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Algebraic quantum groupoids have been developed by two of the authors of this note (AVD and SHW) in a series of papers [A. Van Daele and S. Wang, Weak multiplier Hopf algebras. Preliminaries, motivation and basic examples, Banach Center Publ. 98 (2012) 367–415; A. Van Daele and S. Wang, Weak multiplier Hopf algebras I. The main theory, J. Reine Angew. Math. 705 (2015) 155–209; A. Van Daele and S. Wang, Weak multiplier Hopf algebras II. The source and target algebras, preprint (2014), arXiv: 1403.7906v2 [math.RA]; A. Van Daele and S. Wang, Weak multiplier Hopf algebras III. Integrals and duality, preprint (2017), arXiv: 1701.04951 [math.RA]], see also [A. Van Daele, Algebraic quantum groupoids. An example, preprint (2017), arXiv: 1702.04903 [math.RA]]. By an algebraic quantum groupoid, we understand a regular weak multiplier Hopf algebra with enough integrals. Regular multiplier Hopf algebroids are obtained also by two authors of this note (TT and AVD) in [T. Timmermann and A. Van Daele, Regular multiplier Hopf algebroids. Basic theory and examples, Commun. Algebra 46 (2017) 1926–1958]. Integral theory and duality for those have been studied by one author here (TT) in [T. Timmermann, Integration on algebraic quantum groupoids, Int. J. Math. 27 (2016) 1650014, arXiv: 1507.00660 [QA]; T. Timmermann, On duality of algebraic quantum groupoids, Adv. Math. 309 (2017) 692–746, arXiv: 1605.06384 [math.QA]]. In these papers, the term algebraic quantum groupoid is used for a regular multiplier Hopf algebroid with a single faithful integral. Finally, again two authors of us (TT and AVD) have investigated the relation between weak multiplier Hopf algebras and multiplier Hopf algebroids in [T. Timmermann and A. Van Daele, Multiplier Hopf algebroids arising from weak multiplier Hopf algebras, Banach Center Publ. 106 (2015) 73–110]. In the paper, Weak multiplier Hopf algebras III. Integrals and duality [A. Van Daele and S. Wang, Weak multiplier Hopf algebras III. Integrals and duality, preprint (2017), arXiv: 1701.04951 [math.RA]], one of the main results is that the dual of an algebraic quantum groupoid, admits a dual of the same type. In the paper, On duality of algebraic quantum groupoids [T. Timmermann, On duality of algebraic quantum groupoids, Adv. Math. 309 (2017) 692–746, arXiv: 1605.06384 [math.QA]], a result of the same nature is obtained for regular multiplier Hopf algebroids with a single faithful integral. The duality of regular weak multiplier Hopf algebras with a single integral can be obtained from the duality of regular multiplier Hopf algebroids (see [T. Timmermann, On duality of algebraic quantum groupoids, Adv. Math. 309 (2017) 692–746, arXiv: 1605.06384 [math.QA]]). That is however not the obvious way to obtain this result. It is more difficult and less natural than the direct way followed in [A. Van Daele and S. Wang, Weak multiplier Hopf algebras III. Integrals and duality, preprint (2017), arXiv: 1701.04951 [math.RA]]. We will discuss this statement further in the paper. Nevertheless, it is interesting to investigate the relation between the two approaches to duality in greater detail. This is what we do in this paper. We build further on the intimate relation between weak multiplier Hopf algebras and multiplier Hopf algebroids as studied in [T. Timmermann and A. Van Daele, Multiplier Hopf algebroids arising from weak multiplier Hopf algebras, Banach Center Publ. 106 (2015) 73–110]. We now add the presence of integrals. That seems to be done best in a framework of dual pairs. It is in fact more general than the duality of these objects coming with integrals. We are convinced that the material we present in this paper will provide a deeper understanding of the duality of algebraic quantum groupoids, both within the framework of weak multiplier Hopf algebras, as well as more generally for multiplier Hopf algebroids. Finally, we feel it is also appropriate to include some historical comments on the development of these duality theories. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying 2-local derivations on the planar Galilean conformal algebra. We prove that every 2-local derivation on the planar Galilean conformal algebra is a derivation. [ABSTRACT FROM AUTHOR]

In this paper, we generalize some results shown by Das and Peterson. Let M be a II 1 -factor acting on L 2 (M). We consider certain unital normal completely positive maps on B (L 2 (M)) which are identity on M. We investigate their fixed point spaces and obtain a rigidity result. As an application, we show some results of subfactors. [ABSTRACT FROM AUTHOR]

The skein algebra of an oriented three-manifold is a classical limit of the Kauffman bracket skein module and gives the coordinate ring of the S L 2 (ℂ) -character variety. In this paper, we determine the quotient of a polynomial ring which is isomorphic to the skein algebra of a group with three generators and two relators. As an application, we give an explicit formula for the skein algebra of the Borromean rings complement in S 3 . [ABSTRACT FROM AUTHOR]

In this paper, we consider the infinitesimal freeness in the operator-valued framework, and we show that the operator-valued infinitesimal (OVI) free independence is equivalent to the operator-valued free independence over an algebra of 2 × 2 upper triangular matrices. We introduce the notion of OVI cumulants and investigate its properties, and we then deduce that the OVI freeness is equivalent to the vanishing of our mixed cumulants. Moreover, we derive the formula for obtaining the free additive and multiplicative convolutions within the realm of OVI freeness. [ABSTRACT FROM AUTHOR]

We investigate the deformations of pairs (X , L) , where L is a line bundle on a smooth projective variety X , defined over an algebraically closed field of characteristic 0. In particular, we prove that the DG-Lie algebra controlling the deformations of the pair (X , L) is homotopy abelian whenever X has trivial canonical bundle, and so these deformations are unobstructed. [ABSTRACT FROM AUTHOR]

We develop a notion of a non-commutative hull for a left ideal of the L 1 -algebra of a compact quantum group . A notion of non-commutative spectral synthesis for compact quantum groups is proposed as well. It is shown that a certain Ditkin's property at infinity (which includes those where the dual quantum group ̂ has the approximation property) is equivalent to every hull having synthesis. We use this work to extend recent work of White that characterizes the weak * closed ideals of a measure algebra of a compact group to those of the measure algebra of a coamenable compact quantum group. In the sequel, we use this work to study bounded right approximate identities of certain left ideals of L 1 () in relation to coamenability of . [ABSTRACT FROM AUTHOR]

Starting with a vertex-weighted pointed graph (Γ , μ , v 0) , we form the free loop algebra S 0 defined in Hartglass–Penneys' article on canonical C ∗ -algebras associated to a planar algebra. Under mild conditions, 0 is a non-nuclear simple C ∗ -algebra with unique tracial state. There is a canonical polynomial subalgebra A ⊂ 0 together with a Dirac number operator N such that (A , L 2 A , N) is a spectral triple. We prove the Haagerup-type bound of Ozawa–Rieffel to verify ( 0 , A , N) yields a compact quantum metric space in the sense of Rieffel. We give a weighted analog of Benjamini–Schramm convergence for vertex-weighted pointed graphs. As our C ∗ -algebras are non-nuclear, we adjust the Lip-norm coming from N to utilize the finite dimensional filtration of A. We then prove that convergence of vertex-weighted pointed graphs leads to quantum Gromov–Hausdorff convergence of the associated adjusted compact quantum metric spaces. As an application, we apply our construction to the Guionnet–Jones–Shyakhtenko (GJS) C ∗ -algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS C ∗ -algebras of many infinite families of planar algebras converge in quantum Gromov–Hausdorff distance. [ABSTRACT FROM AUTHOR]

For a polynomial W with an isolated singularity, we can consider the Jacobian ring as an invariant of the singularity. If in addition we have a group action on the polynomial ring with W fixed, we are led to consider the twisted Jacobian ring which reflects the equivariant structure as well. Our main result is to show that the twisted Jacobian ring is isomorphic to an endomorphism ring of the "twisted diagonal"matrix factorization. As an application, we suggest a way to investigate Floer theory of Lagrangian submanifolds which represent homological mirror functors. [ABSTRACT FROM AUTHOR]

Both the basic cohomology groups and the reduced cohomology groups of the Schrödinger–Virasoro conformal algebra with trivial coefficients are completely determined. [ABSTRACT FROM AUTHOR]

We describe the subfactor planar algebra of an intermediate subfactor N ⊂ Q ⊂ M of an extremal subfactor N ⊂ M of finite Jones index which is not necessarily irreducible. [ABSTRACT FROM AUTHOR]

We consider the C ∗ -algebra ℰ n , m q , which is a q -twist of two Cuntz–Toeplitz algebras. For the case | q | < 1 , we give an explicit formula which untwists the q -deformation showing that the isomorphism class of ℰ n , m q does not depend on q. For the case | q | = 1 , we give an explicit description of all ideals in ℰ n , m q . In particular, we show that ℰ n , m q contains a unique largest ideal ℳ q . We identify ℰ n , m q / ℳ q with the Rieffel deformation of n ⊗ m and use a K-theoretical argument to show that the isomorphism class does not depend on q. The latter result holds true in a more general setting of multiparameter deformations. [ABSTRACT FROM AUTHOR]

We study Shimura curves of PEL type in the space of polarized abelian varieties A p δ generically contained in the ramified Prym locus. We generalize to ramified double covers, the construction done in [E. Colombo, P. Frediani, A. Ghigi and M. Penegini, Shimura curves in the Prym locus, Commun. Contemp. Math.21(2) (2019) 1850009] in the unramified case and in the case of two ramification points. Namely, we construct families of double covers which are compatible with a fixed group action on the base curve. We only consider the case of one-dimensional families and where the quotient of the base curve by the group is ℙ 1 . Using computer algebra we obtain 184 Shimura curves contained in the (ramified) Prym loci. [ABSTRACT FROM AUTHOR]

An admissible Poisson algebra (or briefly, an adm-Poisson algebra) gives an equivalent presentation with only one operation for a Poisson algebra. We establish a bialgebra theory for adm-Poisson algebras independently and systematically, including but beyond the corresponding results on Poisson bialgebras given in [27]. Explicitly, we introduce the notion of adm-Poisson bialgebras which are equivalent to Manin triples of adm-Poisson algebras as well as Poisson bialgebras. The direct correspondence between adm-Poisson bialgebras with one comultiplication and Poisson bialgebras with one cocommutative and one anti-cocommutative comultiplications generalizes and illustrates the polarization–depolarization process in the context of bialgebras. The study of a special class of adm-Poisson bialgebras which include the known coboundary Poisson bialgebras in [27] as a proper subclass in general, illustrating an advantage in terms of the presentation with one operation, leads to the introduction of adm-Poisson Yang–Baxter equation in an adm-Poisson algebra. It is an unexpected consequence that both the adm-Poisson Yang–Baxter equation and the associative Yang–Baxter equation have the same form and thus it motivates and simplifies the involved study from the study of the associative Yang–Baxter equation, which is another advantage in terms of the presentation with one operation. A skew-symmetric solution of adm-Poisson Yang–Baxter equation gives an adm-Poisson bialgebra. Finally, the notions of an -operator of an adm-Poisson algebra and a pre-adm-Poisson algebra are introduced to construct skew-symmetric solutions of adm-Poisson Yang–Baxter equation and hence adm-Poisson bialgebras. Note that a pre-adm-Poisson algebra gives an equivalent presentation for a pre-Poisson algebra introduced by Aguiar. [ABSTRACT FROM AUTHOR]

Extending a result of the first author and Katsura, we prove that for every UHF algebra A of infinite type, in every uncountable cardinality κ there are 2 κ nonisomorphic approximately matricial C*-algebras with the same K 0 group as A. These algebras are group C*-algebras "twisted" by prescribed canonical commutation relations (CCR), and they can also be considered as nonseparable generalizations of noncommutative tori. [ABSTRACT FROM AUTHOR]

We discuss the structure of the Motzkin algebra M k (D) by introducing a sequence of idempotents and the basic construction. We show that ⋃ k ≥ 1 M k (D) admits a factor trace if and only if D ∈ { 2 cos (π / n) + 1 | n ≥ 3 } ∪ [ 3 , ∞) and the higher commutants of these factors depend on D. Then a family of irreducible bimodules over these factors is constructed. A tensor category with A n fusion rule is obtained from these bimodules. [ABSTRACT FROM AUTHOR]