Mathematics

Let P J be the standard parabolic subgroup of S L n obtained by deleting a subset J of negative simple roots, and let P J = L J U J be the standard Levi decomposition. Following work of the first author, we study the quantum analogue θ: O q ( P J )→ O q ( L J )⊗ O q ( P J ) of an induced coaction and the corresponding subalgebra O q ( P J ) coθ ⊆ O q ( P J ) of coinvariants. It was shown that the smash product algebra O q ( L J )# O q ( P J ) coθ is isomorphic to O q ( P J ) . In view of this, O q ( P J ) coθ -- while it is not a Hopf algebra -- can be viewed as a quantum analogue of the coordinate ring O( U J ) . In this paper we prove that when q∈K is nonzero and not a root of unity, O q ( P J ) coθ is isomorphic to a quantum Schubert cell algebra U + q [w] associated to a parabolic element w in the Weyl group of sl(n) . An explicit presentation in terms of generators and relations is found for these quantum Schubert cells.

We introduce and define the quantum affine (m|n) -superspace (or say quantum Manin superspace) A m|n q and its dual object, the quantum Grassmann superalgebra Ω q (m|n) . Correspondingly, a quantum Weyl algebra W q (2(m|n)) of (m|n) -type is introduced as the quantum differential operators (QDO for short) algebra Diff q ( Ω q ) defined over Ω q (m|n) , which is a smash product of the quantum differential Hopf algebra D q (m|n) (isomorphic to the bosonization of the quantum Manin superspace) and the quantum Grassmann superalgebra Ω q (m|n) . An interested point of this approach here is that even though W q (2(m|n)) itself is in general no longer a Hopf algebra, so are some interesting sub-quotients existed inside. This point of view gives us one of main expected results, that is, the quantum (restricted) Grassmann superalgebra Ω q is made into the U q (g) -module (super)algebra structure, Ω q = Ω q (m|n) for q generic, or $\Omega_q(m|n, \bold 1)$ for q root of unity, and g=gl(m|n) or sl(m|n) , the general or special linear Lie superalgebra. This QDO approach provides us with explicit realization models for some simple U q (g) -modules, together with the concrete information on their dimensions. Similar results hold for the quantum dual Grassmann superalgebra Ω ! q as U q (g) -module this http URL the paper some examples of pointed Hopf algebras can arise from the QDOs, whose idea is an expansion of the spirit noted by Manin in \cite{Ma}, \& \cite{Ma1}.

In his work on crystal bases \cite{Kas}, Kashiwara introduced a certain degeneration of the quantized universal enveloping algebra of a semi-simple Lie algebra g , which he called a quantum boson algebra. In this paper, we construct Kashiwara operators associated to all positive roots and use them to define a variant of Kashiwara's quantum boson algebra. We show that a quasi-classical limit of the positive half of our variant is a Poisson algebra of the form (P≃C[ n ∗ ],{ , } P ) , where n is the positive part of g and { , } P is a Poisson bracket that has the same rank as, but is different from, the Kirillov-Kostant bracket { , } KK on n ∗ . Furthermore, we prove that, in the special case of type A , any linear combination a 1 { , } P + a 2 { , } KK , a 1 , a 2 ∈C , is again a Poisson bracket. In the general case, we establish an isomorphism of P and the Poisson algebra of regular functions on the open Bruhat cell in the flag variety. In type A , we also construct a Casimir function on the open Bruhat cell, together with its quantization, which may be thought of as an analog of the linear function on n ∗ defined by a root vector for the highest root.

For a finite-index II 1 subfactor N?�M , we prove the existence of a universal Hopf ??-algebra (or, a discrete quantum group in the analytic language) acting on M in a trace-preserving fashion and fixing N pointwise. We call this Hopf ??-algebra the quantum Galois group for the subfactor and compute it in some examples of interest, notably for arbitrary irreducible finite-index depth-two subfactors. Along the way, we prove the existence of universal acting Hopf algebras for more general structures (tensors in enriched categories), in the spirit of recent work by Agore, Gordienko and Vercruysse.

We study the quantum geometry of the fuzzy sphere defined as the angular momentum algebra [ x i , x j ]=2ı λ p ϵ ijk x k modulo setting ∑ i x 2 i to a constant, using a recently introduced 3D rotationally invariant differential structure. Metrics are given by symmetric 3×3 matrices g and we show that for each metric there is a unique quantum Levi-Civita connection with constant coefficients, with scalar curvature 1 2 (Tr( g 2 )− 1 2 Tr(g ) 2 )/det(g) . As an application, we construct Euclidean quantum gravity on the fuzzy unit sphere. We also calculate the charge 1 monopole for the 3D differential structure.

We introduce the notion of quantum N -toroidal algebras uniformly as a natural generalization of the quantum toroidal algebras as well as extended quantized GIM algebras of N -fold affinization. We show that the quantum N -toroidal algebras are quotients of the extended quantized GIM algebras of N -fold affinization, which generalizes a well-known result of Berman and Moody for Lie algebras. Moreover, we construct a level-one vertex representation of the quantum N -toroidal algebra for type A .

In this paper, we seek to prove the equality of the q -graded fermionic sums conjectured by Hatayama et al. in its full generality, by extending the results of Di Francesco and Kedem to the non-simply laced case. To this end, we will derive explicit expressions for the quantum Q -system relations, which are quantum cluster mutations that correspond to the classical Q -system relations, and write the identity of the q -graded fermionic sums as a constant term identity. As an application, we will show that these quantum Q -system relations are consistent with the short exact sequence of the Feigin-Loktev fusion product of Kirillov-Reshetikhin modules obtained by Chari and Venkatesh.

We compute the center and Azumaya locus in the simplest non-abelian examples of quantized multiplicative quiver varieties at a root of unity: quantum Weyl algebras of rank N , and quantum differential operators on the quantum group GL 2 . These examples illustrate in elementary terms much more general phenomena explored further in [Ganev-Jordan-Safronov 2019].

We initiate the study of decorated character stacks and their quantizations using the framework of stratified factorization homology. We thereby extend the construction by Fock and Goncharov of (quantum) decorated character varieties to encompass also the stacky points, in a way that is both compatible with cutting and gluing and equivariant with respect to canonical actions of the modular group of the surface. In the cases G=S L 2 ,PG L 2 we construct a system of categorical charts and flips on the quantum decorated character stacks which generalize the well--known cluster structures on the Fock--Goncharov moduli spaces.

We construct canonical strongly bicovariant differential graded algebra structures on all four flavours of cross product Hopf algebras, namely double cross products A\hookrightarrow A\bowtie H\hookleftarrow H , double cross coproducts A\twoheadleftarrow A {\blacktriangleright\!\!\blacktriangleleft} H\twoheadrightarrow H , biproducts A{\buildrel\hookrightarrow\over \twoheadleftarrow}A{\cdot\kern-.33em\triangleright\!\!\!<} B and bicrossproducts A\hookrightarrow A{\blacktriangleright\!\!\triangleleft} H\twoheadrightarrow H on the assumption that the factors have strongly bicovariant calculi \Omega(A),\Omega(H) (or a braided version \Omega(B) ). We use super versions of each of the constructions. Moreover, the latter three quantum groups all coact canonically on one of their factors and we show that this coaction is differentiable. In the case of the Drinfeld double D(A,H)=A^{\rm op}\bowtie H (where A is dually paired to H ), we show that its canonical actions on A,H are differentiable. Examples include are a canonical \Omega(\Bbb C_q[GL_2\ltimes \Bbb C^2]) for the quantum group of affine transformations of the quantum plane and \Omega(\Bbb C_\lambda[{\rm Poinc_{1,1}}]) for the bicrossproduct Poincaré quantum group in 2 dimensions. We also show that \Omega(\Bbb C_q[GL_2]) itself is uniquely determined by differentiability of the canonical coaction on the quantum plane and of the determinant subalgebra.