AI 中文总结
该研究在素特征p>2的代数闭域上，针对ADE型格顶点代数，在det G_L≢0 mod p时建立其与一阶单仿射顶点代数的同构并分类其不可约ℕ分次模，在det G_L≡0 mod p时确定其单商并分类该单商的不可约ℕ分次模。
AI 中文摘要
我们在特征p>2的代数闭域𝔽上研究ADE型格顶点代数及其表示。设L为ADE型根格，G_L为L的格拉姆矩阵。当det G_L≢0 mod p时，我们建立格顶点代数V_{L,𝔽}与同类型的一阶单仿射顶点代数之间的同构。通过该同构，我们对作为ℕ分次顶点代数的V_{L,𝔽}的不可约ℕ分次模进行分类。我们还考虑L为A_n型且det G_L=n+1≡0 mod p的情形，证明V_{L,𝔽}不是单的，并确定其单的ℕ分次商。此外，当n+1=ap（a为正整数且gcd(a,p)=1）时，我们对V_{L,𝔽}的单商的不可约ℕ分次模给出分类。
英文摘要
We study lattice vertex algebras of type ADE over an algebraically closed field $\mathbb{F}$ of prime characteristic $p>2$ and their representations. Let $L$ be a root lattice of type ADE, and $G_{L}$ the Gram matrix of $L$. When $\det G_{L}\not \equiv 0\pmod{p}$, we establish an isomorphism between the lattice vertex algebra $V_{L,\mathbb{F}}$ and the level-one simple affine vertex algebra of the same type. Via this isomorphism, we classify the irreducible $\mathbb{N}$-graded modules of $V_{L,\mathbb{F}}$ viewed as an $\mathbb{N}$-graded vertex algebra. We also consider the case where $L$ is of type $A_n$ with $\det G_L=n+1\equiv 0\pmod{p}$. We show that $V_{L,\mathbb{F}}$ is not simple and determine the simple $\mathbb{N}$-graded quotient of $V_{L,\mathbb{F}}$. Furthermore, when $n+1=ap$ for some $a\in \mathbb{Z}_+$ with $\gcd(a,p)=1$, we give the classification of the irreducible $\mathbb{N}$-graded modules for the simple quotient of $V_{L,\mathbb{F}}$.
Comments28 pages
