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Haisheng Li

Publications and source records attributed to Haisheng Li.

At least 55 records · Page 3Linked to original sources
arXiv2014

$q$-Virasoro algebra and vertex algebras

In this paper, we study a certain deformation $D$ of the Virasoro algebra that was introduced and called $q$-Virasoro algebra by Nigro,in the context of vertex algebras. Among the main results, we prove that for any complex number $\ell$, the category of restricted $D$-modules of level $\ell$ is canonically isomorphic to the category of quasi modules for a certain vertex algebra of affine type. We also prove that the category of restricted $D$-modules of level $\ell$ is canonically isomorphic to the category of $\mathbb{Z}$-equivariant $ϕ$-coordinated quasi modules for the same vertex algebra. In the process, we introduce and employ a certain infinite dimensional Lie algebra which is defined in terms of generators and relations and then identified explicitly with a subalgebra of $\mathfrak{gl}_{\infty}$.

math.QA↗
arXiv2014

$ϕ_ε$-coordinated modules for vertex algebras

We study $ϕ_ε$-coordinated modules for vertex algebras, where $ϕ_ε$ with $ε$ an integer parameter is a family of associates of the one-dimensional additive formal group. As the main results, we obtain a Jacobi type identity and a commutator formula for $ϕ_ε$-coordinated modules. We then use these results to study $ϕ_ε$-coordinated modules for vertex algebras associated to Novikov algebras by Primc.

math.QA↗
arXiv2013

On a category of $gl_{\infty}$-modules

We study a particular category ${\cal{C}}$ of $\gl_{\infty}$-modules and a subcategory ${\cal{C}}_{int}$ of integrable $\gl_{\infty}$-modules. As the main results, we classify the irreducible modules in these two categories and we show that every module in category ${\cal{C}}_{int}$ is semi-simple. Furthermore, we determine the decomposition of the tensor products of irreducible modules in category ${\cal{C}}_{int}$.

math.QA↗
arXiv2013

On quasi modules at infinity for vertex algebras

A theory of quasi modules at infinity for (weak) quantum vertex algebras including vertex algebras was previously developed in \cite{li-infinity}. In this current paper, quasi modules at infinity for vertex algebras are revisited. Among the main results, we extend some technical results, to fill in a gap in the proof of a theorem therein, and we obtain a commutator formula for general quasi modules at infinity and establish a version of the converse of the aforementioned theorem.

math.QA↗
arXiv2013

Associating quantum vertex algebras to Lie algebra $\gl_{\infty}$

In this paper, we present a canonical association of quantum vertex algebras and their $ϕ$-coordinated modules to Lie algebra $\gl_{\infty}$ and its 1-dimensional central extension. To this end we construct and make use of another closely related infinite-dimensional Lie algebra.

math.QA↗
arXiv2012

$G$-covariant $ϕ$-coordinated quasi modules for quantum vertex algebras

This is a paper in a series to study quantum vertex algebras and their relations with various quantum algebras. In this paper, we introduce a notion of T-type quantum vertex algebra and a notion of $G$-covariant $ϕ$-coordinated quasi module for a $T$-type quantum vertex algebra with an automorphism group $G$. We refine and extend several previous results and we obtain a commutator formula for $G$-covariant $ϕ$-coordinated quasi modules. As an illustrating example, we study a special case of the deformed Virasoro algebra $\V_{p,q}$ with $q=-1$, to which we associate a Clifford vertex superalgebra and its $G$-covariant $ϕ$-coordinated quasi modules.

math.QA↗
arXiv2012

On vertex Leibniz algebras

In this paper, we study a notion of what we call vertex Leibniz algebra. This notion naturally extends that of vertex algebra without vacuum, which was previously introduced by Huang and Lepowsky. We show that every vertex algebra without vacuum can be naturally extended to a vertex algebra. On the other hand, we show that a vertex Leibniz algebra can be embedded into a vertex algebra if and only if it admits a faithful module. To each vertex Leibniz algebra we associate a vertex algebra without vacuum which is universal to the forgetful functor. Furthermore, from any Leibniz algebra $\g$ we construct a vertex Leibniz algebra $V_{\g}$ and show that $V_{\g}$ can be embedded into a vertex algebra if and only if $\g$ is a Lie algebra.

math.QA↗
arXiv2012

Toroidal vertex algebras and their modules

We develop a theory of toroidal vertex algebras and their modules, and we give a conceptual construction of toroidal vertex algebras and their modules. As an application, we associate toroidal vertex algebras and their modules to toroidal Lie algebras.

math.QA↗
arXiv2011

Twisted tensor products of nonlocal vertex algebras

In this paper we introduce and study a twisted tensor product construction of nonlocal vertex algebras. Among the main results, we establish a universal property and give a characterization of a twisted tensor product. Furthermore, we give a construction of modules for a twisted tensor product. We also show that smash products studied by one of us before can be realized as twisted tensor products.

math.QA↗
arXiv2010

Vertex $F$-algebras and their $ϕ$-coordinated modules

In this paper, for every one-dimensional formal group $F$ we formulate and study a notion of vertex $F$-algebra and a notion of $ϕ$-coordinated module for a vertex $F$-algebra where $ϕ$ is what we call an associate of $F$. In the case that $F$ is the additive formal group, vertex $F$-algebras are exactly ordinary vertex algebras. We give a canonical isomorphism between the category of vertex $F$-algebras and the category of ordinary vertex algebras. Meanwhile, for every formal group we completely determine its associates. We also study $ϕ$-coordinated modules for a general vertex $\Z$-graded algebra $V$ with $ϕ$ specialized to a particular associate of the additive formal group and we give a canonical connection between $V$-modules and $ϕ$-coordinate modules for a vertex algebra obtained from $V$ by Zhu's change-of-variables theorem.

math.QA↗
arXiv2010

Quantum vertex algebras and their phi-coordinated quasi modules

We develop a theory of $ϕ$-coordinated (quasi) modules for a nonlocal vertex algebra and we establish a conceptual construction of nonlocal vertex algebras and their $ϕ$-coordinated (quasi) modules, where $ϕ$ is what we call an associate of the one-dimensional additive formal group. By specializing $ϕ$ to a particular associate, we obtain a new construction of weak quantum vertex algebras in the sense of \cite{li-qva1}. As an application, we associate weak quantum vertex algebras to quantum affine algebras, and we also associate quantum vertex algebras and $ϕ$-coordinated modules to a certain quantum $βγ$-system.

math.QA↗
arXiv2010

Quantum vertex $F((t))$-algebras and their modules

This is a paper in a series to study vertex algebra-like structures arising from various algebras including quantum affine algebras and Yangians. In this paper, we develop a theory of what we call (weak) quantum vertex $\F((t))$-algebras with $\F$ a field of characteristic zero and $t$ a formal variable, and we give a conceptual construction of (weak) quantum vertex $\F((t))$-algebras and their modules. As an application, we associate weak quantum vertex $\F((t))$-algebras to quantum affine algebras, providing a solution to a problem posed by Frenkel and Jing. We also explicitly construct an example of quantum vertex $\F((t))$-algebras from a certain quantum $βγ$-system.

math.QA↗
arXiv2010

Twisted modules and pseudo-endomorphisms

We exhibit a connection between two constructions of twisted modules for a general vertex operator algebra with respect to inner automorphisms. We also study pseudo-derivations, pseudo-endomorphisms, and twist deformations of ordinary modules by pseudo-endomorphisms, which are intrinsically connected to one of the two constructions.

math.QA↗
arXiv2010

$\hbar$-adic quantum vertex algebras and their modules

This is a paper in a series to study vertex algebra-like structures arising from various algebras including quantum affine algebras and Yangians. In this paper, we study notions of $\hbar$-adic nonlocal vertex algebra and $\hbar$-adic (weak) quantum vertex algebra, slightly generalizing Etingof-Kazhdan's notion of quantum vertex operator algebra. For any topologically free $\C[[\h]]$-module $W$, we study $\hbar$-adically compatible subsets and $\hbar$-adically $§$-local subsets of $(\End W)[[x,x^{-1}]]$. We prove that any $\hbar$-adically compatible subset generates an $\hbar$-adic nonlocal vertex algebra with $W$ as a module and that any $\hbar$-adically $§$-local subset generates an $\hbar$-adic weak quantum vertex algebra with $W$ as a module. A general construction theorem of $\hbar$-adic nonlocal vertex algebras and $\hbar$-adic quantum vertex algebras is obtained. As an application we associate the centrally extended double Yangian of $\sl_{2}$ to $\hbar$-adic quantum vertex algebras.

math.QA↗
arXiv2009

On quantum vertex algebras and their modules

We give a survey on the developments in a certain theory of quantum vertex algebras, including a conceptual construction of quantum vertex algebras and their modules and a connection of double Yangians and Zamolodchikov-Faddeev algebras with quantum vertex algebras.

math.QA↗