Mathematics

This paper concerns the positive part U + q of the quantum group U q ( sl ˆ 2 ) . The algebra U + q has a presentation involving two generators that satisfy the cubic q -Serre relations. We recently introduced an algebra U + q called the alternating central extension of U + q . We presented U + q by generators and relations. The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. In this paper we obtain a presentation of U + q that involves a small subset of the original set of generators and a very manageable set of relations. We call this presentation the compact presentation of U + q .

In this paper we give a complete classification of unitary fusion categories ⊗ -generated by an object of dimension 1+ 5 √ 2 . We show that all such categories arise as certain wreath products of either the Fibonacci category, or of the dual even part of the 2D2 subfactor. As a by-product of proving our main classification result we produce a classification of finite unitarizable quotients of Fib ∗N satisfying a certain symmetry condition.

The q -Onsager algebra O q is defined by two generators W 0 , W 1 and two relations called the q -Dolan/Grady relations. Recently Baseilhac and Kolb obtained a PBW basis for O q with elements denoted { B nδ+ α 0 } ??n=0 ,{ B nδ+ α 1 } ??n=0 ,{ B nδ } ??n=1 . In their recent study of a current algebra A q , Baseilhac and Belliard conjecture that there exist elements { W ?�k } ??k=0 ,{ W k+1 } ??k=0 ,{ G k+1 } ??k=0 ,{ G ~ k+1 } ??k=0 in O q that satisfy the defining relations for A q . In order to establish this conjecture, it is desirable to know how the elements in the second list above are related to the elements in the first list above. In the present paper, we conjecture the precise relationship and give some supporting evidence. This evidence consists of some computer checks on SageMath due to Travis Scrimshaw, and a proof of our conjecture for a homomorphic image of O q called the universal Askey-Wilson algebra.

We consider a deformation of the Robert-Wagner foam evaluation formula, with an eye toward a relation to formal groups. Integrality of the deformed evaluation is established, giving rise to state spaces for planar GL(N) MOY graphs (Murakami-Ohtsuki-Yamada graphs). Skein relations for the deformation are worked out in details in the GL(2) case. These skein relations deform GL(2) foam relations of Beliakova, Hogancamp, Putyra and Wehrli. We establish the Reidemeister move invariance of the resulting chain complexes assigned to link diagrams, giving us a link homology theory.

A classical theorem of Veldkamp describes the center of an enveloping algebra of a Lie algebra of a semi-simple algebraic group in characteristic p. We generalize this result to a class of Lie algebras with a property that they arise as the reduction modulo p≫0 from an algebraic Lie algebra g, such that g has no nontrivial semi-invariants in Sym(g) and Sym(g ) g is a polynomial algebra. As an application, we solve the derived isomorphism problem of enveloping algebras for the above class of Lie algebras.

Let C be the category of finite dimensional modules over the quantum affine algebra U q ( g ˆ ) of a simple complex Lie algebra g . Let C − be the subcategory introduced by Hernandez and Leclerc. We prove the geometric q -character formula conjectured by Hernandez and Leclerc in types A and B for a class of simple modules called snake modules introduced by Mukhin and Young. Moreover, we give a combinatorial formula for the F -polynomial of the generic kernel associated to the snake module. As an application, we show that snake modules correspond to cluster monomials with square free denominators and we show that snake modules are real modules. We also show that the cluster algebras of the category C 1 are factorial for Dynkin types A,D,E .

A class of classical affine W-algebras are shown to be isomorphic as differential algebras to the coordinate rings of double coset spaces of certain prounipotent proalgebraic groups. As an application, integrable Hamiltonian hierarchies associated with them are constructed geometrically, generalizing the corresponding result of Feigin-Frenkel and Enriquez-Frenkel for the principal cases.

For a Kähler manifold X equipped with a prequantum line bundle L , we give a geometric construction of a family of representations of the Berezin-Toeplitz deformation quantization algebra ( C ∞ (X)[[ℏ]], ⋆ BT ) parametrized by points z 0 ∈X . The key idea is to use peak sections to suitably localize the Hilbert spaces H 0 (X, L ⊗m ) around z 0 in the large volume limit.

This is a companion paper to "Ellipsitomic associators". We provide a (m)operadic description of Enriquez's torsor of cyclotomic associators, as well as of its associated cyclotomic Grothendieck-Teichmüller groups.

We follow the approach developed by Beilinson-Lusztig-MacPherson and modified by Fu and the first author to investigate a new realization for the i-quantum groups U^j(n) of type B, building on the multiplication formulas discovered in [BKLW,Lem.~3.2]. This allows us to present U^j(n) via a basis and multiplication formulas by generators. We also establish a surjective algebra homomorphism from a Lusztig type form of U^j(n) to integral q-Schur algebras of type B. Thus, base changes allow us to relate representations of the i-quantum hyperalgebras of U^j(n) to representations of finite orthogonal groups of odd degree in non-defining characteristics. This generalizes part of Dipper--James' type A theory to the type B case.