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Iván Angiono

Publications and source records attributed to Iván Angiono.

At least 19 recordsLinked to original sources
arXiv2026

Contragredient Lie algebras in symmetric categories

We define contragredient Lie algebras in symmetric categories, generalizing the construction of Lie algebras of the form $\mathfrak{g}(A)$ for a Cartan matrix $A$ from the category of vector spaces to an arbitrary symmetric tensor category. The main complication resides in the fact that, in contrast to the classical case, a general symmetric tensor category can admit tori (playing the role of Cartan subalgebras) which are non-abelian and have a sophisticated representation theory. Using this construction, we obtain and describe new examples of Lie algebras in the universal Verlinde category in characteristic $p\geq5$. We also show that some previously known examples can be obtained with our construction.

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arXiv2026

Computing finite Weyl groupoids

We present algorithms to compute generalized root systems of Nichols algebras of diagonal type and of contragredient Lie superalgebras. As a consequence, we obtain an algorithm to compute the Lyndon words in the Kharchenko PBW basis associated to each positive root, along with their corresponding hyperwords. This data is essential for obtaining a minimal presentation of Nichols algebras of diagonal type with a finite root system.

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arXiv2026

Pointed Hopf algebras revisited, with a view from tensor categories

Hopf algebras appear in connection with various problems in Pure Mathematics and Theoretical Physics, mainly through their categories of representations, which are examples of tensor categories. In recent years, there have been major advances in the classification of finite-dimensional Hopf algebras over the complex numbers, especially when restricted to the pointed case. In this survey, we aim to review the main results in this direction, stating the classification theorems recently obtained and the problems still open, and describing the tools needed to solve the problem by means of the so-called Lifting Method of Andruskiewitsch and Schneider. We will highlight a more categorical point of view related to these classification results, which offers for a better description of the so-called liftings up to involving cocycle deformations, and which in turn allows certain problems to be reduced to the associated graded pointed Hopf algebras. We will emphasise this point of view with applications to other contexts related to the aforementioned pointed Hopf algebras, such as finite pointed tensor categories, Hopf algebras that do not satisfy the Chevalley property and their finite-dimensional Nichols algebras, and finally module categories over the categories of representations of pointed Hopf algebras.

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arXiv2025

Zestings of Hopf Algebras

We extend the previously established zesting techniques from fusion categories to general tensor categories. In particular we consider the category of comodules over a Hopf algebra, providing a detailed translation of the categorical zesting construction into explicit Hopf algebraic terms: we show that the associative zesting of the category of comodules yields a coquasi-Hopf algebra whose comodule category is precisely the zested category. We explicitly write the modified multiplication and the associator, as well as the structures involved in the braided case. For pointed Hopf algebras, we derive concrete formulas for constructing zestings and establish a systematic approach for cyclic group gradings, providing explicit parameterizations of the zesting data.

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arXiv2024

Posets of finite GK-dimensional graded pre-Nichols algebras of diagonal type

We classify graded pre-Nichols algebras of diagonal type with finite Gelfand-Kirillov dimension. The characterization is made through an isomorphism of posets with the family of appropriate subsets of the set of positive roots coming from central extensions of Nichols algebras of diagonal type, generalizing the corresponding extensions for small quantum groups in de Concini-Kac-Procesi forms of quantum groups. On the way to achieving this result, we also classify graded quotients of algebras of functions of unipotent algebraic groups attached to semisimple Lie algebras.

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arXiv2023

Semisimplification of contragredient Lie algebras

We describe the structure and different features of Lie algebras in the Verlinde category, obtained as semisimplification of contragredient Lie algebras in characteristic $p$ with respect to the adjoint action of a Chevalley generator. In particular, we construct a root system for these algebras that arises as a parabolic restriction of the known root system for the classical Lie algebra. This gives a lattice grading with simple homogeneous components and a triangular decomposition for the semisimplified Lie algebra. We also obtain a non-degenerate invariant form that behaves well with the lattice grading. As an application, we exhibit concrete new examples of Lie algebras in the Verlinde category.

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arXiv2023

Rank 4 Nichols Algebras of Pale Braidings

We classify finite GK-dimensional Nichols algebras ${\mathscr B}(V)$ of rank 4 such that $V$ arises as a Yetter-Drinfeld module over an abelian group but it is not a direct sum of points and blocks.

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arXiv2023

Poisson orders on large quantum groups

We develop a Poisson geometric framework for studying the representation theory of all contragredient quantum super groups at roots of unity. This is done in a uniform fashion by treating the larger class of quantum doubles of bozonizations of all distinguished pre-Nichols algebras arXiv:1405.6681 belonging to a one-parameter family; we call these algebras \emph{large} quantum groups. We prove that each of these quantum algebras has a central Hopf subalgebra giving rise to a Poisson order in the sense of arXiv:math/0201042.. We describe explicitly the underlying Poisson algebraic groups and Poisson homogeneous spaces in terms of Borel subgroups of complex semisimple algebraic groups of adjoint type. The geometry of the Poisson algebraic groups and Poisson homogeneous spaces that are involved and its applications to the irreducible representations of the algebras $U_{\mathfrak{q}} \supset U_{\mathfrak{q}}^{\geqslant} \supset U_{\mathfrak{q}}^+$ are also described. Besides all (multiparameter) big quantum groups of De Concini--Kac--Procesi and big quantum super groups at roots of unity, our framework also contains the quantizations in characteristic 0 of the 34-dimensional Kac-Weisfeler Lie algebras in characteristic 2 and the 10-dimensional Brown Lie algebras in characteristic 3. The previous approaches to the above problems relied on reductions to rank two cases and direct calculations of Poisson brackets, which is not possible in the super case since there are 13 kinds of additional Serre relations on up to 4 generators. We use a new approach that relies on perfect pairings between restricted and non-restricted integral forms.

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arXiv2022

Finite GK-dimensional Nichols algebras of diagonal type and finite root systems

Let $(V,c)$ be a finite-dimensional braided vector space of diagonal type. We show that the Gelfand Kirillov dimension of the Nichols algebra $\mathfrak{B}(V)$ is finite if and only if the corresponding root system is finite, that is $\mathfrak{B}(V)$ admits a PBW basis with a finite number of generators. This had been conjectured in arXiv:1606.02521 and proved for $\dim V=2,3$ in arXiv:1803.08804, arXiv:2106.10143 respectively.

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arXiv2022

Pointed Hopf algebras over non-abelian groups with non-simple standard braidings

We construct finite-dimensional Hopf algebras whose coradical is the group algebra of a central extension of an abelian group. They fall into families associated to a semisimple Lie algebra together with a Dynkin diagram automorphism. We show conversely that every finite-dimensional pointed Hopf algebra over a non-abelian group with non-simple infinitesimal braiding of rank at least 4 is of this form. We follow the steps of the Lifting Method by Andruskiewitsch--Schneider. Our starting point is the classification of finite-dimensional Nichols algebras over non-abelian groups by Heckenberger--Vendramin, which consist of low rank exceptions and large rank families. We prove that the large rank families are cocycle twists of Nichols algebras constructed by the second author as foldings of Nichols algebras of Cartan type over abelian groups by outer automorphisms. This enables us to give uniform Lie-theoretic descriptions of the large rank families, prove generation in degree one and construct liftings. We also show that every lifting is a cocycle deformation of the corresponding coradically graded Hopf algebra using an explicit presentation by generators and relations of the Nichols algebra. On the level of tensor categories, we construct families of graded extensions of the representation category of a quantum group by a group of diagram automorphism.

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arXiv2022

Corrigendum: Liftings of the Jordan plane

We complete the classification of the pointed Hopf algebras with finite Gelfand-Kirillov dimension that are liftings of the Jordan plane over a nilpotent-by-finite group, correcting the statement in arXiv:1512.09271.

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arXiv2021

Lie algebras arising from Nichols algebras of diagonal type

Let ${\mathcal B}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type with braiding matrix $\mathfrak{q}$, let $\mathcal{L}_{\mathfrak{q}}$ be the corresponding Lusztig algebra as in arXiv:1501.04518 and let $\operatorname{Fr}_{\mathfrak{q}}: \mathcal{L}_{\mathfrak{q}} \to U(\mathfrak{n}^{\mathfrak{q}})$ be the corresponding quantum Frobenius map as in arXiv:1603.09387. We prove that the finite-dimensional Lie algebra $\mathfrak{n}^{\mathfrak{q}}$ is either 0 or else the positive part of a semisimple Lie algebra $\mathfrak{g}^{\mathfrak{q}}$ which is determined for each $\mathfrak{q}$ in the list of arXiv:math/0605795.

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arXiv2021

Finite GK-dimensional pre-Nichols algebras of (super)modular and unidentified type

We show that every finite GK-dimensional pre-Nichols algebra for braidings of diagonal type with connected diagram of modular, supermodular or unidentified type is a quotient of the distinguished pre-Nichols algebra introduced by the first-named author, up to two exceptions. For both of these exceptional cases, we provide a pre-Nichols algebra that substitutes the distinguished one in the sense that it projects onto all finite GK-dimensional pre-Nichols algebras. We build these two substitutes as non-trivial central extensions with finite GK-dimension of the corresponding distinguished pre-Nichols algebra. We describe these algebras by generators and relations, and provide a basis. This work essentially completes the study of eminent pre-Nichols algebras of diagonal type with connected diagram and finite-dimensional Nichols algebra.

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arXiv2021

Cohomology rings of finite-dimensional pointed Hopf algebras over abelian groups

We show that the cohomology ring of a finite-dimensional complex pointed Hopf algebra with an abelian group of group-like elements is finitely generated. Our strategy has three major steps. We first reduce the problem to the finite generation of cohomology of finite dimensional Nichols algebras of diagonal type. For the Nichols algebras we do a detailed analysis of cohomology via the Anick resolution reducing the problem further to specific combinatorial properties. Finally, to check these properties we turn to the classification of Nichols algebras of diagonal type due to Heckenberger. In this paper we complete the verification of these combinatorial properties for major parametric families, including Nichols algebras of Cartan and super types and develop all the theoretical foundations necessary for the case-by-case analysis. The remaining discrete families are addressed in a separate publication. As an application of the main theorem we deduce finite generation of cohomology for other classes of finite-dimensional Hopf algebras, including basic Hopf algebras with abelian groups of characters and finite quotients of quantum groups at roots of one.

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arXiv2020

Finite GK-Dimensional pre-Nichols algebras of super and standard type

We prove that finite GK-dimensional pre-Nichols algebras of super and standard type are quotients of the corresponding distinguished pre-Nichols algebras, except when the braiding matrix is of type super A and the dimension of the braided vector space is three. For these two exceptions we explicitly construct substitutes as braided central extensions of the corresponding pre-Nichols algebras by a polynomial ring in one variable. Via bosonization this gives new examples of finite GK-dimensional Hopf algebras.

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arXiv2020

On the Hopf algebra structure of the Lusztig quantum divided power algebras

We study the Hopf algebra structure of Lusztig's quantum groups. First we show that the zero part is the tensor product of the group algebra of a finite abelian group with the enveloping algebra of an abelian Lie algebra. Second we build them from the plus, minus and zero parts by means of suitable actions and coactions within the formalism presented by Sommerhauser to describe triangular decompositions.

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arXiv2020

On Nichols algebras of infinite rank with finite Gelfand-Kirillov dimension

We classify infinite-dimensional decomposable braided vector spaces arising from abelian groups whose components are either points or blocks such that the corresponding Nichols algebras have finite Gelfand-Kirillov dimension. In particular we exhibit examples with $\operatorname{GKdim} = n$ for any natural number $n$.

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