Mathematics

We study a filtered vector space A d (n) over a field K of characteristic 0 , which consists of Jacobi diagrams of degree d on n oriented arcs for each n,d?? . We consider an action of the automorphism group Aut( F n ) of the free group F n of rank n on the space A d (n) , which is induced by an action of handlebody groups on bottom tangles. The action of Aut( F n ) on A d (n) induces an action of the general linear group GL(n,K) on the associated graded vector space of A d (n) , which is regarded as the vector space B d (n) consisting of open Jacobi diagrams. Moreover, the Aut( F n ) -action on A d (n) induces an action on B d (n) of the associated graded Lie algebra gr(IA(n)) of the IA-automorphism group IA(n) of F n with respect to its lower central series. We use an irreducible decomposition of B d (n) and computation of the gr(IA(n)) -action on B d (n) to study the Aut( F n ) -module structure of A d (n) . In particular, we consider the case where d=2 in detail and give an indecomposable decomposition of A 2 (n) . We also consider a functor A d , which includes the Aut( F n ) -module structure of A d (n) for all n .

The polyhedral realization of crystal base has been introduced by A.Zelevinsky and the second author([T.Nakashima, A.Zelevinsky, Adv. Math. 131, no. 1 (1997)]), which describe the crystal base B(∞) as a polyhedral convex cone in the infinite Z -lattice Z ∞ . To construct the polyhedral realization, we need to fix an infinite sequence ι from the indices of the simple roots. According to this ι , one has certain set of linear functions defining a polyhedral convex cone and under the `positivity condition' on ι , it has been shown that the polyhedral convex cone is isomorphic to the crystal base B(∞) . To confirm the positivity condition for a given ι , we need to obtain the whole feature of the set of linear functions, which requires, in general, a bunch of explicit calculations. In this article, we introduce the notion of the adapted sequence and show that if ι is an adapted sequence then the positivity condition holds for classical Lie algebras. Furthermore, we reveal the explicit forms of the polyhedral realizations associated with arbitrary adapted sequences ι in terms of column tableaux.

The polyhedral realizations for crystal bases of the integrable highest weight modules of U q (g) have been introduced in ([T.Nakashima, J. Algebra, vol.219, no. 2, (1999)]), which describe the crystal bases as sets of lattice points in the infinite Z -lattice Z ∞ given by some system of linear inequalities, where g is a symmetrizable Kac-Moody Lie algebra. To construct the polyhedral realization, we need to fix an infinite sequence ι from the indices of the simple roots. If the pair ( ι , λ ) ( λ : a dominant integral weight) satisfies the `ample' condition then there are some procedure to calculate the sets of linear inequalities. In this article, we show that if ι is an adapted sequence (defined in our paper [Y.Kanakubo, T.Nakashima, arXiv:1904.10919]) then the pair ( ι , λ ) satisfies the ample condition for any dominant integral weight λ in the case g is a classical Lie algebra. Furthermore, we reveal the explicit forms of the polyhedral realizations of the crystal bases B(λ) associated with arbitrary adapted sequences ι in terms of column tableaux. As an application, we will give a combinatorial description of the function ε ∗ i on the crystal base B(∞) .

The objects of study in this paper are Hopf algebras H which are finitely generated algebras over an algebraically closed field and are extensions of a commutative Hopf algebra by a finite dimensional Hopf algebra. Basic structural and homological properties are recalled and classes of examples are listed. Bounds are obtained on the dimensions of simple H -modules, and the structure of H is shown to be severely constrained when the finite dimensional extension is semisimple and cosemisimple.

For g a Kac-Moody algebra of affine type, we show that there is an AutO -equivariant identification between Fun Op g (D) , the algebra of functions on the space of g -opers on the disc, and W⊂ π 0 , the intersection of kernels of screenings inside a vacuum Fock module π 0 . This kernel W is generated by two states: a conformal vector, and a state δ −1 |0⟩ . We show that the latter endows π 0 with a canonical notion of translation T (aff) , and use it to define the densities in π 0 of integrals of motion of classical Conformal Affine Toda field theory. The AutO -action defines a bundle Π over P 1 with fibre π 0 . We show that the product bundles Π⊗ Ω j , where Ω j are tensor powers of the canonical bundle, come endowed with a one-parameter family of holomorphic connections, ∇ (aff) −α T (aff) , α∈C . The integrals of motion of Conformal Affine Toda define global sections [ v j d t j+1 ]∈ H 1 ( P 1 ,Π⊗ Ω j , ∇ (aff) ) of the de Rham cohomology of ∇ (aff) . Any choice of g -Miura oper χ gives a connection ∇ (aff) χ on Ω j . Using coinvariants, we define a map F χ from sections of Π⊗ Ω j to sections of Ω j . We show that F χ ∇ (aff) = ∇ (aff) χ F χ , so that F χ descends to a well-defined map of cohomologies. Under this map, the classes [ v j d t j+1 ] are sent to the classes in H 1 ( P 1 , Ω j , ∇ (aff) χ ) defined by the g -oper underlying χ .

In this paper, we explore a canonical connection between the algebra of q -difference operators V ? q , affine Lie algebra and affine vertex algebras associated to certain subalgebra A of the Lie algebra gl ??. We also introduce and study a category O of V ? q -modules. More precisely, we obtain a realization of V ? q as a covariant algebra of the affine Lie algebra A ??? , where A ??is a 1-dimensional central extension of A . We prove that restricted V q ? -modules of level ??12 correspond to Z -equivariant ? -coordinated quasi-modules for the vertex algebra V A ? ( ??12 ,0) , where A ? is a generalized affine Lie algebra of A . In the end, we show that objects in the category O are restricted V q ? -modules, and we classify simple modules in the category O .

It was shown by Ostrik (2003) and Natale (2017) that a collection of twisted group algebras in a pointed fusion category serve as explicit Morita equivalence class representatives of indecomposable, separable algebras in such categories. We generalize this result by constructing explicit Morita equivalence class representatives of indecomposable, separable algebras in group-theoretical fusion categories. This is achieved by providing the `free functor' Φ from a pointed fusion category to a group-theoretical fusion category with a monoidal structure. Our algebras of interest are then constructed as the image of twisted group algebras under Φ . We also show that twisted group algebras admit the structure of Frobenius algebras in a pointed fusion category, and we establish a Frobenius monoidal structure on Φ as well. As a consequence, our algebras are Frobenius algebras in a group-theoretical fusion category, and like twisted group algebras in the pointed case, they also enjoy several good algebraic properties.

In the framework of (vector valued) quantized holomorphic functions defined on non-commutative spaces, ``quantized hermitian symmetric spaces'', we analyze what the algebras of quantized differential operators with variable coefficients should be. It is an emediate point that even 0 th order operators, given as multiplications by polynomials, have to be specified as e.g. left or right multiplication operators since the polynomial algebras are replaced by quadratic, non-commutative algebras. In the settings we are interested in, there are bilinear pairings which allows us to define differential operators as duals of multiplication operators. Indeed, there are different choices of pairings which lead to quite different results. We consider three different pairings. The pairings are between quantized generalized Verma modules and quantized holomorphically induced modules. It is a natural demand that the corresponding representations can be expressed by (matrix valued) differential operators. We show that a quantum Weyl algebra Wey l q (n,n) introduced by T. Hyashi (Comm. Math. Phys. 1990) plays a fundamental role. In fact, for one pairing, the algebra of differential operators, though inherently depending on a choice of basis, is precisely matrices over Wey l q (n,n) . We determine explicitly the form of the (quantum) holomorphically induced representations and determine, for the different pairings, if they can be expressed by differential operators.

The notion of Ψ ^ -system in linear monoidal categories was introduced by Geer, Kashaev and Turaev. They showed that, under additional assumptions, a Ψ ^ -system gives rise to invariants of 3-manifolds. They conjectured that all quantum groups at odd roots of unity give rise to a Ψ ^ -system and verified this conjecture in the case of the Borel subalgebra of U q ( sl 2 ) . In this paper we construct a Ψ ^ -system in the category of modules of a quantum group related to U q ( sl 3 ) leading to a family of 3-manifolds invariants. These invariants are constructed using the quantum dilogarithm defined by Faddeev and Kashaev and allow an interpretation in terms of shapes variables of ideal hyperbolic tetrahedra.

Dual canonical bases are expected to satisfy a certain (double) triangularity property by Leclerc's conjecture. We propose an analogous conjecture for common triangular bases of quantum cluster algebras. We show that a weaker form of the analogous conjecture is true. Our result applies to the dual canonical bases of quantum unipotent subgroups. It also applies to the t -analogs of q -characters of simple modules of quantum affine algebras.