Mathematics

In this note we discuss the possibility of constructing the cosimplicial complex for the multiplier Hopf algebras and extending the cyclicity operator to obtain the Hopf-cyclic cohomology for them. We show that the definition of modular pairs in involution for multiplier Hopf algebras and provide the definition of Hopf-cyclic cohomology for algebras of functions over discrete groups.

Universal solutions to deformation quantization problems can be conveniently classified by the cohomology of suitable graph complexes. In particular, the deformation quantization of (finite dimensional) Poisson manifolds and Lie bialgebras are characterised by an action of the Grothendieck-Teichmüller group via one-colored directed and oriented graphs, respectively. In this note, we study the action of multi-oriented graph complexes on Lie bialgebroids and their "quasi" generalisations. Using results due to T. Willwacher and M. Živkovi? on the cohomology of (multi)-oriented graphs, we show that the action of the Grothendieck-Teichmüller group on Lie bialgebras and Lie-quasi bialgebras can be generalised to Lie-quasi bialgebroids via graphs with two colors, one of them being oriented. However, this action generically fails to preserve the subspace of Lie bialgebroids. By resorting to graphs with two oriented colors, we instead show the existence of an obstruction to the quantization of a generic Lie bialgebroid in the guise of a new Lie ??-algebra structure non-trivially deforming the "big bracket" for Lie bialgebroids. This exotic Lie ??-structure can be interpreted as the equivalent in d=3 of the Kontsevich-Shoikhet obstruction to the quantization of infinite dimensional Poisson manifolds (in d=2 ). We discuss the implications of these results with respect to a conjecture due to P. Xu regarding the existence of a quantization map for Lie bialgebroids.

We construct quasi-particle bases of principal subspaces of standard modules L(Λ) , where Λ= k 0 Λ 0 + k j Λ j , and Λ j denotes the fundamental weight of affine Lie algebras of type B (1) l , C (1) l , F (1) 4 or G (1) 2 of level one. From the given bases we find characters of principal subspaces.

Let A ^ n be the completion by the degree of a differential operator of the n -th Weyl algebra with generators x 1 ,…, x n , ∂ 1 ,…, ∂ n . Consider n elements X 1 ,…, X n in A ^ n of the form X i = x i + ∑ K=1 ∞ ∑ l=1 n ∑ j=1 n x l p K−1,l ij (∂) ∂ j , where p K−1,l ij (∂) is a degree (K−1) homogeneous polynomial in ∂ 1 ,…, ∂ n , antisymmetric in subscripts i,j . Then for any natural k and any function i:{1,…,k}→{1,…,n} we prove ∑ σ∈Σ(k) X i σ(1) ⋯ X i σ(k) ▹1=k! x i 1 ⋯ x i k , where Σ(k) is the symmetric group on k letters and ▹ denotes the Fock action of the A ^ n on the space of (commutative) polynomials.

Let k be a field of characteristic zero. This paper studies a problem proposed by Joseph F. Ritt in 1950. Precisely, we prove that (1) If p≥2 is an integer, for every integer i∈N , the nilpotency index of the image of T i in the ring k{T}/[ T p ] equals (i+1)p−i . (2) For every pair of integers (i,j) , the nilpotency index of the image of T i U j in the ring k{T}/[TU] equals i+j+1 .

We present a realisation of the universal/simple Bershadsky--Polyakov vertex algebras as subalgebras of the tensor product of the universal/simple Zamolodchikov vertex algebras and an isotropic lattice vertex algebra. This generalises the realisation of the universal/simple affine vertex algebras associated to sl 2 and osp(1|2) given in arXiv:1711.11342. Relaxed highest-weight modules are likewise constructed, conditions for their irreducibility are established, and their characters are explicitly computed, generalising the character formulae of arXiv:1803.01989.

As a quantum affinization, the quantum toroidal algebra is defined in terms of its "left" and "right" halves, which both admit shuffle algebra presentations. In the present paper, we take an orthogonal viewpoint, and give shuffle algebra presentations for the "top" and "bottom" halves instead, starting from the evaluation representation of the quantum affine group and its usual R-matrix. An upshot of this construction is a new topological coproduct on the quantum toroidal algebra which extends the Drinfeld-Jimbo coproduct on the horizontal quantum affine subalgebra.

We consider the Dirac operator of a general metric in the canonical conformal class on the noncommutative two torus, twisted by an idempotent (representing the K -theory class of a general noncommutative vector bundle), and derive a local formula for the Fredholm index of the twisted Dirac operator. Our approach is based on the McKean-Singer index formula, and explicit heat expansion calculations by making use of Connes' pseudodifferential calculus. As a technical tool, a new rearrangement lemma is proved to handle challenges posed by the noncommutativity of the algebra and the presence of an idempotent in the calculations in addition to a conformal factor.

In this paper we prove that the family of colored Jones polynomials of a knot in S 3 determines the family of ADO polynomials of this knot. More precisely, we construct a two variables knot invariant unifying both the ADO and the colored Jones polynomials. On one hand, the first variable q can be evaluated at 2r roots of unity with r∈ N ∗ and we obtain the ADO polynomial over the Alexander polynomial. On the other hand, the second variable A evaluated at A= q n gives the colored Jones polynomials. From this, we exhibit a map sending, for any knot, the family of colored Jones polynomials to the family of ADO polynomials. As a direct application of this fact, we will prove that every ADO polynomial is q-holonomic and is annihilated by the same polynomials as of the colored Jones function. The construction of the unified invariant will use completions of rings and algebra. We will also show how to recover our invariant from Habiro's quantum sl 2 completion studied in arXiv:math/0605313.

In this paper, we introduce a new sequence N ¯ m to find a new estimation of the cardinality N m of the minimal involutive square-free solution of level m . As an application, using the first values of N ¯ m , we improve the estimations of N m obtained by Gateva-Ivanova and Cameron and by Lebed and Vendramin. Following the approach of the first part, in the last section we construct several new counterexamples to the Gateva-Ivanova's Conjecture.