5,841 research outputs found
On projective representations for compact quantum groups
We study actions of compact quantum groups on type I factors, which may be
interpreted as projective representations of compact quantum groups. We
generalize to this setting some of Woronowicz' results concerning Peter-Weyl
theory for compact quantum groups. The main new phenomenon is that for general
compact quantum groups (more precisely, those which are not of Kac type), not
all irreducible projective representations have to be finite-dimensional. As
applications, we consider the theory of projective representations for the
compact quantum groups associated to group von Neumann algebras of discrete
groups, and consider a certain non-trivial projective representation for
quantum SU(2).Comment: 43 page
On the confluence of lambda-calculus with conditional rewriting
The confluence of untyped \lambda-calculus with unconditional rewriting is
now well un- derstood. In this paper, we investigate the confluence of
\lambda-calculus with conditional rewriting and provide general results in two
directions. First, when conditional rules are algebraic. This extends results
of M\"uller and Dougherty for unconditional rewriting. Two cases are
considered, whether \beta-reduction is allowed or not in the evaluation of
conditions. Moreover, Dougherty's result is improved from the assumption of
strongly normalizing \beta-reduction to weakly normalizing \beta-reduction. We
also provide examples showing that outside these conditions, modularity of
confluence is difficult to achieve. Second, we go beyond the algebraic
framework and get new confluence results using a restricted notion of
orthogonality that takes advantage of the conditional part of rewrite rules
Modular properties of matrix coefficients of corepresentations of a locally compact quantum group
We give a formula for the modular operator and modular conjugation in terms
of matrix coefficients of corepresentations of a quantum group in the sense of
Kustermans and Vaes. As a consequence, the modular autmorphism group of a
unimodular quantum group can be expressed in terms of matrix coefficients. As
an application, we determine the Duflo-Moore operators for the quantum group
analogue of the normaliser of SU(1,1) in ).Comment: 22 pages. To appear in Journal of Lie Theor
Projective geometries arising from Elekes-Szab\'o problems
We generalise the Elekes-Szab\'o theorem to arbitrary arity and dimension and
characterise the complex algebraic varieties without power saving. The
characterisation involves certain algebraic subgroups of commutative algebraic
groups endowed with an extra structure arising from a skew field of
endomorphisms. We also extend the Erd\H{o}s-Szemer\'edi sum-product phenomenon
to elliptic curves. Our approach is based on Hrushovski's framework of
pseudo-finite dimensions and the abelian group configuration theorem.Comment: 48 pages. Minor improvements in presentation. To appear in ASEN
On function field Mordell-Lang: the semiabelian case and the socle theorem
We here aim to complete our model-theoretic account of the function field
Mordell-Lang conjecture, avoiding appeal to dichotomy theorems for Zariski
geometries, where we now consider the general case of semiabelian varieties.
The main result is a reduction, using model-theoretic tools, of the semiabelian
case to the abelian case.Comment: 43 pages. Some minor corrections and clarifications were made
following a referee's repor
t-structures are normal torsion theories
We characterize -structures in stable -categories as suitable
quasicategorical factorization systems. More precisely we show that a
-structure on a stable -category is
equivalent to a normal torsion theory on , i.e. to a
factorization system where both classes
satisfy the 3-for-2 cancellation property, and a certain compatibility with
pullbacks/pushouts.Comment: Minor typographical corrections from v1; 25 pages; to appear in
"Applied Categorical Structures
Enriched factorization systems
In a paper of 1974, Brian Day employed a notion of factorization system in
the context of enriched category theory, replacing the usual diagonal lifting
property with a corresponding criterion phrased in terms of hom-objects. We set
forth the basic theory of such enriched factorization systems. In particular,
we establish stability properties for enriched prefactorization systems, we
examine the relation of enriched to ordinary factorization systems, and we
provide general results for obtaining enriched factorizations by means of wide
(co)intersections. As a special case, we prove results on the existence of
enriched factorization systems involving enriched strong monomorphisms or
strong epimorphisms
Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells
Chern-Schwartz-MacPherson (CSM) classes generalize to singular and/or
noncompact varieties the classical total homology Chern class of the tangent
bundle of a smooth compact complex manifold. The theory of CSM classes has been
extended to the equivariant setting by Ohmoto. We prove that for an arbitrary
complex projective manifold , the homogenized, torus equivariant CSM class
of a constructible function is the restriction of the characteristic
cycle of via the zero section of the cotangent bundle of . This
extends to the equivariant setting results of Ginzburg and Sabbah. We
specialize to be a (generalized) flag manifold . In this case CSM
classes are determined by a Demazure-Lusztig (DL) operator. We prove a `Hecke
orthogonality' of CSM classes, determined by the DL operator and its
Poincar{\'e} adjoint. We further use the theory of holonomic
-modules to show that the characteristic cycle of a Verma
module, restricted to the zero section, gives the CSM class of the
corresponding Schubert cell. Since the Verma characteristic cycles naturally
identify with the Maulik and Okounkov's stable envelopes, we establish an
equivalence between CSM classes and stable envelopes; this reproves results of
Rim{\'a}nyi and Varchenko. As an application, we obtain a Segre type formula
for CSM classes. In the non-equivariant case this formula is manifestly
positive, showing that the expansion in the Schubert basis of the CSM class of
a Schubert cell is effective. This proves a previous conjecture by Aluffi and
Mihalcea, and it extends previous positivity results by J. Huh in the Grassmann
manifold case. Finally, we generalize all of this to partial flag manifolds
.Comment: 40 pages; main changes in v2: removed some unnecessary compactness
hypotheses; added remarks 7.2 and 9.6 explaining how orthogonality of
characteristic cycles for transversal Schubert cell stratifications leads to
orthogonality of stable envelopes and that of CSM classe
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