6,406 research outputs found
Coxeter categories and quantum groups
No full textWe define the notion of braided Coxeter category, which is informally a monoidal category carrying compatible, commuting actions of a generalised braid group B_W and Artin’s braid groups B_n on the tensor powers of its objects. The data which defines the action of B_W bears a formal similarity to the associativity constraints in a monoidal category, but is related to the coherence of a family of fiber functors. We show that the quantum Weyl group operators of a quantised Kac–Moody algebra U_{\hbar }{{\mathfrak {g}}}, together with the universal R-matrices of its Levi subalgebras, give rise to a braided Coxeter category structure on integrable, category {\mathcal {O}}-modules for U_{\hbar }{{\mathfrak {g}}}. By relying on the 2-categorical extension of Etingof–Kazhdan quantisation obtained in Appel and Toledano Laredo (Selecta Math NS 24:3529–3617, 2018), we then prove that this structure can be transferred to integrable, category {\mathcal {O}}-representations of {\mathfrak {g}}. These results are used in Appel and Toledano Laredo (arXiv:1512.03041, p 48, 2015) to give a monodromic description of the quantum Weyl group operators of U_{\hbar }{{\mathfrak {g}}}, which extends the one obtained by the second author for a semisimple Lie algebra
Identifying the main constructs for an interdisciplinary workplace management framework
No full textThis second book in the series focuses on the role of workplace management in the organization and the tasks that workplace management needs to consider. The 18 theories that are presented in this book and applied to workplace research discuss management aspects from the organization’s perspective or dive deeper into issues related to people and/or building management. They all emphasize that workplace management is a complex matter that requires more strategic attention in order to add value for various stakeholders. The final chapter of the book describes a first step towards integrating the presented theories into an interdisciplinary framework for developing a grand workplace management theory.Design Conceptualization and Communicatio
Uniqueness of Coxeter structures on Kac–Moody algebras
No full textAbstract. Let g be a symmetrisable Kac–Moody algebra, and Ung the corresponding quantum group. We showed in [1, 2] that the braided Coxeter structure on integrable, category O representations of Ung which underlies the R–matrix actions arising from the Levi subalgebras of Ung and the quantum Weyl group action of the generalised braid group Bg can be transferred to integrable, category O representations of g. We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, R–matrices, and local monodromies. This extends, simplifies and strengthens a similar result of the second author valid when g is semisimple, and is used in [3] to describe the monodromy of the rational Casimir connection of g in terms of the quantum Weyl group operators of Ung. Our main tool is a refinement of Enriquez’s universal algebras, which is adapted to the PROP describing a Lie bialgebra graded by the non–negative roots of g.<br/
De droom van Karel Appel
No full textThe dream of Karel Appel, a museum in Amsterdam. A portrait of Karel Appel in architecture.ArchitectureArchitectur
The Interplay of R-matrices and Quantum Groups
No full textThese are the expanded lecture notes of a mini-course given by the author in Milan in June 2024, during the conference GABY: Groups and Algebras in Bicocca for Young algebraists. We present a concise introduction to the theory of quantum groups, focusing on their ability to produce solutions of the quantum Yang-Baxter equation
Monodromy of the Casimir connection of a symmetrisable Kac-Moody algebra
No full textLet g be a symmetrisable Kac-Moody algebra and V an integrable g-module in
category O. We show that the monodromy of the (normally ordered) rational
Casimir connection on V can be made equivariant with respect to the Weyl group
W of g, and therefore defines an action of the braid group B_W of W on V. We
then prove that this action is canonically equivalent to the quantum Weyl group
action of B_W on a quantum deformation of V, that is an integrable, category
O-module V_h over the quantum group U_h(g) such that V_h/hV_h is isomorphic to
V. This extends a result of the second author which is valid for g semisimple.Comment: Some details added. Minor revisions throughout. Published version.
107 page
Appel à contributions - « Cahiers J.-M. G. Le Clézio »
No full textLes Cahiers J.-M. G. Le Clézio, numéro 12 : « Corps » Appel à contributions ci-joint. Date-limite : 31 octobre 2017. Coordinateurs : Justine Feyereisen (Université Libre de Bruxelles) et Paul Dirkx (Université de Lorraine) URL de référence : http://www.associationleclezio.com/activites/les-cahiers-j-m-g-le-clezio Argumentair
An explicit isomorphism between quantum and classical sl(n)
No full textLet g be a complex, semisimple Lie algebra. Drinfeld showed that the quantum group associated to g is isomorphic as an algebra to the trivial deformation of the universal enveloping algebra of g. In this paper we construct explicitly such an isomorphism when g = sl(n), previously known only for n=2
Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras
No full textLet be a complex simple Lie algebra and the
corresponding quantum affine algebra. We construct a functor between finite-dimensional modules over a quantum symmetric pair of affine
type and an orientifold KLR algebra
arising from a framed quiver with a contravariant involution, providing a
boundary analogue of Kang-Kashiwara-Kim-Oh generalized Schur-Weyl duality. With
respect to their construction, our combinatorial model is further enriched with
the poles of a trigonometric K-matrix intertwining the action of
on finite-dimensional -modules. By
construction, is naturally compatible with the
Kang-Kashiwara-Kim-Oh functor in that, while the latter is a functor of
monoidal categories, is a functor of module categories.
Relying on a suitable isomorphism \`a la Brundan-Kleshchev-Rouquier, we prove
that recovers the Schur-Weyl dualities due to
Fan-Lai-Li-Luo-Wang-Watanabe in quasi-split type .Comment: Final version. Substantial revision: new examples of enhanced
J-quivers of Dynkin type have been added in Section 6; the material of the
former Sections 6,7, and 11 from v2 has been removed and it will be included
in a separate paper. 50 page
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