Mathematics

We show that the triply graded Khovanov-Rozansky homology of knots and links over a field of positive odd characteristic p descends to an invariant in the homotopy category finite-dimensional p -complexes. A p -extended differential on the triply graded homology discovered by Cautis is compatible with the p -DG structure. As a consequence we get a categorification of the Jones polynomial evaluated at an odd prime root of unity

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.

We begin the study of the representation theory of the infinite Temperley-Lieb algebra. We fully classify its finite dimensional representations, then introduce infinite link state representations and classify when they are irreducible or indecomposable. We also define a construction of projective indecomposable representations for T L n that generalizes to give extensions of T L ∞ representations. Finally we define a generalization of the spin chain representation and conjecture a generalization of Schur-Weyl duality.

We construct a coherent Hopf 2-algebra as quantization of a coherent 2-group, which consists of two Hopf coquasigroups and a coassociator. For this constructive method, if we replace Hopf coquasigroups by Hopf algebras, we can construct a strict Hoft 2-algebra, which is a quantisation of 2-group. We also study the crossed comodule of Hopf algebras, which is shown to be a strict Hopf 2-algebra under some conditions. As an example, a quasi coassociative Hopf coquasigroup is employed to build a special coherent Hopf 2-algebra with nontrivial coassociator. Following this we study functions on Cayley algebra basis.

Let E m be the Fomin-Kirillov algebra, and let B S m be the Nichols-Woronowicz algebra model for Schubert calculus on the symmetric group S m which is a quotient of E m , i.e. the Nichols algebra associated to a Yetter-Drinfeld S m -module defined by the set of reflections of S m and a specific one-dimensional representation of a subgroup of S m . It is a famous open problem to prove that E m is infinite dimensional for all m≥6 . In this work, as a step towards a solution of this problem, we introduce a subalgebra of B S m , and prove, under the assumption of finite dimensionality of B S m , that this subalgebra admits unique integrals in a strong sense, and we relate these integrals to integrals in B S m . The techniques we use rely on braided differential calculus as developed by Bazlov and Liu, and on the notion of integrals for Hopf algebras as introduced by Sweedler.

We consider the super Jordan plane, a braided Hopf algebra introduced--to the best of our knowledge--in works of N. Andruskiewitsch, I. Angiono, I. Heckenberger, and its restricted version in odd characteristic introduced by the same authors. We show that their Drinfeld doubles give rise naturally to Hopf superalgebras justifying a posteriori the adjective super given in their work. These Hopf superalgebras are extensions of super commutative ones by the enveloping, respectively restricted enveloping, algebra of osp(1|2) .

We study the Ehresmann--Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the classical gauge groupoid of a principal bundle. When the base algebra is in the centre of the total space algebra, the gauge group of the noncommutative principal bundle is isomorphic to the group of bisections of the bialgebroid. In particular we consider Galois objects (non-trivial noncommutative bundles over a point in a sense) for which the bialgebroid is a Hopf algebra. For these we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples include Galois objects of group Hopf algebras and of Taft algebras.

We study the skein algebra of the genus 2 surface and its action on the skein module of the genus 2 handlebody. We compute this action explicitly, and we describe how the module decomposes over certain subalgebras in terms of polynomial representations of double affine Hecke algebras. Finally, we show that this algebra is isomorphic to the t=q specialisation of the genus two spherical double affine Hecke algebra recently defined by Arthamonov and Shakirov.

We present an overview of the notions of exact sequences of Hopf algebras and tensor categories and their connections. We also present some examples illustrating their main features; these include simple fusion categories and a natural question regarding composition series of finite tensor categories.

We apply the theory of ϕ -coordinated modules, developed by H.-S. Li, to the Etingof--Kazhdan quantum affine vertex algebra associated with the trigonometric R -matrix of type A . We prove, for a certain associate ϕ of the one-dimensional additive formal group, that any ϕ -coordinated module for the level c∈C quantum affine vertex algebra is naturally equipped with a structure of restricted level c module for the quantum affine algebra in type A and vice versa. Moreover, we show that any ϕ -coordinated module is irreducible with respect to the action of the quantum affine vertex algebra if and only if it is irreducible with respect to the corresponding action of the quantum affine algebra. In the end, we discuss relation between the centers of the quantum affine algebra and the quantum affine vertex algebra.