Mathematics

The higher rank Askey-Wilson algebra was recently constructed in the n -fold tensor product of U_q(\mathfrak{sl}_2) . In this paper we prove a class of identities inside this algebra, which generalize the defining relations of the rank one Askey-Wilson algebra. We extend the known construction algorithm by several equivalent methods, using a novel coaction. These allow to simplify calculations significantly. At the same time, this provides a proof of the corresponding relations for the higher rank q -Bannai-Ito algebra.

In this paper, we introduce the definitions of signatures of braided fusion categories, which are proved to be invariants of their Witt equivalence classes. These signature assignments define group homomorphisms on the Witt group. The higher central charges of pseudounitary modular categories can be expressed in terms of these signatures, which are applied to prove that the Ising modular categories have infinitely many square roots in the Witt group. This result is further applied to prove a conjecture of Davydov-Nikshych-Ostrik on the super-Witt group: the torsion subgroup generated by the completely anisotropic s-simple braided fusion categories has infinite rank.

We introduce higher level q -oscillator representations for the quantum affine (super)algebras of type C (1) n , C (2) (n+1) and B (1) (0,n) . These representations are constructed by applying the fusion procedure to the level one q -oscillator representations which were obtained through the studies of the tetrahedron equation. We prove that these higher level q -oscillator representations are irreducible. For type C (1) n and C (2) (n+1) , we compute their characters explicitly in terms of Schur polynomials.

Given a finite modular tensor category, we associate with each compact surface with boundary a cochain complex in such a way that the mapping class group of the surface acts projectively on its cohomology groups. In degree zero, this action coincides with the known projective action of the mapping class group on the space of chiral conformal blocks. In the case that the surface is a torus and the category is the representation category of a factorizable ribbon Hopf algebra, we recover our previous result on the projective action of the modular group on the Hochschild cohomology groups of the Hopf algebra.

Let X be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the S 1 -equivariant homology H ¯ ¯ ¯ ¯ ¯ S 1 ∗ (LX,Q) of the free loop space of X preserves the Hodge decomposition of H ¯ ¯ ¯ ¯ ¯ S 1 ∗ (LX,Q) , making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture proposed in our earlier work.

We construct covariant q -deformed holomorphic structures for all finitely-generated relative Hopf modules over the irreducible quantum flag manifolds endowed with their Heckenberger--Kolb calculi. In the classical limit these reduce to modules of sections of holomorphic homogeneous vector bundles over irreducible flag manifolds. For the case of simple relative Hopf modules, we show that this covariant holomorphic structure is unique. This generalises earlier work of Majid, Khalkhali, Landi, and van Suijlekom for line modules of the Podleś sphere, and subsequent work of Khalkhali and Moatadelro for general quantum projective space.

The algebra L g,n (H) was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and quantizes the character variety of the Riemann surface Σ g,n ∖D ( D is an open disk). In this article we define a holonomy map in that quantized setting, which associates a tensor with components in L g,n (H) to tangles in ( Σ g,n ∖D)×[0,1] , generalizing previous works of Buffenoir-Roche and Bullock-Frohman-Kania-Bartoszynska. We show that holonomy behaves well for the stack product and the action of the mapping class group; then we specialize this notion to links in order to define a generalized Wilson loop map. Thanks to the holonomy map, we give a geometric interpretation of the vacuum representation of L g,0 (H) on L 0,g (H) . Finally, the general results are applied to the case H= U q 2 ( sl 2 ) in relation to skein theory and the most important consequence is that the stated skein algebra of a compact oriented surface with just one boundary edge is isomorphic to L g,n ( U q 2 ( sl 2 )) . Throughout the paper we use a graphical calculus for tensors with coefficients in L g,n (H) which makes the computations and definitions very intuitive.

We show that the reduced SL 2 (C) -twisted Burau representation can be obtained from the quantum group U q ( sl 2 ) for q=i a fourth root of unity and that representations of U q ( sl 2 ) satisfy a type of Schur-Weyl duality with the Burau representation. As a consequence, the SL 2 (C) -twisted Reidemeister torsion of links can be obtained as a quantum invariant. Our construction is closely related to the quantum holonomy invariant of Blanchet, Geer, Patureau-Mirand, and Reshetikhin, and we interpret their invariant as a twisted Conway potential.

Given any modular category C over an algebraically closed field k , we extract a sequence ( M g ) g≥0 of C -bimodules. We show that the Hochschild chain complex CH(C; M g ) of C with coefficients in M g carries a canonical homotopy coherent projective action of the mapping class group of the surface of genus g+1 . The ordinary Hochschild complex of C corresponds to CH(C; M 0 ) . This result is obtained as part of the following more comprehensive topological structure: We construct a symmetric monoidal functor F C :C- Surf c → Ch k with values in chain complexes over k defined on a symmetric monoidal category of surfaces whose boundary components are labeled with projective objects in C . The functor F C satisfies an excision property which is formulated in terms of homotopy coends. In this sense, any modular category gives naturally rise to a modular functor with values in chain complexes. In zeroth homology, it recovers Lyubashenko's mapping class group representations. The chain complexes in our construction are explicitly computable by choosing a marking on the surface, i.e. a cut system and a certain embedded graph. For our proof, we replace the connected and simply connected groupoid of cut systems that appears in the Lego-Teichmüller game by a contractible Kan complex.

Rota-Baxter operators, O -operators on Lie algebras and their interconnected pre-Lie and post-Lie algebras are important algebraic structures with applications in mathematical physics. This paper introduces the notions of a homotopy Rota-Baxter operator and a homotopy O -operator on a symmetric graded Lie algebra. Their characterization by Maurer-Cartan elements of suitable differential graded Lie algebras is provided. Through the action of a homotopy O -operator on a symmetric graded Lie algebra, we arrive at the notion of an operator homotopy post-Lie algebra, together with its characterization in terms of Maurer-Cartan elements. A cohomology theory of post-Lie algebras is established, with an application to 2-term skeletal operator homotopy post-Lie algebras.