop a theory of localization for braid group representations associated with objects in braided fusion categories and, more generally, to Yang-Baxter operators in monoidal categories. The essential problem is to determine when a family of braid representations can be uniformly modelled upon a tensor power of a fixed vector space in such a way that the braid group generators act "locally". Although related to the notion of (quasi-)fiber functors for fusion categories, remarkably, such localizations can exist for representations associated with objects of non-integral dimension. We conjecture that such localizations exist precisely when the object in question has dimension the square-root of an integer and prove several key special cases of the conjecture.
- We study exact module categories over the representation categories of finite-dimensional quasi-Hopf algebras. As a consequence we classify exact module categories over some families of pointed tensor categories with cyclic group of invertible objets of order p, where p is a prime number.
- Oct 27 2010 math.QA arXiv:1010.5283v3We develop a categorical analogue of Clifford theory for strongly graded rings over graded fusion categories. We describe module categories over a fusion category graded by a group $G$ as induced from module categories over fusion subcategories associated with the subgroups of $G$. We define invariant $\C_e$-module categories and extensions of $\C_e$-module categories. The construction of module categories over $\C$ is reduced to determine invariant module categories for subgroups of $G$ and the indecomposable extensions of this modules categories. We associate a $G$-crossed product fusion category to each $G$-invariant $\C_e$-module category and give a criterion for a graded fusion category to be a group-theoretical fusion category. We give necessary and sufficient conditions for an indecomposable module category to be extended.
- Jun 22 2010 math.QA arXiv:1006.3872v1We classify Galois objects for the dual of a group algebra of a finite group over an arbitrary field.
- A graded tensor category over a group $G$ will be called a crossed product tensor category if every homogeneous component has at least one multiplicatively invertible object. Our main result is a description of the crossed product tensor categories, graded monoidal functors, monoidal natural transformations, and braiding in terms of coherent outer $G$-actions over tensor categories.
- We introduce the class of plane valuations at infinity and prove an analogue to the Abhyankar-Moh (semigroup) Theorem for it.
- A graded tensor category over a group $G$ will be called a strongly $G$-graded tensor category if every homogeneous component has at least one multiplicativily invertible object. Our main result is a description of the module categories over a strongly $G$-graded tensor category as induced from module categories over tensor subcategories associated with the subgroups of $G$.
- We give a characterization theorem for non-degenerated plane foliations of degree different from 1 having a rational first integral. Moreover, we prove that the degree $r$ of a non-degenerated foliation as above provides the minimum number, $r+1$, of points in the projective plane through which infinitely many algebraic leaves of the foliation go.
- Let $V$ be a finite set of divisorial valuations centered at a 2-dimensional regular local ring $R$. In this paper we study its structure by means of the semigroup of values, $S_V$, and the multi-index graded algebra defined by $V$, $\gr_V R$. We prove that $S_V$ is finitely generated and we compute its minimal set of generators following the study of reduced curve singularities. Moreover, we prove a unique decomposition theorem for the elements of the semigroup. The comparison between valuations in $V$, the approximation of a reduced plane curve singularity $C$ by families of sets $V^{(k)}$ of divisorial valuations, and the relationship between the value semigroup of $C$ and the semigroups of the sets $V^{(k)}$, allow us to obtain the (finite) minimal generating sequences for $C$ as well as for $V$. We also analyze the structure of the homogeneous components of $\gr_V R$. The study of their dimensions allows us to relate the Poincaré series for $V$ and for a general curve $C$ of $V$. Since the last series coincides with the Alexander polynomial of the singularity, we can deduce a formula of A'Campo type for the Poincaré series of $V$. Moreover, the Poincaré series of $C$ could be seen as the limit of the series of $V^{(k)}$, $k\ge 0$.
- Jun 03 2008 math.AG arXiv:0806.0231v3For a simple complete ideal $\wp$ of a local ring at a closed point on a smooth complex algebraic surface, we introduce an algebraic object, named Poincaré series $P_{\wp}$, that gathers in an unified way the jumping numbers and the dimensions of the vector space quotients given by consecutive multiplier ideals attached to $\wp$. This paper is devoted to prove that $P_{\wp}$ is a rational function giving an explicit expression for it.
- Let $G$ be a finite group and let $\pi: G \to G'$ be a surjective group homomorphism. Consider the cocycle deformation $L = H^{\sigma}$ of the Hopf algebra $H = k^G$ of $k$-valued linear functions on $G$, with respect to some convolution invertible 2-cocycle $\sigma$. The (normal) Hopf subalgebra $k^{G'} \subseteq k^G$ corresponds to a Hopf subalgebra $L' \subseteq L$. Our main result is an explicit necessary and sufficient condition for the normality of $L'$ in $L$.
- We introduce the concept of $\delta$-sequence. A $\delta$-sequence $\Delta$ generates a well-ordered semigroup $S$ in $\mathbb{Z}^2$ or $\mathbb{R}$. We show how to construct (and compute parameters) for the dual code of any evaluation code associated with a weight function defined by $\Delta$ from the polynomial ring in two indeterminates to a semigroup $S$ as above. We prove that this is a simple procedure which can be understood by considering a particular class of valuations of function fields of surfaces, called plane valuations at infinity. We also give algorithms to construct an unlimited number of $\delta$-sequences of the different existing types, and so this paper provides the tools to know and use a new large set of codes.
- Aug 31 2006 math.QA arXiv:math/0608734v2We show that certain twisting deformations of a family of supersolvable groups are simple as Hopf algebras. These groups are direct products of two generalized dihedral groups. Examples of this construction arise in dimensions 60 and p^2q^2, for prime numbers p, q with q dividing p-1. We also show that certain twisting deformation of the symmetric group is simple as a Hopf algebra. On the other hand, we prove that every twisting deformation of a nilpotent group is semisolvable. We conclude that the notions of simplicity and (semi)solvability of a semisimple Hopf algebra are not determined by its tensor category of representations.