169,833 research outputs found
A Global Quantum duality Principle for Subgroups and Homogeneous Spaces
No full textFor a complex or real algebraic group G, with g := Lie(G), quantizations of global type are suitable Hopf algebras F_q[G] or U_q(g) over C[q,q^{−1}]. Any such quantization yields a structure of
Poisson group on G, and one of Lie bialgebra on g: correspondingly, one has dual Poisson groups G^∗ and a dual Lie bialgebra g^∗. In this context, we introduce suitable notions of quantum subgroup and, correspondingly, of quantum homogeneous space, in three versions: weak, proper and strict (also called flat in the literature). The last
two notions only apply to those subgroups which are coisotropic, and
those homogeneous spaces which are Poisson quotients; the first one instead has no restrictions whatsoever.
The global quantum duality principle (GQDP), as developed in [F. Gavarini, The global quantum duality principle, Journ. fur die Reine Angew. Math. 612 (2007), 17–33.], associates with any global quantization of G, or of g, a global quantization of g^∗, or of G^∗. In this paper we present a similar GQDP for quantum subgroups or quantum
homogeneous spaces. Roughly speaking, this associates with every quantum subgroup, resp. quantum homogeneous space, of G, a quantum homogeneous space, resp. a quantum subgroup, of G^∗. The construction is tailored after four parallel paths — according to the
different ways one has to algebraically describe a subgroup or a homogeneous
space — and is “functorial”, in a natural sense.
Remarkably enough, the output of the constructions are always quantizations
of proper type. More precisely, the output is related to the input as follows: the former is the coisotropic dual of the coisotropic interior of the latter — a fact that extends the occurrence of Poisson duality in the original GQDP for quantum groups. Finally, when the input is a strict quantization then the output is strict as well — so the special role of strict quantizations is respected.
We end the paper with some explicit examples of application of our recipes
The global quantum duality principle: theory, examples, and applications
No full textLet R be an integral domain, h non-zero in R such that R/hR is a field, and HA the category of torsionless (or flat) Hopf algebras over R. We call any H in HA "quantized function algebra" (=QFA), resp. "quantized (restricted) universal enveloping algebra" (=QrUEA), at h if H/hH is the function algebra of a connected Poisson group, resp. the (restricted, if R/hR has positive characteristic) universal enveloping algebra of a (restricted) Lie bialgebra.
We establish an "inner" Galois' correspondence on HA, via the definition of two endofunctors, ( )^\vee and ( )', of HA such that:
(a) the image of ( )^\vee, resp. of ( )', is the full subcategory of all QrUEAs, resp. all QFAs, at h; (b) if R/hR has zero characteristic, the restriction of ( )^\vee to QFAs and of ( )' to QrUEAs yield equivalences inverse to each other; (c) if R/hR has zero characteristic, starting from a QFA over a Poisson group, resp. from a QrUEA over a (restricted) Lie bialgebra, the functor ( )^\vee, resp. ( )', gives a QrUEA, resp. a QFA, over the dual Lie bialgebra, resp. the dual Poisson group.
In particular, (a) yields a recipe to produce quantum groups of both types (QFAs or QrUEAs), (b) gives a characterization of them within HA, and (c) gives a "global" version of the "quantum duality principle" after Drinfeld. We then apply our result to Hopf algebras defined over a field k and extended to the polynomial ring k[h]: this yields quantum groups, hence "classical" geometrical symmetries of Poisson type (via specialization) associated to the "generalized symmetry" encoded by the original Hopf algebra over k. Both the main result and the above mentioned application are illustrated via several examples of many different kinds, which are studied in full detail. WARNING: this paper is *NOT meant for publication*! The results presented here are (or will be) published in separate articles; therefore, any reader willing to quote anything from the present paper is kindly invited to ask the author for the precise reference(s)
A PBW basis for lusztig's form of untwisted affine quantum groups
No full textLet \hat{g} be an untwisted affine Kac-Moody algebra over the field C, and let U_q(\hat{g}) be the associated quantum enveloping algebra; let \frak{U}_q(\hat{g}) be the Lusztig’s integer form of U_q(\hat{g}), generated by q-divided powers of Chevalley generators over a suitable subring R of C(q). We prove a Poincaré-Birkhoff-Witt like theorem for \frak{U}_q(\hat{g}), yielding a basis over R made of ordered products of q-divided powers of suitable quantum root vectors
PBW theorems and Frobenius structures for quantum matrices
No full textLet G one of Mat_n(C), GL_n(C) or SL_n(C)}, let O_q(G) be the quantum function algebra -
over Z[q,q^{-1}] - associated to G, and let
O_e(G) be the specialisation of the latter at a root of unity \varepsilon, whose order \ell is odd. There is a quantum Frobenius morphism that embeds O(G), the function algebra of G, in O_e(G) as a central Hopf subalgebra, so that O_e(G) is a module over O(G). When G = SL_n(C), it is known by works of Brown, Gordon, and of Brown, Gordon and Stafford, that (the complexification of) such a module is free, with rank \ell^{dim(G)}. In this note I prove a PBW-like theorem for O_q(G), and I show that - when G Mat_n or GL_n - it yields explicit bases of O_e(G) over O(G). As a direct application, I prove that O_e(GL_n) and O_e(Mat_n) are free Frobenius extensions
over O(GL_n) and O(Mat_n), thus extending some results of Brown, Gordon and Stroppel
Vulnerabilità sismica degli edifici in c.a. - Nota 1: indagini parametriche sulla resistenza alle azioni laterali di telai piani in c.a. tamponati
No full textVulnerabilità sismica degli edifici in c.a. - Nota 2: Portam: un software per la valutazione della resistenza alle azioni laterali di telai spaziali in c.a. tamponati
No full textThe global quantum duality principle
No full textLet R be an integral domain, let h in R be anon-zero element such that k := R/hR is a field, and let \HA be the category of torsionless (or flat) Hopf algebras over R. We call an object H in \HA a "quantized function algebra" (in short, a QFA), resp. "quantized restricted universal enveloping
algebra" (in short, a QrUEA), at h if — roughly speaking — the quotient H/hH is the function algebra of a connected Poisson group, resp. the (restricted, if R/hR 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, ( )^\vee and ( )', of \HA such that H^\vee is a QrUEA and H' is a QFA (for all H in \HA). In addition: (a) the image of ( )^\vee, resp. of ( )', is the full subcategory of all QrUEAs, resp. of all QFAs; (b)
if p := Char(k) = 0, the restrictions of ( )^\vee to QFAs and of ( )' to QrUEA 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 ( )^\vee, 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 [Ga2]–[Ga4]
Sui parametri relativi alla definizione dei terremoti di progetto negli studi di pericolosità sismica
No full textThe global quantum duality principle: a survey through examples
No full textLet R be a 1-dimensional integral domain, let h (non-zero) be a prime element, and let \HA be the category of torsionless Hopf algebras over R. We call H in \HA a "quantized function algebra" (=QFA), resp. "quantized restricted universal enveloping algebras" (=QrUEA), at h if H/hH is the function algebra of a connected Poisson group, resp. the (restricted, if R/hR has positive characteristic) universal enveloping algebra of a (restricted) Lie bialgebra. An "inner" Galois correspondence on \HA is established via the definition of two endofunctors, ( )^\vee and ( )', of \HA such that: (a) the image of ( )^\vee, resp. of ( )', is the full subcategory of all QrUEAs, resp. QFAs, at h; (b) if p := Char(R/hR) = 0, the restrictions of ( )^\vee to QFAs and of ( )' to QrUEAs yield equivalences inverse to each other; (c) if p = 0, then starting from a QFA over a Poisson group G, resp. from a QrUEA over a Lie bialgebra g, the functor ( )^\vee, resp. ( )', gives a QrUEA, resp. a QFA, over the dual Lie bialgebra, resp. a dual Poisson group. In particular, (a) provides a machine to produce quantum groups of both types (either QFAs or QrUEAs), (b) gives a characterization of them among objects of \HA, and (c) gives a "global" version of the so-called "quantum duality principle" (after Drinfeld's, cf. [Dr]).
These notes draw a sketch of the construction leading to the "global quantum duality principle". Besides, the principle itself, and in particular the above mentioned application, is illustrated by means of several examples. There are no proofs, but all (of them, and any other detail) can be found instead in arXiv:math/0303019v8 [math.QA]
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