Irreducible modules over the universal central extension of the planar Galilean conformal algebra
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912425711763456 |
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| author | Gao, Dongfang |
| author_facet | Gao, Dongfang |
| contents | In this paper, we study the representation theory of the universal central extension $\mathcal{G}$ of the infinite-dimensional Galilean conformal algebra, introduced by Bagchi-Gopakumar, in $(2+1)$ dimensional space-time, which was named the planar Galilean conformal algebra by Aizawa. More precisely, we construct a family of Whittaker modules $W_{ψ_{m,n}}$ over $\mathcal{G}$ while the necessary and sufficient conditions for these modules to be irreducible are given when $m\in\mathbb{Z}_+, n\in\mathbb{N}$ and $m,n$ have the same parity. Moreover, the irreducible criteria of the tensor product modules $Ω(λ,η,σ,0)\otimes R, Ω(λ,η,0,σ)\otimes R$ over $\mathcal{G}$ are obtained, where $Ω(λ,η,σ,0), Ω(λ,η,0,σ)$ are $\mathcal{U}(\mathfrak{h})$-free modules of rank one over $\mathcal{G}$ and $R$ is an irreducible restricted module over $\mathcal{G}$. Also, the isomorphism classes of these tensor product modules are determined. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_10196 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Irreducible modules over the universal central extension of the planar Galilean conformal algebra Gao, Dongfang Representation Theory Mathematical Physics 17B10, 17B65, 17B66, 17B68, 17B70 In this paper, we study the representation theory of the universal central extension $\mathcal{G}$ of the infinite-dimensional Galilean conformal algebra, introduced by Bagchi-Gopakumar, in $(2+1)$ dimensional space-time, which was named the planar Galilean conformal algebra by Aizawa. More precisely, we construct a family of Whittaker modules $W_{ψ_{m,n}}$ over $\mathcal{G}$ while the necessary and sufficient conditions for these modules to be irreducible are given when $m\in\mathbb{Z}_+, n\in\mathbb{N}$ and $m,n$ have the same parity. Moreover, the irreducible criteria of the tensor product modules $Ω(λ,η,σ,0)\otimes R, Ω(λ,η,0,σ)\otimes R$ over $\mathcal{G}$ are obtained, where $Ω(λ,η,σ,0), Ω(λ,η,0,σ)$ are $\mathcal{U}(\mathfrak{h})$-free modules of rank one over $\mathcal{G}$ and $R$ is an irreducible restricted module over $\mathcal{G}$. Also, the isomorphism classes of these tensor product modules are determined. |
| title | Irreducible modules over the universal central extension of the planar Galilean conformal algebra |
| topic | Representation Theory Mathematical Physics 17B10, 17B65, 17B66, 17B68, 17B70 |
| url | https://arxiv.org/abs/2506.10196 |