The structure of the observable algebra determined by a Hopf *-subalgebra in Hopf spin models

Let H be a finite dimensional Hopf C*-algebra, H1 a Hopf*-subalgebra of H. This paper focuses on the observable algebra AH1 determined by H1 in nonequilibrium Hopf spin models, in which there is a copy of H1 on each lattice site, and a copy of ? on each link, where ? denotes the dual of H. Furthermore, using the iterated twisted tensor product of finite +*-algebras, one can prove that the observable algebraAH1 is *-isomorphic to the C*-inductive limit ... o H1 o ? o H1 o ? o H1 o ... .

On the adjoint representation of a hopf algebra

We consider the adjoint representation of a Hopf algebra $H$ focusing on the locally finite part, $H_{{\textrm ad\,fin}}$, defined as the sum of all finite-dimensional subrepresentations. For virtually cocommutative $H$ (i.e., $H$ is finitely generated as module over a cocommutative Hopf subalgebra), we show that $H_{{\textrm ad\,fin}}$ is a Hopf subalgebra of $H$. This is a consequence of the fact, proved here, that locally finite parts yield a tensor functor on the module category of any virtually pointed Hopf algebra. For general Hopf algebras, $H_{{\textrm ad\,fin}}$ is shown to be a left coideal subalgebra. We also prove a version of Dietzmann's Lemma from group theory for Hopf algebras.

BLM realization for 𝒰ℤ(𝔤𝔩 ̂n)

In 1990, Beilinson–Lusztig–MacPherson (BLM) discovered a realization for quantum [Formula: see text] via a geometric setting of quantum Schur algebras. We will generalize their result to the classical affine case. More precisely, we first use Ringel–Hall algebras to construct an integral form [Formula: see text] of [Formula: see text], where [Formula: see text] is the universal enveloping algebra of the loop algebra [Formula: see text]. We then establish the stabilization property of multiplication for the classical affine Schur algebras. This stabilization property leads to the BLM realization of [Formula: see text] and [Formula: see text]. In particular, we conclude that [Formula: see text] is a [Formula: see text]-Hopf subalgebra of [Formula: see text]. As a bonus, this method leads to an explicit [Formula: see text]-basis for [Formula: see text], and it yields explicit multiplication formulas between generators and basis elements for [Formula: see text]. As an application, we will prove that the natural algebra homomorphism from [Formula: see text] to the affine Schur algebra over [Formula: see text] is surjective.

Projectivity of Hopf algebras over subalgebras with semilocal central localizations

The purpose of this paper is to extend the class of pairs A, H where H is a Hopf algebra over a field and A a right coideal subalgebra for which H is proved to be either projective or flat as an A-module. The projectivity is obtained under the assumption that H is residually finite dimensional, A has semilocal localizations with respect to a central subring, and there exists a Hopf subalgebra B of H such that the antipode of B is bijective and B is a finitely generated A-module. The flatness of H over A is shown to hold when H is a directed union of residually finite dimensional Hopf subalgebras, and there exists a Hopf subalgebra of H whose center contains A. More general projectivity and flatness results are established for (co)equivariant modules over an H-(co)module algebra under similar assumptions.

PBW THEOREMS AND FROBENIUS STRUCTURES FOR QUANTUM MATRICES

Let $G \in \{{\it Mat}_n(\C), {GL}_n(\C), {SL}_n(\C)\}$, let $\Oqg$ be the quantum function algebra – over $\Z [q,q^{-1}]$ – associated to G, and let $\Oeg$ be the specialisation of the latter at a root of unity ϵ, whose order ℓ is odd. There is a quantum Frobenius morphism that embeds $\Og,$ the function algebra of G, in $\Oeg$ as a central Hopf subalgebra, so that $\Oeg$ is a module over $\Og$. When $G = {SL}_n(\C)$, it is known by [3], [4] that (the complexification of) such a module is free, with rank ℓdim(G). In this note we prove a PBW-like theorem for $\Oqg$, and we show that – when G is Matn or GLn – it yields explicit bases of $\Oeg $ over $ \Og$ over $\Og,$. As a direct application, we prove that $\Oegl$ and $\Oem$ are free Frobenius extensions over $\Ogl$ and $\Om$, thus extending some results of [5].

DUALITY FOR MULTIPARAMETER QUANTUM DEFORMATION OF THE SUPERGROUP GL(m/n)

We find the Hopf superalgebra [Formula: see text], which is in duality with the multiparameter quantum deformation GL u q (m/n) of the supergroup GL (m/n). Naturally [Formula: see text] is a multiparameter deformation of the superalgebra U ( gl (m/n)). We show that as a commutation algebra we have the classical structure, namely a split into two subalgebras: [Formula: see text], where [Formula: see text] is isomorphic to the standard one-parameter deformation U u( sl (m/n)), and [Formula: see text] is central in [Formula: see text] for m ≠ n. However, as a coalgebra [Formula: see text] cannot be split in this way, as only [Formula: see text] is a Hopf subalgebra, while [Formula: see text] is not a Hopf subalgebra unless m = n = 1 or some special relations between the parameters exist. These special relations are established and used to obtain explicit multiparameter Hopf superalgebra deformations of U ( sl (m/n)).