Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1512.06978

Loop Schrödinger-Virasoro Lie conformal algebra

Haibo Chen·Jianzhi Han·Yucai Su·Ying Xu
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we introduce two kinds of Lie conformal algebras associated with the loop Schrödinger-Virasoro Lie algebra and the extended loop Schrödinger-Virasoro Lie algebra, respectively. The conformal derivations, the second cohomology groups of these two conformal algebras are completely determined. And nontrivial free conformal modules of rank one and Z-graded free intermediate series modules over these two conformal algebras are also classified in the present paper

Explore related subjects

math.QA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Haibo Chen, Jianzhi Han, Yucai Su, Ying Xu. 2015-12-22. Loop Schrödinger-Virasoro Lie conformal algebra. https://arxiv.org/abs/1512.06978

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Nodal degeneration of chiral algebras II: Local structure and chiral Zhu algebras

Given a universal factorization algebra $\mathcal{A}$, we constructed in our previous paper a derived associative algebra $\mathfrak{Z}_{\mathcal{A}}^0$, together with chiral modules $\\mathfrak{Z}_{\mathcal{A}}^+$ and $\mathfrak{Z}_{\mathcal{A}}^-$ associated to a puncture, and a chiral bimodule $\mathfrak{Z}_{\mathcal{A}}$ associated to a node. Furthermore, these constructions assemble to a factorization $\mathcal{A}$-module over any family of nodal punctured curves. In this paper, we show that in the case where $\mathcal{A}$ is constructed from a quasi-conformal vertex algebra $V$, the zeroth homology algebra $H^0\mathfrak{Z}_{\mathcal{A}}$ is naturally isomorphic to Zhu's associative algebra $A(V)$, and we identify $H^0\mathfrak{Z}_{\mathcal{A}}$ with the bimodule underlying the mode-transition algebra of Damiolini-Gibney-Krashen. We also give an explicit description of the smoothing module $H^0\tilde{\mathfrak{Z}}_{\mathcal{C}}$ which describes the deformation of $H^0\mathfrak{Z}_{\mathcal{A}}$ which we attach to a smoothing family of a nodal curve. We therefore get a geometric interpretation of the Zhu algebra and the mode-transition algebra, as the integration of a factorization algebra over a certain compactification of configuration spaces of punctured nodal curves.

math.QA↗
arXiv2026

Braided Hopf algebroids and Lie algebroids

We construct braided Hopf algebroids in a braided monoidal category over a field $k$ and study its properties using graphical representation. Then we study Ehresmann-Schauenburg Hopf algebroids assocaited to braided Hopf Galois extensions. We introduce braided Lie-Rinehart algebras which generalize the definition in \cite{ALP24, ALP23, ALP25} and study its universal enveloping algebra under symmetric condition. In particular, we introduce Lie-Rinehart algebras (braided Lie-algebroids) associated with braided Hopf algebroids in the category of the module of a triangular Hopf algebra. We then focus on the case for braided Ehresmann-Schauenburg Hopf algebroids and study their right invariant vector fields, which turn out to be isomorphic to the $H$-equivariant vector fields on the quantum principal bundle (Hopf Galois extension) as Lie-Rinehart algebras. Finally, we introduce braided jet Hopf algebroids associate to braided Hopf algebroids in the case that the source and target subalgebra belong to the braided center. Moreover, we show there is a dual pairing between the $k$-order universal enveloping algebra of the braided right invariant vector fields and the $k$-th jet space of a braided Hopf algebroid and further more a skew pairing between the universal enveloping algebra of its braided right invariant vector fields and its jet Hopf algebroid under a finiteness condition.

math.QA↗
arXiv2026

Affine quantum Schur--Weyl duality

Let $\mathpzc K$ be an arbitrary commutative ring containing an invertible element $\varepsilon$. Let ${\mathcal H}_{\!\vartriangle\!}(r)_{\mathpzc K}$ be the extended affine Hecke algebra of type $A$ with Hecke parameter $\varepsilon$, let $Ω_{\mathpzc K}^{\otimes r}$ be the affine tensor space, and let ${\mathcal S}_{\!\vartriangle\!}(n,r)_{\mathpzc K}$ be the corresponding affine quantum Schur algebra. We first prove that the natural right action of ${\mathcal H}_{\!\vartriangle\!}(r)_{\mathpzc K}$ on $Ω_{\mathpzc K}^{\otimes r}$ is always faithful. Assume further that $\mathpzc K$ is a field of characteristic $0$ and that $\varepsilon$ is not a root of unity. We prove that, for any $n\geq 2$, the natural algebra homomorphism $ξ_r:{\mathcal H}_{\!\vartriangle\!}(r)_{\mathpzc K}\rightarrow\operatorname{End}_{{\mathcal S}_{\!\vartriangle\!}(n,r)_{\mathpzc K}}(Ω_{\mathpzc K}^{\otimes r})^{\mathrm{op}}$ is an isomorphism. This proves Conjecture~3.8.8 of \cite{DDF}. As an application, we prove the conjecture formulated in \cite[5.2.4]{DDF} concerning the center of the affine quantum Schur algebra. We also prove that ${\mathcal S}_{\!\vartriangle\!}(n,r)_{\mathpzc K}$ is left and right Noetherian whenever $\mathpzc K$ is a Noetherian commutative ring, which verify a conjecture in \cite[Rem. 1.7]{DY}.

math.QA↗