Categorification of Virasoro-Magri Poisson vertex algebra

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• 作者：Seok-Jin Kang^a ; ¹ ; ^{sjkang@math.snu.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Uhi Rinn Suh^b ; ² ; ^{uhrisu1@math.snu.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Poisson vertex algebra ; Categorification ; Group of symmetric algebras
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 April 2016
• 年：2016
• 卷：452
• 期：Complete
• 页码：133-156
• 全文大小：430 K

文摘

Let Σ be the direct sum of algebra of symmetric groups CΣ_n$\mathbb{C}{\mathrm{\Sigma }}_n$, n∈Z_≥0$n\in {\mathbb{Z}}_{\ge 0}$. We show that the Grothendieck group K₀(Σ)$K_0(\mathrm{\Sigma })$ of the category of finite dimensional modules of Σ is isomorphic to the differential algebra of polynomials $\mathbb{Z}[{\partial }^{n}x|n\in {\mathbb{Z}}_{\ge }0]$. Moreover, we define m  -th products (m∈Z_≥0$m\in {\mathbb{Z}}_{\ge 0}$) on K₀(Σ)$K_0(\mathrm{\Sigma })$ which make the algebra K₀(Σ)$K_0(\mathrm{\Sigma })$ isomorphic to an integral form of the Virasoro–Magri Poisson vertex algebra. Also, we investigate relations between K₀(Σ)$K_0(\mathrm{\Sigma })$ and K₀(N)$K_0(N)$ where K₀(N)$K_0(N)$ is the direct sum of Grothendieck groups K₀(N_n)$K_0(N_n)$, n≥0$n\ge 0$, of finitely generated projective N_n$N_n$-modules. Here N_n$N_n$ is the nil-Coxeter algebra generated by n−1$n-1$ elements.