147 research outputs found
Differential and holomorphic differential operators on noncommutative algebras
Abstract This paper deals with sheaves of differential operators on noncommutative algebras, in a manner related to the classical theory of D-modules. The sheaves are defined by quotienting the tensor algebra of vector fields (suitably deformed by a covariant derivative). As an example we can obtain enveloping algebra like relations for Hopf algebras with differential structures which are not bicovariant. Symbols of differential operators are defined, but not studied. These sheaves are shown to be in the center of a category of bimodules with flat bimodule covariant derivatives. Also holomorphic differential operators are considered
Notes on two-parameter quantum groups, (II)
This paper is the sequel to [HP1] to study the deformed structures and
representations of two-parameter quantum groups
associated to the finite dimensional simple Lie algebras \mg. An equivalence
of the braided tensor categories \O^{r,s} and \O^{q} is explicitly
established.Comment: 21 page
The Serre spectral sequence of a noncommutative fibration for de Rham cohomology
For differential calculi on noncommutative algebras, we construct a twisted
de Rham cohomology using flat connections on modules. This has properties
similar, in some respects, to sheaf cohomology on topological spaces. We also
discuss generalised mapping properties of these theories, and relations of
these properties to corings. Using this, we give conditions for the Serre
spectral sequence to hold for a noncommutative fibration. This might be better
read as giving the definition of a fibration in noncommutative differential
geometry. We also study the multiplicative structure of such spectral
sequences. Finally we show that some noncommutative homogeneous spaces satisfy
the conditions to be such a fibration, and in the process clarify the
differential structure on these homogeneous spaces. We also give two explicit
examples of differential fibrations: these are built on the quantum Hopf
fibration with two different differential structures.Comment: LaTeX, 33 page
Fusion in the entwined category of Yetter--Drinfeld modules of a rank-1 Nichols algebra
We rederive a popular nonsemisimple fusion algebra in the braided context,
from a Nichols algebra. Together with the decomposition that we find for the
product of simple Yetter-Drinfeld modules, this strongly suggests that the
relevant Nichols algebra furnishes an equivalence with the triplet W-algebra in
the (p,1) logarithmic models of conformal field theory. For this, the category
of Yetter-Drinfeld modules is to be regarded as an \textit{entwined} category
(the one with monodromy, but not with braiding).Comment: 36 pages, amsart++, times, xy. V3: references added, an unnecessary
assumption removed, plus some minor change
Highest weight categories arising from Khovanov's diagram algebra II: Koszulity
This is the second of a series of four articles studying various
generalisations of Khovanov's diagram algebra. In this article we develop the
general theory of Khovanov's diagrammatically defined "projective functors" in
our setting. As an application, we give a direct proof of the fact that the
quasi-hereditary covers of generalised Khovanov algebras are Koszul.Comment: Minor changes, extra sections on Kostant modules and rigidity of cell
modules adde
Automorphisms of associative algebras and noncommutative geometry
A class of differential calculi is explored which is determined by a set of
automorphisms of the underlying associative algebra. Several examples are
presented. In particular, differential calculi on the quantum plane, the
-deformed plane and the quantum group GLpq(2) are recovered in this way.
Geometric structures like metrics and compatible linear connections are
introduced.Comment: 28 pages, some references added, several amendments of minor
importance, remark on modular group in section 8 omitted, to appear in J.
Phys.
The Quantum Affine Origin of the AdS/CFT Secret Symmetry
We find a new quantum affine symmetry of the S-matrix of the one-dimensional
Hubbard chain. We show that this symmetry originates from the quantum affine
superalgebra U_q(gl(2|2)), and in the rational limit exactly reproduces the
secret symmetry of the AdS/CFT worldsheet S-matrix.Comment: 22 page
Secret Symmetries in AdS/CFT
We discuss special quantum group (secret) symmetries of the integrable system
associated to the AdS/CFT correspondence. These symmetries have by now been
observed in a variety of forms, including the spectral problem, the boundary
scattering problem, n-point amplitudes, the pure-spinor formulation and quantum
affine deformations.Comment: 20 pages, pdfLaTeX; Submitted to the Proceedings of the Nordita
program `Exact Results in Gauge-String Dualities'; Based on the talk
presented by A.T., Nordita, 15 February 201
Structure of the string R-matrix
By requiring invariance directly under the Yangian symmetry, we rederive
Beisert's quantum R-matrix, in a form that carries explicit dependence on the
representation labels, the braiding factors, and the spectral parameters u_i.
In this way, we demonstrate that there exist a rewriting of its entries, such
that the dependence on the spectral parameters is purely of difference form.
Namely, the latter enter only in the combination u_1-u_2, as indicated by the
shift automorphism of the Yangian. When recasted in this fashion, the entries
exhibit a cleaner structure, which allows to spot new interesting relations
among them. This permits to package them into a practical tensorial expression,
where the non-diagonal entries are taken care by explicit combinations of
symmetry algebra generators.Comment: 9 pages, LaTeX; typos correcte
Classification of bicovariant differential calculi on the Jordanian quantum groups GL_{g,h}(2) and SL_{h}(2) and quantum Lie algebras
We classify all 4-dimensional first order bicovariant calculi on the
Jordanian quantum group GL_{h,g}(2) and all 3-dimensional first order
bicovariant calculi on the Jordanian quantum group SL_{h}(2). In both cases we
assume that the bicovariant bimodules are generated as left modules by the
differentials of the quantum group generators. It is found that there are 3
1-parameter families of 4-dimensional bicovariant first order calculi on
GL_{h,g}(2) and that there is a single, unique, 3-dimensional bicovariant
calculus on SL_{h}(2). This 3-dimensional calculus may be obtained through a
classical-like reduction from any one of the three families of 4-dimensional
calculi on GL_{h,g}(2). Details of the higher order calculi and also the
quantum Lie algebras are presented for all calculi. The quantum Lie algebra
obtained from the bicovariant calculus on SL_{h}(2) is shown to be isomorphic
to the quantum Lie algebra we obtain as an ad-submodule within the Jordanian
universal enveloping algebra U_{h}(sl(2)) and also through a consideration of
the decomposition of the tensor product of two copies of the deformed adjoint
module. We also obtain the quantum Killing form for this quantum Lie algebra.Comment: 33 pages, AMSLaTeX, misleading remark remove
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