1,721,003 research outputs found
The global quantum duality principle: theory, examples, and applications
Full text linkLet 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)
PBW theorems and Frobenius structures for quantum matrices
Get PDFLet 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
Poisson Geometrical Symmetries Associated to Non-Commutative Formal Diffeomorphisms
Full text linkLet G^dif be the group of all formal power series starting with x with coefficients in a field k of zero characteristic (with the composition product), and let F[G^dif] be its function algebra. In [BF] a non-commutative, non-cocommutative graded Hopf algebra H^dif was introduced via a direct process of ‘‘disabelianisation’’ of F[G^dif], taking the like presentation of the latter as an algebra but dropping the commutativity constraint. In this paper we apply a general method to provide four one-parameter deformations of H^dif, which are quantum groups whose semiclassical limits are Poisson geometrical symmetries such as Poisson groups or Lie bialgebras, namely two quantum function algebras and two quantum universal enveloping algebras. In particular the two Poisson groups are extensions of G^dif, isomorphic as proalgebraic Poisson varieties but not as proalgebraic groups
Geometrical Meaning of R-matrix action for Quantum Groups at Roots of 1
Full text linkThe present work splits in two parts: first, we perform a straightforward generalization of results from [Reshetikhin, N. "Quasitriangularity of quantum groups at
roots of 1", Commun. Math. Phys. 170 (1995), 79-99], proving that quantum groups U_q^M(g) and their unrestricted specializations at roots of 1, in particular the function algebra F[H] of the Poisson group H dual to G, are braided. Second, as a main contribution, we prove the convergence of the (specialized) R-matrix action to a birational automorphism of a 2\ell-fold ramified covering of {Spec(U_\varepsilon^M(g))}^{\times 2} when
\varepsilon is a primitive \ell-th root of 1, and of a 2-fold ramified covering of H, thus giving a geometric content to the notion of triangularity (or braiding) for quantum groups at roots of 1
A Global Version of the Quantum Duality Principle
Full text linkThe “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
Algebraic supergroups of Cartan type
Full text linkI present a construction of connected affine algebraic supergroups G_V associated with simple Lie superalgebras g of Cartan type and with g-modules V. Conversely, I prove that every connected affine algebraic supergroup whose tangent Lie superalgebra is of Cartan type
is necessarily isomorphic to one of the supergroups G_V that I introduced. In particular, the supergroup associated in this way with g = W(n) and its standard representation is described
The crystal duality principle: from Hopf algebras to geometrical symmetries
Full text linkWe give functorial recipes to get, out of any Hopf algebra over a field, two pairs of Hopf algebras with some geometrical content. If the ground field has characteristic zero, the first pair is made by a function algebra F[G_+] over a connected Poisson group and a universal enveloping algebra U(g_−) over a Lie bialgebra g_− . In addition, the Poisson group as a variety is an affine space, and the Lie bialgebra as a Lie algebra is graded. Forgetting these last details, the second pair is of the same type, namely (F[K_+],U(k_−)) for some Poisson group K_+ and some Lie bialgebra k_− . When the Hopf algebra H we start from is already of geometric type the result involves Poisson duality. The first Lie bialgebra associated to H = F[G] is g∗ - with g := Lie(G) - and the first Poisson group associated to H = U(g) is of type G∗ , i.e. it has g as cotangent Lie bialgebra. If the ground field has positive characteristic, the same recipes give similar results, but the Poisson groups obtained have dimension 0 and height 1, and restricted universal enveloping algebras are obtained. We show how these geometrical Hopf algebras are linked to the initial one via 1-parameter deformations, and explain how these results follow from quantum group theory. We examine in detail the case of group algebras
On the radical of Brauer algebras
Full text linkThe radical of the Brauer algebra B_f(x) is known to be non-trivial when the parameter x is an integer subject to certain conditions (with respect to f). In these cases, we display a wide family of elements in the radical, which are explicitly described by means of the diagrams of the usual basis of B_f(x). The proof is by direct approach for
x=0, and via classical Invariant Theory in the other cases, exploiting then the well-known representation of Brauer algebras as centralizer algebras of orthogonal or symplectic groups acting on tensor powers of their standard representation. This also gives a great part of the radical of the generic indecomposable B_f(x)-modules.
We conjecture that this part is indeed the whole radical in the case of modules, and it is the whole part in a suitable step of the standard filtration in the case of the algebra. As an application, we find some
more precise results for the module of pointed chord diagrams, and for the Temperley-Lieb algebra - realised inside B_f(1) - acting on it
Chevalley supergroups of type D(2,1;a)
Full text linkWe present a construction “a` la Chevalley” of connected affine supergroups associated with Lie superalgebras of type D(2,1;a), for any possible value of the parameter a. This
extends the results in [R. Fioresi, F. Gavarini, Chevalley Supergroups, Memoirs AMS 215 (2012), no. 1014] (and [R. Fioresi, F. Gavarini, On the construction of Chevalley supergroups, Supersymmetry in Mathematics & Physics (UCLA Los Angeles, U.S.A. 2010), 101–123, Lecture Notes in Mathematics
2027, Springer-Verlag, Berlin-Heidelberg, 2011]) where all other simple Lie superalgebras of classical type were considered. The case of simple Lie superalgebras of Cartan type being dealt with in [F. Gavarini, Algebraic supergroups of Cartan type, Forum Mathematicum (to appear), 92
pages], so this work completes the program of constructing connected affine supergroups associated with any simple Lie superalgebra
The crystal duality principle: from general symmetries to geometrical symmetries
Full text linkWe give functorial recipes to get, out of any Hopf algebra over a field, two pairs of Hopf algebras with some geometrical content. If the ground field has characteristic zero, the first pair is made by a function algebra F[G_+] over a connected Poisson group and a universal enveloping algebra U(g_−) over a Lie bialgebra g_− . In addition, the Poisson group as a variety is an affine space, and the Lie bialgebra as a Lie algebra is graded. Forgetting these last details, the second pair is of the same type, namely (F[K_+],U(k_−)) for some Poisson group K_+ and some Lie bialgebra k_− . When the Hopf algebra H we start from is already of geometric type the result involves Poisson duality. The first Lie bialgebra associated to H = F[G] is g∗ - with g := Lie(G) - and the first Poisson group associated to H = U(g) is of type G∗ , i.e. it has g as cotangent Lie bialgebra. If the ground field has positive characteristic, the same recipes give similar results, but the Poisson groups obtained have dimension 0 and height 1, and restricted universal enveloping algebras are obtained. We show how these geometrical Hopf algebras are linked to the initial one via 1-parameter deformations, and explain how these results follow from quantum group theory. We examine in detail the case of group algebras
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