Mathematics

The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In general, there are more quantum symmetries than symmetries and it is a non-trivial task to determine when this is the case for a given graph: The question is whether or not the algebra associated to the quantum automorphism group is commutative. We use Gröbner base computations in order to tackle this problem; the implementation uses GAP and the SINGULAR package LETTERPLACE. We determine the existence of quantum symmetries for all connected, undirected graphs without multiple edges and without self-edges, for up to seven vertices. As an outcome, we infer within our regime that a classical automorphism group of order one or two is an obstruction for the existence of quantum symmetries.

For the standard Drinfeld-Jimbo quantum group U q (g) associated with a simple Lie algebra g , we construct explicit generators of the centre Z( U q (g)) , and determine the relations satisfied by the generators. For g of type A n (n??) , D 2k+1 (k??) or E 6 , the centre Z( U q (g)) is isomorphic to a quotient of a polynomial algebra in multiple variables, which is described in a uniform manner for all cases. For g of any other type, Z( U q (g)) is generated by n= rank (g) algebraically independent elements.

We study a potential method for constructing the Rozansky--Witten TQFT as an extended (1+1+1) -TQFT. We construct a 2 -category consisting of schemes, complexes of sheaves and sheaf morphisms and show that there are (1+1) -TQFTs valued in the truncation of this category which have state spaces that agree with the Rozansky--Witten TQFT. However, we also show that if such a TQFT is based on a reduced Noetherian scheme, it cannot be extended upwards to a (1+1+1) -TQFT.

In his PhD thesis, Goosen combined the string-net and the generators-and-relations formalisms for arbitrary once-extended 3-dimensional TQFTs. In this paper we work this out in detail for the simplest non-trivial example, where the underlying spherical fusion category is the category of Z/2Z -graded vector spaces. This allows us to give an elementary string-net description of the linear maps associated to 3-dimensional bordisms. The string-net formalism also simplifies the description of the mapping class group action in the resulting TQFT. We conclude the paper by performing some example calculations from this viewpoint.

In this paper, we explore natural connections among the representations of the extended affine Lie algebra s l N ˆ ( C q ) with C q = C q [ t ±1 0 , t ±1 1 ] an irrational quantum 2-torus, the simple affine vertex algebra L s l ∞ ˆ (ℓ,0) with ℓ a positive integer, and Levi subgroups G of G L ℓ (C) . First, we give a canonical isomorphism between the category of integrable restricted s l N ˆ ( C q ) -modules of level ℓ and that of equivariant quasi L s l ∞ ˆ (ℓ,0) -modules. Second, we classify irreducible N -graded equivariant quasi L s l ∞ ˆ (ℓ,0) -modules. Third, we establish a duality between irreducible N -graded equivariant quasi L s l ∞ ˆ (ℓ,0) -modules and irreducible regular G -modules on certain fermionic Fock spaces. Fourth, we obtain an explicit realization of every irreducible N -graded equivariant quasi L s l ∞ ˆ (ℓ,0) -module. Fifth, we completely determine the following branchings: 1 The branching from L s l ∞ ˆ (ℓ,0)⊗ L s l ∞ ˆ ( ℓ ′ ,0) to L s l ∞ ˆ (ℓ+ ℓ ′ ,0) for quasi modules. 2 The branching from s l N ˆ ( C q ) to its Levi subalgebras. 3 The branching from s l N ˆ ( C q ) to its subalgebras s l N ˆ ( C q [ t ± M 0 0 , t ± M 1 1 ]) .

We compute higher Frobenius-Schur indicators of pq-dimensional pointed Hopf algebras in characteristic p through their associated graded Hopf algebras. These indicators are gauge invariants for the monoidal categories of representations of these algebras.

We study a certain class of bulk-boundary systems in the Batalin-Vilkovisky (BV) formalism. We construct factorization algebras of observables for such bulk-boundary systems, and show that these factorization algebras have a natural Poisson bracket of cohomological degree 1.

In [BK], it is shown that the Turaev-Viro invariants defined for a spherical fusion category A extends to invariants of 3-manifolds with corners. In [Kir], an equivalent formulation for the 2-1 part of the theory (2-manifolds with boundary) is described using the space of "stringnets with boundary conditions" as the vector spaces associated to 2-manifolds with boundary. Here we construct a similar theory for the 3-2 part of the 4-3-2 theory in [CY1993].

We develop a method of quantization for free field theories on manifolds with boundary where the bulk theory is topological in the direction normal to the boundary and a local boundary condition is imposed. Our approach is within the Batalin-Vilkovisky formalism. At the level of observables, the construction produces a stratified factorization algebra that in the bulk recovers the factorization algebra developed by Costello and Gwilliam. The factorization algebra on the boundary stratum enjoys a perturbative bulk-boundary correspondence with this bulk factorization algebra. A central example is the factorization algebra version of the abelian Chern-Simons/Wess-Zumino-Witten correspondence, but we examine higher dimensional generalizations that are related to holomorphic truncations of string theory and M -theory and involve intermediate Jacobians.

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.