A Global Version of the Quantum Duality Principle

The “quantum duality principle” states that a quantisation of a Lie bialgebra provides also a quantisation of the dual formal Poisson group and, conversely, a quantisation of a formal Poisson group yields a quantisation of the dual Lie bialgebra as well. We extend this to a much more general result: namely, for any principal ideal domain R and for each prime p ∈ R we establish an “inner” Galois’ correspondence on the category HA of torsionless Hopf algebras over R, using two functors (from HA to itself) such that the image of the first and the second is the full subcategory of those Hopf algebras which are commutative and cocommutative, modulo p, respectively (i.e., they are “quantum function algebras” (=QFA) and “quantum universal enveloping algebras” (=QUEA), at p, respectively). In particular we provide a machine to get two quantum groups — a QFA and a QUEA — out of any Hopf algebra H over a field k: apply the functors to k[ν] ⊗k H for p = ν . A relevant example occurring in quantum electro-dynamics is studied in some detail

Quantum function algebras as quantum enveloping algebras

Inspired by a result in [Ga], we locate two $k[q,q^{-1}]$-integer forms of $F_q[SL(n+1)]$, along with a presentation by generators and relations, and prove that for $q=1$ they specialize to $U({\mathfrak{h}})$, where ${\mathfrak{h}}$ is the Lie bialgebra of the Poisson Lie group $H$ dual of $SL(n+1)$; moreover, we explain the relation with [loc. cit.]. In sight of this, we prove two PBW-like theorems for $F_q[SL(n+1)]$, both related to the classical PBW theorem for $U({\mathfrak{h}})$.Comment: 27 pages, AMS-TeX C, Version 3.0 - Author's file of the final version, as it appears in the journal printed version, BUT for a formula in Subsec. 3.5 and one in Subsec. 5.2 - six lines after (5.1) - that in this very pre(post)print have been correcte

On the global quantum duality principle

The “quantum duality principle” states that a quantisation of a Lie bialgebra provides also a quantisation of the dual formal Poisson group and, conversely, a quantisation of a formal Poisson group yields a quantisation of the dual Lie bialgebra as well. We extend this to a much more general result: namely, for any principal ideal domain R and for each prime p ∈ R we establish an “inner” Galois’ correspondence on the category HA of torsionless Hopf algebras over R, using two functors (from HA to itself) such that the image of the first and the second is the full subcategory of those Hopf algebras which are commutative and cocommutative, modulo p, respectively (i.e., they are “quantum function algebras” (=QFA) and “quantum universal enveloping algebras” (=QUEA), at p, respectively). In particular, this yields a machine to get two quantum groups — a QFA and a QUEA — out of any Hopf algebra H over a field k: just apply the functors to k[ν] ⊗k H for R := k[ν] and p := ν

Quantization of Projective Homogeneous Spaces and Duality Principle

We introduce a general recipe to construct quantum projective homogeneous spaces, with a particular interest for the examples of the quantum Grassmannians and the quantum generalized flag varieties. Using this construction, we extend the quantum duality principle to quantum projective homogeneous spaces.Comment: Final version (after correcting the journal's proofs), to appear in "Journal of Noncommutative Geometry

A formula for the logarithm of the KZ associator

We prove that the logarithm of a group-like element in a free algebra coincides with its image by a certain linear map. We use this result and the formula of Le and Murakami for the Knizhnik-Zamolodchikov (KZ) associator Φ to derive a formula for log(Φ) in terms of MZV's (multiple zeta values)

A quantum homogeneous space of nilpotent matrices

A quantum deformation of the adjoint action of the special linear group on the variety of nilpotent matrices is introduced. New non-embedded quantum homogeneous spaces are obtained related to certain maximal coadjoint orbits, and known quantum homogeneous spaces are revisited.Comment: 12 page

Test, Reliability and Functional Safety Trends for Automotive System-on-Chip

This paper encompasses three contributions by industry professionals and university researchers. The contributions describe different trends in automotive products, including both manufacturing test and run-time reliability strategies. The subjects considered in this session deal with critical factors, from optimizing the final test before shipment to market to in-field reliability during operative life

A 2-categorical extension of Etingof–Kazhdan quantisation

Let k be a field of characteristic zero. Etingof and Kazhdan constructed a quantisation U_h(b) of any Lie bialgebra b over k, which depends on the choice of an associator Phi. They prove moreover that this quantisation is functorial in b. Remarkably, the quantum group U_h(b) is endowed with a Tannakian equivalence F_b from the braided tensor category of Drinfeld-Yetter modules over b, with deformed associativity constraints given by Phi, to that of Drinfeld-Yetter modules over U_h(b). In this paper, we prove that the equivalence F_b is functorial in b.Comment: Small revisions in Sections 2 and 6. An appendix added on the equivalence between admissible Drinfeld-Yetter modules over a QUE and modules over its quantum double. To appear in Selecta Math. 71 page