# Papers and preprints

## Contragredient Lie algebras in symmetric categories

Joint with Iván Angiono and Julia Plavnik

Submitted

We define contragredient Lie algebras in symmetric categories, generalizing the construction of Lie algebras of the form 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≥5. We also show that some previously known examples can be obtained with our construction.

## Semisimplification of contragredient Lie algebras

Joint with Iván Angiono and Julia Plavnik

Orbita Mathematicae, to appear

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.

## On finite group scheme-theoretical categories, 1

Joint with Shlomo Gelaki

Submitted

Let 𝒞 denote a finite group scheme-theoretical category over an algebraically closed field of characteristic p≥0 as introduced by the first author. For any indecomposable exact module category over 𝒞, we classify its simple objects and provide an expression for their projective covers, in terms of double cosets and projective representations of certain closed subgroup schemes, which upgrades a result by Ostrik for group-theoretical fusion categories. As a byproduct, we describe the simples and indecomposable projectives of 𝒞, and parametrize the Brauer-Picard group of Coh(G) for any finite connected group scheme G. Finally, we apply our results to describe the blocks of the center of Coh(G), which is a group scheme-theoretical category.

## Finite-dimensional quantum groups of type super A and non-semisimple modular categories

Joint with Robert Laugwitz

Submitted

We construct a series of finite-dimensional quantum groups as braided Drinfeld doubles of Nichols algebras of type Super A, for an even root of unity, and classify ribbon structures for these quantum groups. Ribbon structures exist if and only if the rank is even and all simple roots are odd. In this case, the quantum groups have a unique ribbon structure which comes from a non-semisimple spherical structure on the positive Borel Hopf subalgebra. Hence, the categories of finite-dimensional modules over these quantum groups provide examples of non-semisimple modular categories. In the rank-two case, we explicitly describe all simple modules of these quantum groups. We finish by computing link invariants, based on generalized traces, associated to a four-dimensional simple module of the rank-two quantum group. These knot invariants distinguish certain knots indistinguishable by the Jones or HOMFLYPT polynomials.

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

Joint with Iván Angiono and Simon Lentner

Proceedings of the London Mathematical Society, Volume 117, Issue 4 (2023), 1185-1245.

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.

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

Joint with Iván Angiono and Emiliano Campagnolo

Journal of Noncommutative Geometry Volume 17 (2023), 499–525.

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.

## Finite GK-dimensional pre-Nichols algebras of super and standard type.

Joint with Iván Angiono and Emiliano Campagnolo

Journal of Pure and Applied Algebra Volume 28 Issue 2 (2024), 107464.

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.

## Finite GK-dimensional pre-Nichols algebras of quantum linear spaces and of Cartan type.

Transactions American Mathematical Society Series B Volume 8 (2021), 296–329.

We describe all finite-dimensional pointed Hopf algebras whose infinitesimal braiding is a fixed Yetter-Drinfeld module decomposed as the sum of two simple objects: a point and the one of transpositions of the symmetric group in three letters. We give a presentation by generators and relations of the corresponding Nichols algebra and show that Andruskiewitsch-Schneider Conjecture holds for this kind of pointed Hopf algebras.

## Pointed Hopf algebras over non abelian groups with decomposable braidings, I.

Joint with Iván Angiono

Journal of Algebra, Volume 549 (2020), 78–111.

We describe all finite-dimensional pointed Hopf algebras whose infinitesimal braiding is a fixed Yetter-Drinfeld module decomposed as the sum of two simple objects: a point and the one of transpositions of the symmetric group in three letters. We give a presentation by generators and relations of the corresponding Nichols algebra and show that Andruskiewitsch-Schneider Conjecture holds for this kind of pointed Hopf algebras.