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In this paper, we study Z -graded vertex superalgebras over a general field of an odd prime characteristic. In particular, we study affine vertex superalgebras V g ˆ (ℓ , 0) and a family of quotient vertex superalgebras V g ˆ χ (ℓ , 0) with g assumed to be a restricted Lie superalgebra. Among the main results, we obtain a necessary and sufficient condition for a g ˆ -module of level ℓ to be a V g ˆ χ (ℓ , 0) -module. Furthermore, we study the special case with g = g 1 ¯ and introduce a quotient vertex superalgebra L g ˆ (ℓ , 0) of V g ˆ (ℓ , 0). As the main results, we prove that for ℓ ≠ 0 , vertex superalgebra L g ˆ (ℓ , 0) is simple, the adjoint module is the only irreducible N -graded module up to equivalence, and every L g ˆ (ℓ , 0) -module is isomorphic to the direct sum of some copies of L g ˆ (ℓ , 0). [ABSTRACT FROM AUTHOR]

This paper is to study vertex superalgebras over a field of prime characteristic. For a general 1 2 Z -graded vertex superalgebra V , Zhu's A (V) theory is examined with some suitable modifications. Among the main results, Zhu's algebras of certain affine vertex superalgebras and Neveu-Schwarz vertex superalgebras, and irreducible modules for certain quotient vertex superalgebras associated to p -center, are determined. Irreducible modules for Clifford vertex superalgebras are also determined and a complete reducibility theorem is obtained. [ABSTRACT FROM AUTHOR]

In this paper, twisted modules for modular affine vertex algebras V g ˆ (ℓ , 0) and for their quotient vertex algebras V g ˆ 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 1 T N -graded irreducible σ -twisted V g ˆ 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 h ˆ [ σ ] is actually isomorphic to the untwisted Heisenberg Lie algebra h ˆ , unlike in the case of characteristic zero. Furthermore, for any nonzero level ℓ , irreducible σ -twisted L h ˆ (ℓ , 0) -modules are explicitly classified and the complete reducibility of every σ -twisted L h ˆ (ℓ , 0) -module is obtained. [ABSTRACT FROM AUTHOR]

Based on a work of Dong and Griess, we study a class of vertex algebras over fields of prime characteristic obtained from integral forms in lattice vertex operator algebras over complex field. In particular, the generating set is given, a bilinear form is obtained, and the simplicity is studied. Finally, the vertex operator algebra structure is characterized. [ABSTRACT FROM AUTHOR]