### Abstract

In this paper, twisted modules for modular affine vertex algebras V_{gˆ}(ℓ,0) and for their quotient vertex algebras V_{gˆ} ^{χ}(ℓ,0) with g a restricted Lie algebra are studied. Let σ be an automorphism of g and let T be a positive integer relatively prime with the characteristic p such that σ^{T}=1. It is proved that [Formula presented]-graded irreducible σ-twisted V_{gˆ} ^{0}(ℓ,0)-modules are in one-to-one correspondence with irreducible modules for the restricted enveloping algebra u(g_{0}), where g_{0} is the subalgebra of σ-fixed points in g. It is also proved that when g=h is abelian, the twisted Heisenberg Lie algebra hˆ[σ] is actually isomorphic to the untwisted Heisenberg Lie algebra hˆ, unlike in the case of characteristic zero. Furthermore, for any nonzero level ℓ, irreducible σ-twisted L_{hˆ}(ℓ,0)-modules are explicitly classified and the complete reducibility of every σ-twisted L_{hˆ}(ℓ,0)-module is obtained.

Original language | English (US) |
---|---|

Pages (from-to) | 380-414 |

Number of pages | 35 |

Journal | Journal of Algebra |

Volume | 541 |

DOIs | |

State | Published - Jan 1 2020 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Keywords

- Twisted module
- Vertex operator algebra

### Cite this

*Journal of Algebra*,

*541*, 380-414. https://doi.org/10.1016/j.jalgebra.2019.08.031

}

*Journal of Algebra*, vol. 541, pp. 380-414. https://doi.org/10.1016/j.jalgebra.2019.08.031

**Twisted modules for affine vertex algebras over fields of prime characteristic.** / Li, Haisheng; Mu, Qiang.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Twisted modules for affine vertex algebras over fields of prime characteristic

AU - Li, Haisheng

AU - Mu, Qiang

PY - 2020/1/1

Y1 - 2020/1/1

N2 - In this paper, twisted modules for modular affine vertex algebras Vgˆ(ℓ,0) and for their quotient vertex algebras Vgˆ χ(ℓ,0) with g a restricted Lie algebra are studied. Let σ be an automorphism of g and let T be a positive integer relatively prime with the characteristic p such that σT=1. It is proved that [Formula presented]-graded irreducible σ-twisted Vgˆ 0(ℓ,0)-modules are in one-to-one correspondence with irreducible modules for the restricted enveloping algebra u(g0), where g0 is the subalgebra of σ-fixed points in g. It is also proved that when g=h is abelian, the twisted Heisenberg Lie algebra hˆ[σ] is actually isomorphic to the untwisted Heisenberg Lie algebra hˆ, unlike in the case of characteristic zero. Furthermore, for any nonzero level ℓ, irreducible σ-twisted Lhˆ(ℓ,0)-modules are explicitly classified and the complete reducibility of every σ-twisted Lhˆ(ℓ,0)-module is obtained.

AB - In this paper, twisted modules for modular affine vertex algebras Vgˆ(ℓ,0) and for their quotient vertex algebras Vgˆ χ(ℓ,0) with g a restricted Lie algebra are studied. Let σ be an automorphism of g and let T be a positive integer relatively prime with the characteristic p such that σT=1. It is proved that [Formula presented]-graded irreducible σ-twisted Vgˆ 0(ℓ,0)-modules are in one-to-one correspondence with irreducible modules for the restricted enveloping algebra u(g0), where g0 is the subalgebra of σ-fixed points in g. It is also proved that when g=h is abelian, the twisted Heisenberg Lie algebra hˆ[σ] is actually isomorphic to the untwisted Heisenberg Lie algebra hˆ, unlike in the case of characteristic zero. Furthermore, for any nonzero level ℓ, irreducible σ-twisted Lhˆ(ℓ,0)-modules are explicitly classified and the complete reducibility of every σ-twisted Lhˆ(ℓ,0)-module is obtained.

KW - Twisted module

KW - Vertex operator algebra

UR - http://www.scopus.com/inward/record.url?scp=85072965832&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85072965832&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2019.08.031

DO - 10.1016/j.jalgebra.2019.08.031

M3 - Article

AN - SCOPUS:85072965832

VL - 541

SP - 380

EP - 414

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -