Year: 2007
Type: article
Abstract: Let R be an integral domain, let ħ ∈ R \{0} be such that 𝕂 := R /ħ R is a field, and let ℋ𝒜 be the category of torsionless (or flat) Hopf algebras over R . We call H ∈ ℋ𝒜 a ‘ quantized function algebra ’ (= QFA), resp. ‘ quantized restricted universal enveloping algebra ’ (= QrUEA), at ħ if—roughly speaking— H /ħ H is the function algebra of a connected Poisson group, resp. the (restricted, if R /ħ R has positive characteristic) universal enveloping algebra of a (restricted) Lie bialgebra. Extending a result of Drinfeld, we establish an ‘inner’ Galois' correspondence on ℋ𝒜, via two endofunctors, ( ) ∨ and ( )′, of ℋ𝒜 such that H ∨ is a QrUEA and H ′ is a QFA (for all H ∈ ℋ𝒜). In addition: (a) the image of ( ) ∨ , resp. of ( )′, is the full subcategory of all QrUEAs, resp. of all QFAs; (b) if p := Char(𝕂) = 0, the restrictions ( ) ∨ | QFAs and ( )′| QrUEAs yield equivalences inverse to each other; (c) if p = 0, starting from a QFA over a Poisson group G , resp. from a QrUEA over a Lie bialgebra 𝔤, the functor ( ) ∨ , resp. ( )′, gives a QrUEA, resp. a QFA, over the dual Lie bialgebra, resp. the dual Poisson group. Several, far-reaching applications are developed in detail in [ F. Gavarini , The global quantum duality principle: theory, examples, and applications, preprint 2003, http://arxiv.org/abs/math.QA/0303019] and [ F. Gavarini , The Crystal Duality Principle: from Hopf Algebras to Geometrical Symmetries, J. Algebra 285 (2005), 399–437] and [ F. Gavarini , Poisson geometrical symmetries associated to non-commutative formal diffeomorphisms, Commun. Math. Phys. 253 (2005), 121–155].
Author: Fabio Gavarini
Institution: University of Rome Tor Vergata
Language: German
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2283126564