It is known that symmetric orbits in ${\bf g}^*$ for any simple Lie algebra ${\bf g}$ are equiped with a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the reduced Sklyanin bracket associated to the "canonical" R-matrix. We realize quantization of this Poisson pencil on $\cp$ type orbits (i.e. orbits in $sl(n+1)^*$ whose real compact form is $ CP^n$) by means of q-deformed Verma modules.
In our previous paper, we constructed an explicit GL(n)-equivariant quantization of the Kirillov--Kostant-Souriau bracket on a semisimple coadjoint orbit. In the present paper, we realize that quantization as a subalgebra of endomorphisms of a generalized Verma module. As a corollary, we obtain an explicit description of the annihilators of generalized Verma modules over U(gl(n)). As an application, we construct real forms of the quantum orbits and classify finite dimensional representations. We compute the non-commutative Connes index for basic homogenous vector bundles over the quantum orbit…
Some natural hidden symmetries in the Verma modules over the Virasoro algebra are constructed in terms of geometric quantization. Their differential geometric meaning is established and their expression via $q_R$-conformal symmetries in the Verma modules over the Lie algebra $sl(2,C)$ is found. The analysis and the unraveling of the algebraic structure of these families of hidden symmetries are performed.
We prove the existence of a homomorphism of vertex algebras, from the vacuum Verma module over the loop algebra of an untwisted affine algebra, whose construction is analogous to that of the Feigin-Frenkel homomorphism from the vacuum Verma module at critical level over an affine algebra.
We study representations of nilpotent type nontrivial liftings of quantum linear spaces and their Drinfel'd quantum doubles. We construct a family of Verma- type modules in both cases and prove a parametrization theorem for simple modules. We compute the Loewy and socle series of Verma modules under a mild restriction on the datum of a lifting. We find bases and dimensions of simple modules.
Imaginary Verma modules, parabolic imaginary Verma modules, and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated, and several results are generalized from the affine Lie algebras. In particular, imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.
We generalize the results of [KMST] concerning equivariant quantization by means of Verma modules $M(λ)$ for generic weight $λ$ to the case of general $λ$. We consider the relationship between the Shapovalov form on an irreducible highest weight module of a semisimple complex Lie algebra, fusion elements, and equivariant quantization. We also discuss some limiting properties of fusion elements. [KMST] E. Karolinsky and A. Stolin, Dynamical Yang-Baxter equations, quasi-Poisson homogeneous spaces, and quantization, Lett. Math. Phys., 71 (2005), p.179-197; e-print math.QA/0309203.
The Drinfeld double D of the bosonization of a finite-dimensional Nichols algebra B(V) over a finite non-abelian group G is called a quantum group at a non-abelian group. We introduce Verma modules over such a quantum group D and prove that a Verma module has simple head and simple socle. This provides two bijective correspondences between the set of simple modules over D and the set of simple modules over the Drinfeld double D(G). As an example, we describe the lattice of submodules of the Verma modules over the quantum group at the symmetric group S3 attached to the 12-dimensional Fomin-Kiri…
We obtain a family of strict $\hat G$-invariant products on the space of holomorphic functions on a semisimple coadjoint orbit of a complex connected semisimple Lie group $\hat G$. By restriction, we also obtain strict $G$-invariant products $*_\hbar$ on a space $A(O)$ of certain analytic functions on a semisimple coadjoint orbit $O$ of a real connected semisimple Lie group $G$. The space $A(O)$ endowed with one of the products $*_\hbar$ is a Fréchet algebra, and the formal expansion of the products around $\hbar = 0$ determines a formal deformation quantization of $O$, which is of Wick type i…
Let M be a manifold with an action of a Lie group G, $\A$ the function algebra on M. The first problem we consider is to construct a $U_h(\g)$ invariant quantization, $\A_h$, of $\A$, where $U_h(\g)$ is a quantum group corresponding to G. Let s be a G invariant Poisson bracket on M. The second problem we consider is to construct a $U_h(\g)$ invariant two parameter (double) quantization, $\A_{t,h}$, of $\A$ such that $\A_{t,0}$ is a G invariant quantization of $s$. We call $\A_{t,h}$ a $U_h(\g)$ invariant quantization of the Poisson bracket s. In the paper we study the cases when G is a simple …
In this paper we describe a multiparameter deformation of the function algebra of a semisimple coadjoint orbit. In the first section we use the representation of the Lie algebra on a generalized Verma module to quantize the Kirillov bracket on the family of semisimple coadjoint orbits of a given orbit type. In the second section we extend this construction to define a deformation in the category of representations of the quantized enveloping algebra. In an earlier paper we used cohomological methods to prove the existence of a two parameter family quantizing a compatible pair of Poisson bracke…
We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras.
Let $\mathcal{D}$ be the Drinfeld double of the bosonization ${\mathfrak B}(V)\#\Bbbk G$ of a finite-dimensional Nichols algebra ${\mathfrak B}(V)$ over a finite group $G$. It is known that the simple $\mathcal{D}$-modules are parametrized by the simple modules over $\mathcal{D}(G)$, the Drinfeld double of $G$. This parametrization can be obtained by considering the head $\mathsf{L}(λ)$ of the Verma module $\mathsf{M}(λ)$ for every simple $\mathcal{D}(G)$-module $λ$. In the present work, we show that the projective $\mathcal{D}$-modules are filtered by Verma modules and the BGG Reciprocity $[\…
The quantum double for the quantized BRST superalgebra is studied. The corresponding R-matrix is explicitly constucted. The Hopf algebras of the double form an analytical variety with coordinates described by the canonical deformation parameters. This provides the possibility to construct the nontrivial quantization of the proper time supergroup cotangent bundle. The group-like classical limit for this quantization corresponds to the generic super Lie bialgebra of the double.
Hopf algebra quantizations of 4-dimensional and 6-dimensional real classical Drinfel'd doubles are studied by following a direct "analytic" approach. The full quantization is explicitly obtained for most of the Drinfel'd doubles, except a small number of them for which the dual Lie algebra is either sl(2) or so(3). In the latter cases, the classical r-matrices underlying the Drinfel'd double quantizations contain known standard ones plus additional twists. Several new four and six-dimensional quantum algebras are presented and some general features of the method are emphasized.
Using the quantum double method, we construct a universal R-matrix for the quantization of the Lie superalgebra D(2,1,x). We use this result to construct a link invariant and show it coincides with a specialization of Kauffman's Dubrovnik polynomial.
For $\g=sl(n)$ we construct a two parametric $U_h(\g)$-invariant family of algebras, $(S\g)_{t,h}$, which defines a quantization of the function algebra $S\g$ on the coadjoint representation and in the parameter $t$ gives a quantization of the Lie bracket. The family induces a two parametric deformation of the function algebra of any maximal orbit which is a quantization of the Kirillov-Kostant-Souriau bracket in the parameter $t$. In addition we construct a quantum de Rham complex on $\g^*$.
We continue the study of the Drinfeld double of the Jordan plane, denoted by $\mathcal D$ and introduced in arXiv:2002.02514. The simple finite-dimensional modules were computed in arXiv:2108.13849; it turns out that they factorize through $U(\spl_2(\Bbbk))$. Here we introduce the Verma modules and the category $\mathfrak O$ for $\mathcal D$, which have a resemblance to the similar ones in Lie theory but induced from indecomposable modules of the 0-part of the triangular decomposition. Accordingly, there is the notion of highest weight rank (hw-rk). We classify the indecomposable modules of hw…
We describe a natural $q$-deformation of Fock and Goncharov's canonical basis for the algebra of regular functions on a cluster variety associated to a quiver of type $A$. We then describe an extension of this construction involving a cluster variety called the symplectic double.
We show that the double quantization of Seiberg-Witten spectral curve for $Γ$-quiver gauge theory defines the generating current of W$(Γ)$-algebra in the free field realization. We also show that the partition function is given as a correlator of the corresponding W$(Γ)$-algebra, which is equivalent to the AGT relation under the gauge/quiver (spectral) duality.
It is well known that (in suitable codimension) the spaces of long knots in $\mathbb{R}^n$ modulo immersions are double loop spaces. Hence the homology carries a natural Gerstenhaber structure, given by the Gerstenhaber structure on the Hochschild homology of the $n$-Poisson operad. In this paper, we compute the latter Gerstenhaber structure in terms of hairy graphs, and show that it is not quite trivial combinatorially. Curiously, the construction makes essential use of methods and results of deformation quantization, and thus provides a bridge between two previously little related subjects i…
We study one and two parameter quantizations of the function algebra on a semisimple orbit in the coadjoint representation of a simple Lie group subject to the condition that the multiplication on the quantized algebra is invariant under action of the Drinfeld-Jimbo quantum group. We prove that the corresponding Poisson bracket must be the sum of the so-called R-matrix bracket and an invariant bracket. We classify such brackets for all semisimple orbits and show that they form a family of dimension equal to the rank equal to the second cohomology group of the orbit and then we quantize these b…
The quantization problem for the trace-bracket algebra, derived from double Poisson brackets, is discussed. We obtain a generalization of the boundary YBE (or so-called ABCD-algebra) for the quantization of quadratic trace-brackets. A dynamical deformation is proposed on the lines of Gervais-Neveu-Felder dynamical quantum algebras.
Star products on the classical double group of a simple Lie group and on corresponding symplectic grupoids are given so that the quantum double and the "quantized tangent bundle" are obtained in the deformation description. "Complex" quantum groups and bicovariant quantum Lie algebras are discused from this point of view. Further we discuss the quantization of the Poisson structure on symmetric algebra $S(g)$ leading to the quantized enveloping algebra $U_{h}(g)$ as an example of biquantization in the sense of Turaev. Description of $U_{h}(g)$ in terms of the generators of the bicovariant diff…
We construct by geometric methods a noncommutative model E of the algebra of regular functions on the universal (2-fold) cover M of certain nilpotent coadjoint orbits O for a complex simple Lie algebra g. Here O is the dense orbit in the cotangent bundle of the generalized flag variety X associated to a complexified Cartan decomposition g=(p^+)+k+(p^-) where p^+- are Jordan algebras by the TKK construction. We obtain E as the algebra of g-finite differential operators on a smooth Lagrangian subvariety in M where g is given by differential operators twisted according to a critical parameter. Af…
For any finite-dimensional Hopf algebra $A$ there exists a natural associative algebra homomorphism $D(A) \to H(A)$ between its Drinfeld double $D(A)$ and its Heisenberg double $H(A)$. We construct this homomorphism using a pair of commuting quantum moment maps, and then use it to provide a homomorphism of certain reflection equation algebras. We also explain how the quantization of the Grothendieck-Springer resolution arises in this context.
We define even dimensional quantum spheres Sigma_q^2n that generalize to higher dimension the standard quantum two-sphere of Podle's and the four-sphere Sigma_q^4 obtained in the quantization of the Hopf bundle. The construction relies on an iterated Poisson double suspension of the standard Podle's two-sphere. The Poisson spheres that we get have the same symplectic foliation consisting of a degenerate point and a symplectic plane and, after quantization, have the same C^*-algebraic completion. We investigate their K-homology and K-theory by introducing Fredholm modules and projectors.
Let $ G^τ$ be a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfeld structure of Poisson group; let $ H^τ$ be its dual Poisson group. By means of Drinfeld's double construction and dualization via formal Hopf algebras, we construct new quantum groups $ U_{q,ϕ}^M ({\frak h}) $ --- dual of $ U_{q,ϕ}^{M'} ({\frak g}) $ --- which yield infinitesimal quantization of $ H^τ$ and $ G^τ$; we study their specializations at roots of 1 (in particular, their classical limits), thus discovering new quantum Frobenius morphisms. The whole description dualize for $…
Let $ G^τ$ be a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfel'd structure of Poisson group, let $ H^τ$ be its dual Poisson group. By means of quantum double construction and dualization via formal Hopf algebras, we construct new quantum groups $ U_{q,φ}^M(\frak{h}) $ --- dual of $ U_{q,φ}^{M'}(\frak{g}) $ --- which yield infinitesimal quantization of $ H^τ$ and $ G^τ\, $, we study their specializations at roots of 1 (in particular, their classical limits), thus discovering new quantum Frobenius morphisms. The whole description dualize for $ H^…
We define generalized double affine Hecke algebras (GDAHA) of higher rank, attached to a non-Dynkin star-like graph D. This generalizes GDAHA of rank 1 defined in math.QA/0406480 and math.QA/0409261. If the graph is extended D4, then GDAHA is the algebra defined by Sahi in q-alg/9710032, which is a generalization of the Cherednik algebra of type BCn. We prove the formal PBW theorem for GDAHA, and parametrize its irreducible representations in the case when D is affine (i.e. extended D4, E6, E7, E8) and q=1. We formulate a series of conjectures regarding algebraic properties of GDAHA. We expect…
We build representations of the affine and double affine braid groups and Hecke algebras of type $C^\vee C_n$, based upon the theory of quantum symmetric pairs $(U,B)$. In the case $U=U_q(gl_N)$, our constructions provide a quantization of the representations constructed by Etingof, Freund and Ma in arXiv:0801.1530, and also a type $BC$ generalization of the results in arXiv:0805.2766.
We propose a double quantization of four-dimensional ${\cal N}=2$ Seiberg-Witten geometry, for all classical gauge groups and a wide variety of matter content. This can be understood as a set of certain non-perturbative Schwinger-Dyson identities, following the program initiated by Nekrasov [arXiv:1512.05388]. The construction relies on the computation of the instanton partition function of the gauge theory on the so-called $Ω$-background on $\mathbb{R}^4$, in the presence of half-BPS codimension 4 defects. The two quantization parameters are identified as the two parameters of this background…
In the structure theory of quantized enveloping algebras, the algebra isomorphisms determined by Lusztig led to the first general construction of PBW bases of these algebras. Also, they have important applications to the representation theory of these and related algebras. In the present paper the Drinfel'd double for a class of graded Hopf algebras is investigated. Various quantum algebras, including multiparameter quantizations of semisimple Lie algebras and of Lie superalgebras, are covered by the given definition. For these Drinfel'd doubles Lusztig maps are defined. It is shown that these…
