Mathematics

Recently Sarah Bockting-Conrad introduced the double lowering operator ψ for a tridiagonal pair. Motivated by ψ we consider the following problem about polynomials. Let F denote an algebraically closed field. Let x denote an indeterminate, and let F[x] denote the algebra consisting of the polynomials in x that have all coefficients in F . Let N denote a positive integer or ∞ . Let { a i } N−1 i=0 , { b i } N−1 i=0 denote scalars in F such that ∑ i−1 h=0 a h ≠ ∑ i−1 h=0 b h for 1≤i≤N . For 0≤i≤N define polynomials τ i , η i ∈F[x] by τ i = ∏ i−1 h=0 (x− a h ) and η i = ∏ i−1 h=0 (x− b h ) . Let V denote the subspace of F[x] spanned by { x i } N i=0 . An element ψ∈End(V) is called double lowering whenever ψ τ i ∈F τ i−1 and ψ η i ∈F η i−1 for 0≤i≤N , where τ −1 =0 and η −1 =0 . We give necessary and sufficient conditions on { a i } N−1 i=0 , { b i } N−1 i=0 for there to exist a nonzero double lowering map. There are four families of solutions, which we describe in detail.

We describe the double Yangian of the general linear Lie algebra gl N by following a general scheme of Drinfeld. This description is based on the construction of the universal R -matrix for the Yangian. To make the exposition self contained, we include the proofs of all necessary facts about the Yangian itself. In particular, we describe the centre of the Yangian by using its Hopf algebra structure, and provide a proof of the analogue of the Poincaré-Birkhoff-Witt theorem for the Yangian based on its representation theory. This proof extends to the double Yangian, thus giving a description of its underlying vector space.

This paper addresses a Hom-associative algebra built as a direct sum of a given Hom-associative algebra (A,⋅,α) and its dual ( A ∗ ,∘, α ∗ ), endowed with a non-degenerate symmetric bilinear form B, where ⋅ and ∘ are the products defined on A and A ∗ , respectively, and α and α ∗ stand for the corresponding algebra homomorphisms. Such a double construction, also called Hom-Frobenius algebra, is interpreted in terms of an infinitesimal Hom-bialgebra. The same procedure is applied to characterize the double construction of biHom-associative algebras, also called biHom-Frobenius algebra. Finally, a double construction of Hom-dendriform algebras, also called double construction of Connes cocycle or symplectic Hom-associative algebra, is performed. Besides, the concept of biHom-dendriform algebras is introduced and discussed. Their bimodules and matched pairs are also constructed, and related relevant properties are given.

In this article we prove that double quasi-Poisson algebras, which are non-commutative analogues of quasi-Poisson manifolds, naturally give rise to pre-Calabi-Yau algebras. This extends one of the main results in [11] (see also [10]), where a relationship between pre-Calabi-Yau algebras and double Poisson algebras was found. However, a major difference between the pre-Calabi-Yau algebra constructed in the mentioned articles and the one constructed in this work is that the higher multiplications indexed by even integers of the underlying A ∞ -algebra structure of the pre-Calabi-Yau algebra associated to a double quasi-Poisson algebra do not vanish, but are given by nice cyclic expressions multiplied by explicitly determined coefficients involving the Bernoulli numbers.

We exhibit new examples of double quasi-Poisson brackets, based on some classification results and the method of fusion. This method was introduced by Van den Bergh for a large class of double quasi-Poisson brackets which are said differential, and our main result is that it can be extended to arbitrary double quasi-Poisson brackets. We also provide an alternative construction for the double quasi-Poisson brackets of Van den Bergh associated to quivers, and of Massuyeau-Turaev associated to the fundamental groups of surfaces.

For a couple of associative algebras we define the notion of their double and give a set of examples. Also, we discuss applications of such doubles to representation theory of certain quantum algebras and to a new type of Noncommutative Geometry.

The most common geometric interpretation of the Yang-Baxter equation is by braids, knots and relevant Reidemeister moves. So far, cubes were used for connections with the third Reidemeister move only. We will show that there are higher-dimensional cube complexes solving the D -state Yang-Baxter equation for arbitrarily large D . More precisely, we introduce explicit constructions of cube complexes covered by products of n trees and show that these cube complexes lead to new solutions of the Yang-Baxter equations.

Given any quantum cluster algebra arising from a quantum unipotent subgroup of symmetrizable Kac-Moody type, we verify the quantization conjecture in full generality that the quantum cluster monomials are contained in the dual canonical basis after rescaling.

Let S be the spinor representation of U_q\mathfrak{so}_N , for N odd and q^2 not a rooot of unity. We show that the commutant of its action on S^{\otimes n} is given by a representation of the nonstandard quantum group U'_{-q^2}\mathfrak{so}_n . For N even, an analogous statement also holds for S=S_+\oplus S_- the direct sum of the irreducible spinor representations of U'_q\mathfrak{so}_N , with the commutant given by U'_{-q}\mathfrak{o}_n , a \mathbb{Z}/2 -extension of U'_{-q}\mathfrak{so}_n . Similar statements also hold for fusion tensor categories with q a root of unity.

We consider the Knizhnik-Zamolodchikov and dynamical operators, both differential and difference, in the context of the ( gl k , gl n ) -duality for the space of polynomials in kn anticommuting variables. We show that the Knizhnik-Zamolodchikov and dynamical operators naturally exchange under the duality.