14 research outputs found
A 4-sphere with non central radius and its instanton sheaf
We build an SU(2)-Hopf bundle over a quantum toric four-sphere whose radius
is non central. The construction is carried out using local methods in terms of
sheaves of Hopf-Galois extensions. The associated instanton bundle is presented
and endowed with a connection with anti-selfdual curvature.Comment: minor changes, appendix section extended. To appear in Letters in
Mathematical Physics. 22 pages, no figure
Infinitesimal 2-braidings and differential crossed modules
We categorify the notion of an infinitesimal braiding in a linear strict
symmetric monoidal category, leading to the notion of a (strict) infinitesimal
2-braiding in a linear symmetric strict monoidal 2-category. We describe the
associated categorification of the 4-term relation, leading to six categorified
relations. We prove that any infinitesimal 2-braiding gives rise to a flat and
fake flat 2-connection in the configuration space of particles in the
complex plane, hence to a categorification of the Knizhnik-Zamolodchikov
connection. We discuss infinitesimal 2-braidings in a 2-category naturally
assigned to every differential crossed module, leading to the notion of a
quasi-invariant tensor in a differential crossed module. Finally we prove that
quasi-invariant tensors exist in the differential crossed module associated to
the String Lie-2-algebra.Comment: v3 - the introduction has been expanded, overall improvements in the
presentation. Final version, to appear in Adv. Mat
Categorifying the Knizhnik-Zamolodchikov Connection via an Infinitesimal 2-Yang-Baxter Operator in the String Lie-2-Algebra
We construct a flat (and fake-flat) 2-connection in the configuration space
of indistinguishable particles in the complex plane, which categorifies the
-Knizhnik-Zamolodchikov connection obtained from the adjoint
representation of . This will be done by considering the adjoint
categorical representation of the string Lie 2-algebra and the notion of an
infinitesimal 2-Yang-Baxter operator in a differential crossed module.
Specifically, we find an infinitesimal 2-Yang-Baxter operator in the string Lie
2-algebra, proving that any (strict) categorical representation of the string
Lie-2-algebra, in a chain-complex of vector spaces, yields a flat and (fake
flat) 2-connection in the configuration space, categorifying the
-Knizhnik-Zamolodchikov connection. We will give very detailed
explanation of all concepts involved, in particular discussing the relevant
theory of 2-connections and their two dimensional holonomy, in the specific
case of 2-groups derived from chain complexes of vector spaces.Comment: The main result was considerably sharpened. Title, abstract and
introduction updated. 50 page
The quantum Cartan algebra associated to a bicovariant differential calculus
We associate to any (suitable) bicovariant differential calculus on a quantum
group a Cartan Hopf algebra which has a left, respectively right,
representation in terms of left, respectively right, Cartan calculus operators.
The example of the Hopf algebra associated to the differential calculus
on is described.Comment: 20 pages, no figures. Minor corrections in the example in Section 4
Connected components of compact matrix quantum groups and finiteness conditions
We introduce the notion of identity component of a compact quantum group and
that of total disconnectedness. As a drawback of the generalized Burnside
problem, we note that totally disconnected compact matrix quantum groups may
fail to be profinite. We consider the problem of approximating the identity
component as well as the maximal normal (in the sense of Wang) connected
subgroup by introducing canonical, but possibly transfinite, sequences of
subgroups. These sequences have a trivial behaviour in the classical case. We
give examples, arising as free products, where the identity component is not
normal and the associated sequence has length 1.
We give necessary and sufficient conditions for normality of the identity
component and finiteness or profiniteness of the quantum component group. Among
them, we introduce an ascending chain condition on the representation ring,
called Lie property, which characterizes Lie groups in the commutative case and
reduces to group Noetherianity of the dual in the cocommutative case. It is
weaker than ring Noetherianity but ensures existence of a generating
representation. The Lie property and ring Noetherianity are inherited by
quotient quantum groups. We show that A_u(F) is not of Lie type. We discuss an
example arising from the compact real form of U_q(sl_2) for q<0.Comment: 43 pages. Changes in the introduction. The relation between our and
Wang's notions of central subgroup has been clarifie
Instantons and vortices on noncommutative toric varieties
We elaborate on the quantization of toric varieties by combining techniques from toric geometry, isospectral deformations and noncommutative geometry in braided monoidal categories, and the construction of instantons thereon by combining methods from non- commutative algebraic geometry and a quantized twistor theory. We classify the real structures on a toric noncommutative deformation of the Klein quadric and use this to derive a new noncommutative four-sphere which is the unique deformation compat- ible with the noncommutative twistor correspondence. We extend the computation of equivariant instanton partition functions to noncommutative gauge theories with both adjoint and fundamental matter fields, finding agreement with the classical results in all instances. We construct moduli spaces of noncommutative vortices from the moduli of invariant instantons, and derive corresponding equivariant partition functions which also agree with those of the classical limit
Connected components of compact matrix quantum groups and finiteness conditions
We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of approximating the identity component as well as the maximal normal connected subgroup by introducing
canonical, transfinite, sequences of subgroups, which have a trivial behaviour
in the classical case. We give examples, arising as free products, where the identity component is not normal, in the sense of Wang, and the associated sequence has length .
We give necessary and sufficient conditions for normality of the identity component and finiteness or profiniteness of the quantum component group. Among them, we introduce an ascending chain condition on the representation ring, called Lie property, which characterizes Lie groups in the commutative case and reduces to group Noetherianity of the dual in the cocommutative case. It is weaker than ring Noetherianity but ensures existence of a generating representation. The Lie property and ring Noetherianity are inherited by quotient quantum groups. We show that is not of Lie type. We discuss an example arising from the compact real form of for