Andrew Green, Melanie Vollmar, Matthias Blum
et al.
Join us for a live panel discussion as part of our ongoing webinar series, Large Language Models and their applications in bioinformatics. This interactive QA session offers attendees the opportunity to engage with speakers from the series, ask new questions, and revisit key discussions.
Through the Erasmus+ QA-SURE project, this study examines the institutional development of internal Quality Assurance (QA) offices and systems in higher education institutions (HEIs) in Kosovo and Albania. The initiative concentrated on creating or improving specialised QA offices at three HEIs in Albania and two in Kosovo, as opposed to more general national QA system changes. The study emphasises the ways in which training, participatory governance, strategic planning, and infrastructure support aided in the development of institutional capacity, drawing on ESG 2015. Results show advancements in digitisation, student and stakeholder interaction, and quality culture. The report makes the case that system alignment with European standards and sustained academic success are made possible by institution-driven QA systems.
We study solutions of the Bethe Ansatz equations for the cyclotomic Gaudin model of (Vicedo B., Young C.A.S., arXiv:1409.6937). We give two interpretations of such solutions: as critical points of a cyclotomic master function, and as critical points with cyclotomic symmetry of a certain \extended" master function. In finite types, this yields a correspondence between the Bethe eigenvectors and eigenvalues of the cyclotomic Gaudin model and those of an \extended" non-cyclotomic Gaudin model. We proceed to define populations of solutions to the cyclotomic Bethe equations, in the sense of (Mukhin E., Varchenko A., Commun. Contemp. Math. 6 (2004), 111{163, math.QA/0209017), for dia- gram automorphisms of Kac{Moody Lie algebras. In the case of type A with the diagram automorphism, we associate to each population a vector space of quasi-polynomials with specified ramification conditions. This vector space is equipped with a Z2-gradation and a non-degenerate bilinear form which is (skew-)symmetric on the even (resp. odd) graded subspace. We show that the population of cyclotomic critical points is isomorphic to the variety of isotropic full flags in this space.
In this paper we continue to study Belavin-Drinfeld cohomology introduced in arXiv:1303.4046 [math.QA] and related to the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra. Here we compute Belavin-Drinfeld cohomology for all non-skewsymmetric $r$-matrices from the Belavin-Drinfeld list for simple Lie algebras of type $B$, $C$, and $D$.
The Poisson sigma model is a widely studied two-dimensional topological field theory. This note shows that boundary conditions for the Poisson sigma model are related to coisotropic submanifolds (a result announced in [math.QA/0309180]) and that the corresponding reduced phase space is a (possibly singular) dual pair between the reduced spaces of the given two coisotropic submanifolds. In addition the generalization to a more general tensor field is considered and it is shown that the theory produces Lagrangian evolution relations if and only if the tensor field is Poisson.
A 2-category was introduced in arXiv:0803.3652 [math.QA] that categorifies Lusztig's integral version of quantum sl(2). Here we construct for each positive integer N arepresentation of this 2-category using the equivariant cohomology of iterated flag varieties. This representation categorifies the irreducible (N+1)-dimensional representation of quantum sl(2).
We complete the calculation of the twisted cyclic homology of the quantised coordinate ring of SL(2) that we began in math.QA/0405249. In particular, a nontrivial cyclic 3-cocycle is constructed which also has a nontrivial class in Hochschild cohomology and thus should be viewed as a noncommutative geometry analogue of a volume form.
Mikhail Khovanov in math.QA/9908171 defined, for a diagram of an oriented classical link, a collection of groups numerated by pairs of integers. These groups were constructed as homology groups of certain chain complexes. The Euler characteristics of these complexes are coefficients of the Jones polynomial of the link. The goal of this note is to rewrite this construction in terms more friendly to topologists. A version of Khovanov homology for framed links is introduced. For framed links whose Kauffman brackets are involved in a skein relation, these homology groups are related by an exact sequence.
Resume We study real and complex Manin triples for a complex reductive Lie algebra g . First, we generalize results of E. Karolinsky (1996, Math. Phys. Anal. Geom3, 545–563; 1999, Preprint math.QA.9901073) on the classification of Lagrangian subalgebras. Then we show that, if g is noncommutative, one can attach to each Manin triple in g another one for a strictly smaller reductive complex Lie subalgebra of g . This gives a powerful tool for induction. Then we classify complex Manin triples in terms of what we call generalized Belavin–Drinfeld data. This generalizes, by other methods, the classification of A. Belavin and V. G. Drinfeld of certain r-matrices, i.e., the solutions of modified triangle equations for constants (cf. A. Belavin and V. G. Drinfeld, “Triangle Equations and Simple Lie Algebras,” Mathematical Physics Reviews, Vol. 4, pp. 93–165, Harwood Academic, Chur, 1984, Theorem 6.1). We get also results for real Manin triples. In passing, we retrieve a result of A. Panov (1999, Preprint math.QA.9904156) which classifies certain Lie bialgebra structures on a real simple Lie algebra.
We define and study the Kirillov--Reshetikhin modules for algebras of type $G_2$. We compute the graded character of these modules and verify that they are in accordance with the conjectures in math.QA/981202 and math.QA/0102113. These results give the first complete description of families of Kirillov--Reshetikhin modules whose isotypical components have multiplicity bigger than one.
We study the geometric aspects of two exotic bialgebras S03 and S14 introduced in math.QA/0206053. These bialgebras are obtained by the Faddeev–Reshetikhin–Takhtajan RTT prescription with non-triangular R-matrices which are denoted R03 and R14 in the classification of Hietarinta, and they are not deformations of either GL(2) or GL(1|1). We give the spectral decomposition which involves two, respectively, three, projectors. These projectors are then used to provide the baxterization procedure with one, respectively, two, parameters. Further, the projectors are used to construct the noncommutative planes together with the corresponding differentials following the Wess–Zumino prescription. In all these constructions there appear nonstandard features which are noted. Such features show the importance of systematic study of all bialgebras of four generators.
Abstract In [I. Gelfand, V. Retakh, S. Serconek, R.L. Wilson, On a class of algebras associated to directed graphs, Selecta Math. (N.S.) 11 (2005), math.QA/0506507 ] I. Gelfand and the authors of this paper introduced a new class of algebras associated to directed graphs. In this paper we show that these algebras are Koszul for a large class of layered graphs.
We define generalized double affine Hecke algebras (GDAHA) of higher rank, attached to a non-Dynkin star-like graph D. This generalizes GDAHA of rank 1 defined in math.QA/0406480 and math.QA/0409261. If the graph is extended D4, then GDAHA is the algebra defined by Sahi in q-alg/9710032, which is a generalization of the Cherednik algebra of type BCn. We prove the formal PBW theorem for GDAHA, and parametrize its irreducible representations in the case when D is affine (i.e. extended D4, E6, E7, E8) and q=1. We formulate a series of conjectures regarding algebraic properties of GDAHA. We expect that, similarly to how GDAHA of rank 1 provide quantizations of del Pezzo surfaces (as shown in math.QA/0406480), GDAHA of higher rank provide quantizations of deformations of Hilbert schemes of these surfaces. The proofs are based on the study of the rational version of GDAHA (which is closely related to the algebras studied in math.QA/0401038), and differential equations of Knizhnik-Zamolodchikov type.
Informally, a homotopy monoid is a monoid-like structure in which properties such as associativity only hold `up to homotopy' in some consistent way. This short paper comprises a rigorous definition of homotopy monoid and a brief analysis of some examples. It is a much-abbreviated version of the paper `Homotopy Algebras for Operads' (math.QA/0002180), and does not assume any knowledge of operads.