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arXiv2001

On families of triangular Hopf algebras

Following the ideas of our previous works math.QA/0008232 (joint with Andruskiewitsch) and math.QA/0101049, we study families of triangular Hopf algebras obtained by twisting finite supergroups by a twist lying entirely in the odd part. These families are parametrized by data (G,V,u,B), where G is a finite group, V its finite dimensional representation, u a central element of G of order 2 acting by -1 on V, and B an element of S^2V. We fix the discrete data G,V,u, and find the set of isomorphism classes of the members of the family as Hopf algebras, in terms of the continuous parameter B. This set is often infinite, which provides examples of nontrivial continuous families of triangular Hopf algebras. The lowest dimension in which such a family occurs is 32, in which case we get 3 families which are dual to the 3 families of pointed Hopf algebras of dimension 32 constructed recently by Grana. Furthermore, we show that if (S^2V)^G=0 then such continuous families are nontrivial not only up to a Hopf algebra isomorphism, but also up to twisting of the multiplication. Thus, they provide counterexamples to Masuoka's weakened Kaplansky's 10th conjecture, which claims that up to twisting, there are finitely many types of Hopf algebras in each dimension. Finally, we study the algebra structure of the duals of our families, and show they are direct sums of Clifford algebras. Since there are finitely many types of Clifford algebras in each dimension, this allows us to construct nontrivial families of rigid tensor structures on the abelian category of modules over a finite dimensional algebra, with a fixed Grothendieck ring.

math.QA↗
arXiv2003

Generalized twisted coHom objects

A generalization of the concept of twisted internal coHom object in the category of conic quantum spaces (c.f. math.QA/0112233) was outlined in math.QA/0202205. The aim of this article is to discuss in more detail this generalization.

math.QA↗
arXiv2003

On Hopf algebras of dimension pq

In this paper we contribute to the classification of Hopf algebras of dimension pq, where p,q are distinct prime numbers. More precisely, we prove that if p and q are odd primes with p<q<2p+3, then any complex Hopf algebra of dimension pq is semisimple and hence isomorphic to either a group algebra or to the dual of a group algebra by the previous works math.QA/9801129 and math.QA/9801128.

math.QA↗
arXiv2012

A quantum duality principle for subgroups and homogeneous spaces

We develop a quantum duality principle for subgroups of a Poisson group and its dual, in two formulations. Namely, in the first one we provide functorial recipes to produce quantum coisotropic subgroups in the dual Poisson group out of any quantum subgroup (in a tautological sense) of the initial Poisson group, while in the second one similar recipes are given only starting from coisotropic subgroups. In both cases this yields a Galois-type correspondence, where a quantum coisotropic subgroup is mapped to its complementary dual; moreover, in the first formulation quantum coisotropic subgroups are characterized as being the fixed points in this Galois' reciprocity. By the natural link between quantum subgroups and quantum homogeneous spaces then we argue a quantum duality principle for homogeneous spaces too, where quantum coisotropic spaces are the fixed elements in a suitable Galois' reciprocity. As an application, we provide an explicit quantization of the space of Stokes matrices with the Poisson structure given by Dubrovin and Ugaglia. The *paper is actually under revision* to fix some minor, technical issues. The geometric objects considered here (Poisson groups, subgroups and homogeneous spaces) are "global" - as opposed to "formal" - and quantizations are considered as standard - i.e.non topological - Hopf algebras over the ring of Laurent polynomials (or other non-topological rings). Instead, a "local/formal" version of this work is developed in math.QA/0412465 - which is in final form - due to appear in "Advances in Mathematics". The example of Stokes matrices mentioned above is considered in math.QA/0412465 as well.

math.QA↗
arXiv2004

A Formality Theorem for Hochschild Chains

We prove Tsygan's formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold M using the Fedosov resolutions proposed in math.QA/0307212 and the formality quasi-isomorphism for Hochschild chains of R[[y_1, ..., y_d]] proposed in paper math.QA/0010321 by Shoikhet. This result allows us to describe traces on the quantum algebra of functions on an arbitrary Poisson manifold.

math.QA↗
arXiv2005

Monomials of q and q,t-characters for non simply-laced quantum affinizations

Nakajima introduced the morphism of q,t-characters for finite dimensional representation of simply-laced quantum affine algebras : it is a t-deformation of the Frenkel-Reshetikhin's morphism of q-characters (sum of monomials in infinite variables). In math.QA/0212257 we generalized the construction of q,t-characters for non simply-laced quantum affine algebras. First in this paper we prove a conjecture of math.QA/0212257 : the monomials of q and q,t-characters of standard representations are the same in non simply-laced cases (the simply-laced cases were treated by Nakajima) and the coefficients are non negative. In particular these q,t-characters can be considered as t-deformations of q-characters. In the proof we show that for quantum affine algebras of type A, B, C and quantum toroidal algebras of type A^{(1)} the l-weight spaces of fundamental representations are of dimension 1. Eventually we show and use a generalization of results of Frenkel-Reshetikhin, -Mukhin and Nakajima : for general quantum affinizations we prove that the l-weights of a l-highest weight simple module are lower than the highest l-weight in the sense of monomials.

math.QA↗
arXiv2004

New deformations of group algebras of Coxeter groups

We define new deformations of group algebras of Coxeter groups W and of subgroups of even elements in them, by deforming the braid relations. We show that these deformations are algebraically flat iff they are formally flat, and that this happens iff the group W has no finite parabolic subgroups of rank 3. The proof uses the theory of constructible sheaves on cell complexes attached to W. We explain the connection of our deformations with the Hecke algebras of orbifolds defined by the first author in math.QA/0406499 and with generalized double affine Hecke algebras defined by the authors and A. Oblomkov in math.QA/0406480. The paper is dedicated to the memory of Walter Feit.

math.QA↗
arXiv2005

Normal Hopf subalgebras, depth two and Galois extensions

Let $S$ be the left $R$-bialgebroid of a depth two extension with centralizer $R$ as defined in math.QA/0108067. We show that the left endomorphism ring of depth two extension, not necessarily balanced, is a left $S$-Galois extension of $A^{\rm op}$. Looking to examples of depth two, we establish that a Hopf subalgebra is normal if and only if it is a Hopf-Galois extension. We find a class of examples of the alternative Hopf algebroids in math.QA/0302325. We also characterize finite weak Hopf-Galois extensions using an alternate Galois canonical mapping with several corollaries: that these are depth two and that surjectivity of the Galois mapping implies its bijectivity.

math.QA↗
arXiv2005

The Endomorphism Ring Theorem for Galois and D2 extensions

Let $S$ be the left bialgebroid $\End {}_BA_B$ over the centralizer $R$ of a right D2 algebra extension $A \| B$, which is to say that its tensor-square is isomorphic as $A$-$B$-bimodules to a direct summand of a finite direct sum of $A$ with itself. We prove that its left endomorphism algebra is a left $S$-Galois extension of $A^{\rm op}$. As a corollary, endomorphism ring theorems for D2 and Galois extensions are derived from the D2 characterization of Galois extension (cf. math.QA/0502188 and math.QA/0409589). We note the converse that a Frobenius extension satisfying a generator condition is D2 if its endomorphism algebra extension is D2.

math.QA↗
arXiv2006

Quasicoherent sheaves on complex noncommutative two-tori

We introduce the notion of a quasicoherent sheaf on a complex noncommutative two-torus $T$ as an ind-object in the category of holomorphic vector bundles on $T$. Extending the results of math.QA/0211262 and math.QA/0308136 we prove that the derived category of quasicoherent sheaves on $T$ is equivalent to the derived category of usual quasicoherent sheaves on the corresponding elliptic curve. We define the rank of a quasicoherent sheaf that can take arbitrary nonnegative real values. We study the category $\Qcoh(η_T)$ obtained by taking the quotient of the category of quasicoherent sheaves by the subcategory of objects of rank zero (called torsion sheaves). We show that projective objects of finite rank in $\Qcoh(η_T)$ are classified up to an isomorphism by their rank. We also prove that the subcategory of objects of finite rank in $\Qcoh(η_T)$ is equivalent to the category of finitely presented modules over a semihereditary algebra.

math.QA↗
arXiv2007

Quantum duality principle for coisotropic subgroups and Poisson quotients

We develop a quantum duality principle for coisotropic subgroups of a (formal) Poisson group and its dual: namely, starting from a quantum coisotropic subgroup (for a quantization of a given Poisson group) we provide functorial recipes to produce quantizations of the dual coisotropic subgroup (in the dual formal Poisson group). By the natural link between subgroups and homogeneous spaces, we argue a quantum duality principle for Poisson homogeneous spaces which are Poisson quotients, i.e. have at least one zero-dimensional symplectic leaf. Only bare results are presented, while detailed proofs can be found in math.QA/0412465, or in the reference [3]. The last section contains new, unpublished material about examples and applications, which is not included in math.QA/0412465 (nor in its printed version).

math.QA↗
arXiv2006

Kirillov--Reshetikhin modules associated to G_2

We define and study the Kirillov--Reshetikhin modules for algebras of type $G_2$. We compute the graded character of these modules and verify that they are in accordance with the conjectures in math.QA/981202 and math.QA/0102113. These results give the first complete description of families of Kirillov--Reshetikhin modules whose isotypical components have multiplicity bigger than one.

math.RT↗
arXiv2008

Smallness problem for quantum affine algebras and quiver varieties

The geometric small property (Borho-MacPherson) of projective morphisms implies a description of their singularities in terms of intersection homology. In this paper we solve the smallness problem raised by Nakajima (math.QA/0105173) for certain resolutions of quiver varieties (analogs of the Springer resolution) : for Kirillov-Reshetikhin modules of simply-laced quantum affine algebras, we characterize explicitly the Drinfeld polynomials corresponding to the small resolutions. We use an elimination theorem for monomials of Frenkel-Reshetikhin q-characters that we establish for non necessarily simply-laced quantum affine algebras. We also refine results of (math.QA/0501202) and extend the main result to general simply-laced quantum affinizations, in particular to quantum toroidal algebras (double affine quantum algebras).

math.QA↗
arXiv1998

On Function Theory in Quantum Disc: Invariant Kernels

In our earlier work math.QA/9808015 some results on integral representations of functions in quantum disc were announced. It was then shown in math.QA/9808037 that the validity of those results is related to the invariance of kernels of some integral operators. We introduce here a method which allows us to prove the invariance of the above kernels.

math.QA↗
arXiv1999

The Whittaker model of the center of the quantum group and Hecke algebras

In 1978 Kostant suggested the Whittaker model of the center of the universal enveloping algebra U(g) of a complex simple Lie algebra g. The main result is that the center of U(g) is isomorphic to a commutative subalgebra in U(b), where b is a Borel subalgebra in g. This observation is used in the theory of principal series representations of the corresponding Lie group G and in the proof of complete integrability of the quantum Toda lattice. In this thesis we generalize the Kostant's construction to quantum groups. In our construction we use quantum analogues of regular nilpotent elements defined in math.QA/9812107. We also show that the Whittaker model has a natural homological interpretation in terms of Hecke algebras introduced by the author in math.QA/9805134. The new Whittaker model is applied to the deformed quantum Toda lattice recently studied by Etingof. We give new proofs of his results which resemble the original Kostant's proofs for the quantum Toda lattice. Finally, we study the ``quasi-classical'' limit of the Whittaker model for U_h(g). Using the cross-section theorem proved in q-alg/9702016 we establish a relation between the Whittaker model and the set of conjugacy classes of regular elements in the corresponding Lie group $G$.

math.QA↗
arXiv1999

Quantum deformation of Whittaker modules and Toda lattice

In 1978 Kostant suggested the Whittaker model of the center of the universal enveloping algebra U(g) of a complex simple Lie algebra g. The main result is that the center of U(g) is isomorphic to a commutative subalgebra in U(b), where b is a Borel subalgebra in g. This observation is used in the theory of principal series representations of the corresponding Lie group G and in the proof of complete integrability of the quantum Toda lattice. In this paper we generalize the Kostant's construction to quantum groups. In our construction we use quantum analogues of regular nilpotent elements defined in math.QA/9812107. Using the Whittaker model of the center of 5the algebra U_h(g) we define quantum deformations of Whittaker modules. The new Whittaker model is also applied to the deformed quantum Toda lattice recently studied by Etingof in math.QA/9901053. We give new proofs of his results which resemble the original Kostant's proofs for the quantum Toda lattice.

math.QA↗
arXiv1998

On q-Analogues of Bounded Symmetric Domains and Dolbeault Complexes

A very well known result by Harish-Chandra claims that any Hermitian symmetric space of non-compact type admits a canonical embedding into a complex vector space $V$. The image of this embedding is a bounded symmetric domain in $V$. This work provides a construction of q-analogues of a polynomial algebra on $V$ and the differential algebra of exterior forms on $V$. A way of producing a q-analogue of the bounded function algebra in a bounded symmetric domain is described. All the constructions are illustrated by detailed calculations in the case of the simplest Hermitian symmetric space $SU(1,1)/U(1)$. The development of these ideas can be found in math.QA/9803110 and math.QA/9809038 .

q-alg↗