Search

This is a paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In three previous papers, we introduce the notion of formal manifolds and study their basic theory, focusing on function spaces and Poincare's lemma. In this paper, we further explore the foundational framework of formal manifolds, including the local structure of constant rank morphisms (such as inverse function theorem and constant rank theorems) as well as the theory of formal submanifolds.

This is a paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In two previous papers, we develop the basic theory of formal manifolds, including generalizations of vector-valued distributions and generalized functions on smooth manifolds to the setting of formal manifolds. In this paper, we establish Poincar\'e's lemma for de Rham complexes with coefficients in formal functions, formal generalized functions, compactly supported formal densities, or compactly supported formal distributions., Comment: The preprint arXiv:2401.01535v1 of ours was split into three separate papers. This is the third paper about de Rham complexes and Poincare's Lemma

This is a paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In a previous paper, we introduce the notion of formal manifolds and develop the foundational framework of formal manifolds. In this paper, we study various function spaces on formal manifolds, including generalizations of vector-valued generalized functions and vector-valued distributions on smooth manifolds to the setting of formal manifolds., Comment: The preprint arXiv:2401.01535v1 of ours was split into three separate papers. This is the second paper about vector-valued generalized functions and vector-valued distributions

In this paper, we associate the quantum toroidal algebra $\mathcal{E}_N$ of type $\mathfrak{gl}_N$ with quantum vertex algebra through equivariant $\phi$-coordinated quasi modules. More precisely, for every $\ell\in \mathbb{C}$, by deforming the universal affine vertex algebra of $\mathfrak{sl}_\infty$, we construct an $\hbar$-adic quantum $\Z$-vertex algebra $V_{\widehat{\mathfrak{sl}}_{\infty},\hbar}(\ell,0)$. Then we prove that the category of restricted $\mathcal{E}_N$-modules of level $\ell$ is canonically isomorphic to that of equivariant $\phi$-coordinated quasi $V_{\widehat{\mathfrak{sl}}_{\infty},\hbar}(\ell,0)$-modules.

This is the first paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In this paper, we lay the foundations for this study by introducing the notion of formal manifolds in the context of differential geometry, inspired by the notion of formal schemes in algebraic geometry. We develop the basic theory for formal manifolds, and establish a fully faithful contravariant functor from the category of formal manifolds to the category of topological $\mathbb{C}$-algebras. We also prove the existence of finite products in the category of formal manifolds by studying vector-valued formal functions., Comment: First version of this paper was split into three separate papers. Vector-valued generalized functions and vector-valued distributions were separated into "Function spaces on Formal Manifolds" preprint, while de Rham complexes and Poincare's Lemma were separated into "Poincare's Lemma for formal manifolds"preprint. All the rest about the foundations of formal manifolds stayed in v3 of this preprint

In this paper, we construct and study various dual pairs acting on the oscillator modules of the symplectic toroidal Lie algebras coordinated by irrational quantum tori. This extends the classical Howe dual pairs to the toroidal setup.

In this paper, we associate the TKK algebra $\widehat{\mathcal{G}}(\mathcal{J})$ with vertex algebras through twisted modules. Firstly, we prove that for any complex number $\ell$, the category of restricted $\widehat{\mathcal{G}}(\mathcal{J})$-modules of level $\ell$ is canonically isomorphic to the category of $\sigma$-twisted $V_{\widehat{\mathcal{C}_{\mathfrak g}}}(\ell,0)$-modules, where $V_{\widehat{\mathcal{C}_{\mathfrak g}}}(\ell,0)$ is a vertex algebra arising from the toroidal Lie algebra of type $C_2$ and $\sigma$ is an isomorphism of $V_{\widehat{\mathcal{C}_{\mathfrak g}}}(\ell,0)$ induced from the involution of this toroidal Lie algebra. Secondly, we prove that for any nonnegative integer $\ell$, the integrable restricted $\widehat{\mathcal{G}}(\mathcal{J})$-modules of level $\ell$ are exactly the $\sigma$-twisted modules for the quotient vertex algebra $L_{\widehat{\mathcal{C}_{\mathfrak g}}}(\ell,0)$ of $V_{\widehat{\mathcal{C}_{\mathfrak g}}}(\ell,0)$. Finally, we classify the irreducible $\frac{1}{2}{\mathbb{N}}$-graded $\sigma$-twisted $L_{\widehat{\mathcal{C}_{\mathfrak g}}}(\ell,0)$-modules., Comment: 19 pages

In this paper, we give a unified construction of vertex algebras arising from infinite-dimensional Lie algebras, including the affine Kac-Moody algebras, Virasoro algebras, Heisenberg algebras and their higher rank analogs, orbifolds and deformations. We define a notion of what we call quasi vertex Lie algebra to unify these Lie algebras. Starting from any (maximal) quasi vertex Lie algebra $\mathfrak{g}$, we construct a corresponding vertex Lie algebra ${\mathfrak{g}}_0$, and establish a canonical isomorphism between the category of restricted $\mathfrak{g}$-modules and that of equivariant $\phi$-coordinated quasi $V_{{\mathfrak{g}}_0}$-modules, where $V_{{\mathfrak{g}}_0}$ is the universal enveloping vertex algebra of ${\mathfrak{g}}_0$. This unified all the previous constructions of vertex algebras from infinite-dimensional Lie algebras and shed light on the way to associate vertex algebras with Lie algebras., Comment: 34 pages

In this paper, for every $\epsilon\in \mathbb{Z}$, we introduce an extension of the 2-toroidal Lie algebra by certain derivations. Based on the $\phi_\epsilon$-coordinated modules theory for vertex algebras, we give an explicit realization of a class of irreducible highest weight modules for this extended toroidal Lie algebra. When $\epsilon=1$, this affords a realization of certain irreducible modules for the toroidal extended affine Lie algebras first constructed by Billig.

For any nullity $2$ extended affine Lie algebra $\mathcal{E}$ of maximal type and $\ell\in\mathbb{C}$, we prove that there exist a vertex algebra $V_{\mathcal{E}}(\ell)$ and an automorphism group $G$ of $V_{\mathcal{E}}(\ell)$ equipped with a linear character $\chi$, such that the category of restricted $\mathcal{E}$-modules of level $\ell$ is canonically isomorphic to the category of $(G,\chi)$-equivariant $\phi$-coordinated quasi $V_{\mathcal{E}}(\ell)$-modules. Moreover, when $\ell$ is a nonnegative integer, there is a quotient vertex algebra $L_{\mathcal{E}}(\ell)$ of $V_{\mathcal{E}}(\ell)$ modulo by a $G$-stable ideal, and we prove that the integrable restricted $\mathcal{E}$-modules of level $\ell$ are exactly the $(G,\chi)$-equivariant $\phi$-coordinated quasi $L_{\mathcal{E}}(\ell)$-modules., Comment: 33 pages

To give a unified treatment on the association of Lie algebras and vertex algebras, we study $(G,\chi_\phi)$-equivariant $\phi$-coordinated quasi modules for vertex algebras, where $G$ is a group with $\chi_\phi$ a linear character of $G$ and $\phi$ is an associate of the one-dimensional additive formal group. The theory of $(G,\chi_\phi)$-equivariant $\phi$-coordinated quasi modules for nonlocal vertex algebra is established in \cite{JKLT}. In this paper, we concentrate on the context of vertex algebras. We establish several conceptual results, including a generalized commutator formula and a general construction of vertex algebras and their $(G,\chi_\phi)$-equivariant $\phi$-coordinated quasi modules. Furthermore, for any conformal algebra $\mathcal{C}$, we construct a class of Lie algebras $\widehat{\mathcal{C}}_\phi[G]$ and prove that restricted $\widehat{\mathcal{C}}_\phi[G]$-modules are exactly $(G,\chi_\phi)$-equivariant $\phi$-coordinated quasi modules for the universal enveloping vertex algebra of $\mathcal{C}$. As an application, we determine the $(G,\chi_\phi)$-equivariant $\phi$-coordinated quasi modules for affine and Virasoro vertex algebras., Comment: 31 pages

In this paper, we study nullity-2 toroidal extended affine Lie algebras in the context of vertex algebras and their $\phi$-coordinated modules. Among the main results, we introduce a variant of toroidal extended affine Lie algebras, associate vertex algebras to the variant Lie algebras, and establish a canonical connection between modules for toroidal extended affine Lie algebras and $\phi$-coordinated modules for these vertex algebras. Furthermore, by employing some results of Billig, we obtain an explicit realization of irreducible modules for the variant Lie algebras.

In this paper we introduce a new quantum algebra which specializes to the $2$-toroidal Lie algebra of type $A_1$. We prove that this quantum toroidal algebra has a natural triangular decomposition, a (topological) Hopf algebra structure and a vertex operator realization.

In this paper, for an arbitrary Kac-Moody Lie algebra $\mathfrak g$ and a diagram automorphism $\mu$ of $\mathfrak g$ satisfying certain natural linking conditions, we introduce and study a $\mu$-twisted quantum affinization algebra $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$ of $\mathfrak g$. When $\mathfrak g$ is of finite type, $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$ is Drinfeld's current algebra realization of the twisted quantum affine algebra. When $\mu=\mathrm{id}$ and $\mathfrak g$ in affine type, $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$ is the quantum toroidal algebra introduced by Ginzburg, Kapranov and Vasserot. As the main results of this paper, we first prove a triangular decomposition for $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$. Second, we give a simple characterization of the affine quantum Serre relations on restricted $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$-modules in terms of "normal order products". Third, we prove that the category of restricted $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$-modules is a monoidal category and hence obtain a topological Hopf algebra structure on the "restricted completion" of $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$. Last, we study the classical limit of $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$ and abridge it to the quantization theory of extended affine Lie algebras. In particular, based on a classification result of Allison-Berman-Pianzola, we obtain the $\hbar$-deformation of all nullity $2$ extended affine Lie algebras., Comment: 66 pages. Final version

In this paper, we explore natural connections among the representations of the extended affine Lie algebra $\widehat{sl_N}(\mathbb{C}_q)$ with $\mathbb{C}_q=\mathbb{C}_q[t_0^{\pm1},t_1^{\pm1}]$ an irrational quantum 2-torus, the simple affine vertex algebra $L_{\widehat{sl_{\infty}}}(\ell,0)$ with $\ell$ a positive integer, and Levi subgroups $G$ of $GL_\ell(\mathbb{C})$. First, we give a canonical isomorphism between the category of integrable restricted $\widehat{sl_N}(\mathbb{C}_q)$-modules of level $\ell$ and that of equivariant quasi $L_{\widehat{sl_{\infty}}}(\ell,0)$-modules. Second, we classify irreducible $\mathbb{N}$-graded equivariant quasi $L_{\widehat{sl_{\infty}}}(\ell,0)$-modules. Third, we establish a duality between irreducible $\mathbb{N}$-graded equivariant quasi $L_{\widehat{sl_{\infty}}}(\ell,0)$-modules and irreducible regular $G$-modules on certain fermionic Fock spaces. Fourth, we obtain an explicit realization of every irreducible $\mathbb{N}$-graded equivariant quasi $L_{\widehat{sl_{\infty}}}(\ell,0)$-module. Fifth, we completely determine the following branchings: 1 The branching from $L_{\widehat{sl_{\infty}}}(\ell,0)\otimes L_{\widehat{sl_{\infty}}}(\ell',0)$ to $L_{\widehat{sl_{\infty}}}(\ell+\ell',0)$ for quasi modules. 2 The branching from $\widehat{sl_N}(\mathbb{C}_q)$ to its Levi subalgebras. 3 The branching from $\widehat{sl_N}(\mathbb{C}_q)$ to its subalgebras $\widehat{sl_N}(\mathbb{C}_q[t_0^{\pm M_0},t_1^{\pm M_1}])$., Comment: 54 Pages

Let $\mathfrak{g}$ be the derived subalgebra of a Kac-Moody Lie algebra of finite type or affine type, $\mu$ a diagram automorphism of $\mathfrak{g}$ and $L(\mathfrak{g},\mu)$ the loop algebra of $\mathfrak{g}$ associated to $\mu$. In this paper, by using the vertex algebra technique, we provide a general construction of current type presentations for the universal central extension $\widehat{\mathfrak{g}}[\mu]$ of $L(\mathfrak{g},\mu)$. The construction contains the classical limit of Drinfeld's new realization for (twisted and untwisted) quantum affine algebras ([Dr]) and the Moody-Rao-Yokonuma presentation for toroidal Lie algebras ([MRY]) as special examples. As an application, when $\mathfrak{g}$ is of simply-laced type, we prove that the classical limit of the $\mu$-twisted quantum affinization of the quantum Kac-Moody algebra associated to $\mathfrak{g}$ introduced in [CJKT1] is the universal enveloping algebra of $\widehat{\mathfrak{g}}[\mu]$.

Let $\fg$ be an affine Kac-Moody algebra, and $\mu$ a diagram automorphism of $\fg$. In this paper, we give an explicit realization for the universal central extension $\wh\fg[\mu]$ of the twisted loop algebra of $\fg$ related to $\mu$, which provides a Moody-Rao-Yokonuma presentation for the algebra $\wh\fg[\mu]$ when $\mu$ is non-transitive, and the presentation is indeed related to the quantization of toroidal Lie algebras.

In this paper we generalize Drinfeld's twisted quantum affine algebras to construct twisted quantum algebras for all simply-laced generalized Cartan matrices and present their vertex representation realizations., Comment: 15 pages

In this paper, we classify the irreducible integrable modules with finite dimensional weight spaces and non-trivial $\widetilde{\mathfrak g}_c$-action for the nullity $2$ toroidal extended affine Lie algebra $\widetilde{\mathfrak g}$, where $\widetilde{\mathfrak g}_c$ is the core of $\widetilde{\mathfrak g}$.

Let $\fg$ be any untwisted affine Kac-Moody algebra, $\mu$ any fixed complex number, and $\wt\fg(\mu)$ the corresponding toroidal extended affine Lie algebra of nullity two. For any $k$-tuple $\bm{\lambda}=({\lambda}_1, \cdots, {\lambda}_k)$ of weights of $\fg$, and $k$-tuple $\bm{a}=(a_1,\cdots, a_k)$ of distinct non-zero complex numbers, we construct a class of modules $\wt V(\bm{\lambda},\bm{a})$ for the extended affine Lie algebra $\wt\fg(\mu)$. We prove that the $\wt\fg(\mu)$-module $\wt V(\bm{\lambda},\bm{a})$ is completely reducible. We also prove that the $\wt\fg(\mu)$-module $\wt V(\bm{\lambda},\bm{a})$ is integrable when all weights $\lambda_i$ in $\bm{\lambda}$ are dominant integral. Thus, we obtain a new class of irreducible integrable weight modules for the toroidal extended affine Lie algebra $\wt\fg(\mu)$.

Let $\rk$ be a local field of characteristic zero. Let $\pi$ be an irreducible admissible smooth representation of $\GL_{2n}(\rk)$. We prove that for all but countably many characters $\chi$ of $\GL_n(\rk)\times \GL_n(\rk)$, the space of $\chi$-equivariant (continuous in the archimedean case) linear functionals on $\pi$ is at most one dimensional. Using this, we prove the uniqueness of twisted Shalika models.