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Haijun Tan

Publications and source records attributed to Haijun Tan.

7 recordsLinked to original sources
arXiv2026

$\mathfrak{so}_n(\mathbb{C})$-modules which are free over an abelian nilradical

In this paper, we classify the category of $\mathfrak{so}_n(\mathbb{C})$-modules whose restrictions to the universal enveloping algebra of the abelian nilradical of a maximal parabolic subalgebra are free of rank $1$. We show that they generically are simple, and give an explicit description of submodules in the non-simple case. This category embodies a certain connection between weight modules and non-weight modules.

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arXiv2025

On the boundary Carrollian conformal algebra

We initiate the mathematical study of the boundary Carrollian conformal algebra (BCCA), an infinite-dimensional Lie algebra recently discovered in the context of Carrollian physics. The BCCA is an intriguing object from both physical and mathematical perspectives, since it is a filtered but not graded Lie algebra. In this paper, we first construct some modules for the BCCA and one of its subalgebras, which we call $\mathcal{O}$, by restriction of well-known modules of the BMS$_3$ and Witt algebras respectively. Along the way, we prove the irreducibility criteria for the so-called ``induced modules'' of the BMS$_3$ algebra (which we prefer to call massive modules to avoid ambiguity) and show that this is the same criteria for the irreducibility of the Verma modules of the BMS$_3$ algebra. Interestingly, the modules generated by the action of the BCCA on the generating vector of the massive modules are also irreducible under the same criteria. When this criteria holds, every massive module decomposes into a direct sum of two BCCA-submodules, each of which we conjecture to be indecomposable. Meanwhile, restricting Verma modules to the BCCA and $\mathcal{O}$ leads to free or ``almost free'' modules, which are not particularly interesting from a representation-theoretic viewpoint. This motivates the construction of BCCA modules intrinsically. To do this, we go through some structure theory on the BCCA to define a new basis and a decreasing filtration on the algebra, using which we construct Whittaker modules over the BCCA and the subalgebra $\mathcal{O}$ and prove criteria for their irreducibility.

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arXiv2021

Simple restricted modules over the Heisenberg-Virasoro algebra as VOA modules

In this paper, we determine all simple restricted modules over the mirror Heisenberg-Virasoro algebra ${\mathfrak{D}}$, and the twisted Heisenberg-Virasoro algebra $\bar\mathfrak{D}$ with nonzero level. As applications, we characterize simple Whittaker modules and simple highest weight modules over ${\mathfrak{D}}$. A vertex-algebraic interpretation of our result is the classification of simple weak twisted and untwisted modules over the Heisenberg-Virasoro vertex operator algebras $\mathcal V^{c} \cong V_{Vir}^{c}\otimes M(1)$. We also present a few examples of simple restricted ${\mathfrak{D}}$-modules and $\bar\mathfrak{D}$-modules induced from simple modules over finite dimensional solvable Lie algebras, that are not tensor product modules of Virasoro modules and Heisenberg modules. This is very different from the case of simple highest weight modules over $\mathfrak{D}$ and $\bar\mathfrak{D}$ which are always tensor products of simple Virasoro modules and simple Heisenberg modules.

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arXiv2014

$\W_n^+$- and $W_n$-module structures on $U(h)$

Let $\h_n$ be the Cartan subalgebra of the Witt algebras $\W_n^+=\text{Der}\C[t_1, t_2, ..., t_n]$ and $\W_n=\text{Der}\C[t_1^{\pm 1},t_2^{\pm 1},\cdots,t_n^{\pm1}]$ where $1\le n\le \infty$. In this paper, we classify the modules over $\W_n^+$ and over $\W_n$ which are free $U(\h_n)$-modules of rank $1$. These are the $\W_n^+$-modules $Ω(Λ_{n},a, S) $ for some $Λ_n=(λ_1,\cdots,λ_n) \in (\C^*)^n, a\in \C$, and $S\subset \{1,2,..., n\}$; and $\W_n$-modules $Ω(Λ_n,a)$ for some $Λ_n\in (\C^*)^n$ and some $a\in \C.$

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arXiv2013

Irreducible modules over Witt algebras $\mathcal{W}_n$ and over $\mathfrak{sl}_{n+1}(\mathbb{C})$

In this paper, by using the "twisting technique" we obtain a class of new modules $A_b$ over the Witt algebras $\mathcal{W}_n$ from modules $A$ over the Weyl algebras $\mathcal{K}_n$ (of Laurent polynomials) for any $b\in\mathbb{C}$. We give the necessary and sufficient conditions for $A_b$ to be irreducible, and determine the necessary and sufficient conditions for two such irreducible $\mathcal{W}_n$-modules to be isomorphic. Since $\sl_{n+1}(\mathbb{C})$ is a subalgebra of $\mathcal{W}_n$, all the above irreducible $\mathcal{W}_n$-modules $A_b$ can be considered as $\sl_{n+1}(\mathbb{C})$-modules. For a class of such $\sl_{n+1}(\mathbb{C})$-modules, denoted by $\Omega_{1-a}(\lambda_1,\lambda_2,\cdots,\lambda_n)$ where $a\in\mathbb{C}, \lambda_1,\lambda_2,\cdots,\lambda_n \in \mathbb{C}^*$, we determine the necessary and sufficient conditions for these $\sl_{n+1}(\mathbb{C})$-modules to be irreducible. If the $\sl_{n+1}(\mathbb{C})$-module $\Omega_{1-a}(\lambda_1,\lambda_2,\cdots,\lambda_n)$ is reducible, we prove that it has a unique nontrivial submodule $W_{1-a}(\lambda_1, \lambda_2,...\lambda_n)$ and the quotient module is the finite dimensional $\sl_{n+1}(\mathbb{C})$-module with highest weight $m\Lambda_n$ for some non-negative integer $m\in \Z_+$. The necessary and sufficient conditions for two $\mathfrak{sl}_{n+1}(\mathbb{C})$-modules $\Omega_{1-a}(\lambda_1,\lambda_2,\cdots,\lambda_n)$ and $W_{1-a}(\lambda_1, \lambda_2,...\lambda_n)$ to be isomorphic are also determined. The irreducible $\mathfrak{sl}_{n+1}(\mathbb{C})$-modules $\Omega_{1-a}(\lambda_1, \lambda_2,...\lambda_n)$ and $W_{1-a}(\lambda_1, \lambda_2,...\lambda_n)$ are new.

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arXiv2013

Irreducible Virasoro modules from tensor products

In this paper, we obtain a class of irreducible Virasoro modules by taking tensor products of the irreducible Virasoro modules $\Omega(\lambda,b)$ defined in [LZ], with irreducible highest weight modules $V(\theta,h)$ or with irreducible Virasoro modules Ind$_{\theta}(N)$ defined in [MZ2]. We determine the necessary and sufficient conditions for two such irreducible tensor products to be isomorphic. Then we prove that the tensor product of $\Omega(\lambda,b)$ with a classical Whittaker module is isomorphic to the module $\mathrm{Ind}_{\theta,\lambda}(\mathbb{C_\mathbf{m}})$ defined in [MW]. As a by-product we obtain the necessary and sufficient conditions for the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C_\mathbf{m}})$ to be irreducible. We also generalize the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C_\mathbf{m}})$ to $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ for any non-negative integer $ n$ and use the above results to completely determine when the modules $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are irreducible. The submodules of $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are studied and an open problem in [GLZ] is solved. Feigin-Fuchs' Theorem on singular vectors of Verma modules over the Virasoro algebra is crucial to our proofs in this paper.

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