Mathematics

We study pre-Nichols algebras of quantum linear spaces and of Cartan type with finite GK-dimension. We prove that out of a short list of exceptions involving only roots of order 2, 3, 4, 6, any such pre-Nichols algebra is a quotient of the distinguished pre-Nichols algebra introduced by Angiono generalizing the De Concini-Procesi quantum groups. There are two new examples, one of which can be thought of as G 2 at a third root of one.

Let e be an arbitrary even nilpotent element in the general linear Lie superalgebra gl M|N and let W e be the associated finite W -superalgebra. Let Y m|n be the super Yangian associated to the Lie superalgebra gl m|n . A subalgebra of Y m|n , called the shifted super Yangian and denoted by Y m|n (σ) , is defined and studied. Moreover, an explicit isomorphism between W e and a quotient of Y m|n (σ) is established.

The Suzuki algebra A μλ Nn was introduced by Suzuki Satoshi in 1998, which is a class of cosemisimple Hopf algebras. In this paper, the author gives a complete set of simple Yetter-Drinfeld modules over the Suzuki algebra A μλ N2n and investigates the Nichols algebras over those simple Yetter-Drinfeld modules. The involved finite dimensional Nichols algebras of diagonal type are of Cartan type A 1 , A 1 × A 1 , A 2 , Super type A 2 (q; I 2 ) and the Nichols algebra ufo(8) . And the 4m and m 2 -dimensional Nichols algebras which discovered in \cite[section 3.7]{Andruskiewitsch2018} can be realized in the category of Yetter-Drinfeld modules over A μλ Nn . Furthermore, we obtain that dimB( V abe )=∞ under the condition b 2 =(ae ) −1 , b∈ G m for m≥5 , by using a result of Masuoka.

The representations of the quantum toroidal algebras have been widely studied by many authors. However, no one has constructed some finite dimensional modules for them while q is generic. In this paper, for all g -generic q , if g is not of type A 1 , we prove that the quantum toroidal algebra U q ( g tor ) has no nontrivial finite dimensional simple module.

We prove that the Drinfeld double of an arbitrary finite group scheme has finitely generated cohomology. That is to say, for G any finite group scheme, and D(G) the Drinfeld double of the group ring kG, we show that the self-extension algebra of the trivial representation for D(G) is a finitely generated algebra, and that for each D(G)-representation V the extensions from the trivial representation to V form a finitely generated module over the aforementioned algebra. As a corollary, we find that all categories rep(G)*_M dual to rep(G) are of also of finite type (i.e. have finitely generated cohomology), and we provide a uniform bound on their Krull dimensions. This paper completes earlier work of E. M. Friedlander and the author.

We provide finite presentations for stated skein algebras and deduce that those algebras are Koszul and that they are isomorphic to the quantum moduli algebras appearing in lattice gauge field theory, generalizing previous results of Bullock, Frohman, Kania-Bartoszynska and Faitg.

Finite real spectral triples are defined to characterise the non-commutative geometry of a fuzzy torus. The geometries are the non-commutative analogues of flat tori with moduli determined by integer parameters. Each of these geometries has four different Dirac operators, corresponding to the four unique spin structures on a torus. The spectrum of the Dirac operator is calculated. It is given by replacing integers with their quantum integer analogues in the spectrum of the corresponding commutative torus.

We prove an analog of Deligne's theorem for finite symmetric tensor categories C with the Chevalley property over an algebraically closed field k of characteristic 2 . Namely, we prove that every such category C admits a symmetric fiber functor to the symmetric tensor category D of representations of the triangular Hopf algebra $(k[\dd]/(\dd^2),1\ot 1 + \dd\ot \dd)$. Equivalently, we prove that there exists a unique finite group scheme G in D such that C is symmetric tensor equivalent to $\Rep_{\mathcal{D}}(G)$. Finally, we compute the group H 2 inv (A,K) of equivalence classes of twists for the group algebra K[A] of a finite abelian p -group A over an arbitrary field K of characteristic p>0 , and the Sweedler cohomology groups H i Sw (O(A),K) , i≥1 , of the function algebra O(A) of A .

We analyse the rack structure of conjugacy classes in simple Suzuki and Ree groups and determine which classes are kthulhu. Combining this results with abelian rack techniques, we show that the only finite-dimensional complex pointed Hopf algebras over the simple Suzuki and Ree groups are their group algebras.

In [arXiv:1711.07391] we have defined quantum groups U υ (sl(R)) and U υ (sl( S 1 )) , which can be interpreted as continuous generalizations of the quantum groups of the Kac-Moody Lie algebras of finite, respectively affine type A . In the present paper, we define the Fock space representation F R of the quantum group U υ (sl(R)) as the vector space generated by real pyramids (a continuous generalization of the notion of partition). In addition, by using a variant of the "folding procedure" of Hayashi-Misra-Miwa, we define an action of U υ (sl( S 1 )) on F R .