We study S
2
-orbifolds of N = 1 and N = 2 superconformal vertex algebras. For generic values of the central charge, we determine types of these orbifolds and prove that they are W-algebras. For some ...special rational values of the central charge, we get new examples of rational N = 1 vertex superalgebras. We also investigate S
2
-orbifold of the Heisenberg-Virasoro vertex algebra.
In this paper we study the integral form of the lattice vertex algebra VL. We show that divided powers of general vertex operators preserve the integral lattice spanned by Schur functions indexed by ...partition-valued functions. We also show that the Garland operators, counterparts of divided powers of Heisenberg elements in affine Lie algebras, also preserve the integral form. These construe analogs of the Kostant Z-forms for the enveloping algebras of simple Lie algebras and the algebraic affine Lie groups in the situation of the lattice vertex algebras.
Quadratic duality for chiral algebras Gui, Zhengping; Li, Si; Zeng, Keyou
Advances in mathematics (New York. 1965),
August 2024, 2024-08-00, Volume:
451
Journal Article
Peer reviewed
We introduce a notion of quadratic duality for chiral algebras. This can be viewed as a chiral version of the usual quadratic duality for quadratic associative algebras. We study the relationship ...between this duality notion and the Maurer-Cartan equations for chiral algebras, which turns out to be parallel to the associative algebra case. We also present some explicit examples.
Hopf actions on vertex algebras Dong, Chongying; Ren, Li; Yang, Chao
Journal of algebra,
04/2024, Volume:
644
Journal Article
Peer reviewed
In this article, we investigate Hopf actions on vertex algebras. Our first main result is that every finite-dimensional Hopf algebra that inner faithfully acts on a given π2-injective vertex algebra ...must be a group algebra. Secondly, under suitable assumptions, we establish a Schur-Weyl type duality for semisimple Hopf actions on Hopf modules of vertex algebras.
Let k be a field of characteristic zero. This paper studies a problem proposed by Joseph F. Ritt in 1950. Precisely, we prove that(1)If p⩾2 is an integer, for every integer i∈N, the nilpotency index ...of the image of Ti in the ring k{T}/Tp equals (i+1)p−i.(2)For every pair of integers (i,j), the nilpotency index of the image of TiUj in the ring k{T}/TU equals i+j+1.
The coset (commutant) construction is a fundamental tool to construct vertex operator algebras from known vertex operator algebras. The aim of this paper is to provide a fundamental example of the ...commutants of vertex algebras outside vertex operator algebras. Namely, the commutant
C
of the
principal subalgebra
W
of the
A
1
lattice vertex operator algebra
V
A
1
is investigated. An explicit minimal set of generators of
C
, which consists of infinitely many elements and strongly generates
C
, is introduced. It implies that the algebra
C
is not finitely generated. Furthermore, Zhu’s Poisson algebra of
C
is shown to be isomorphic to an infinite-dimensional algebra
C
x
1
,
x
2
,
…
/
(
x
i
x
j
|
i
,
j
=
1
,
2
,
…
)
. In particular, the associated variety of
C
consists of a point. Moreover,
W
and
C
are verified to form a dual pair in
V
A
1
.
Given a vertex operator algebra V, one can attach a graded Poisson algebra called the C2-algebra R(V). The associate Poisson scheme provides an important invariant for V and has been studied by ...Arakawa as the associated variety. In this article, we define and examine the cohomological variety of a vertex algebra, a notion cohomologically dual to that of the associated variety, which measures the smoothness of the associated scheme at the vertex point. We study its basic properties and then construct a closed subvariety of the cohomological variety for rational affine vertex operator algebras constructed from finite dimensional simple Lie algebras. We also determine the cohomological varieties of the simple Virasoro vertex operator algebras. These examples indicate that, although the associated variety for a rational C2-cofinite vertex operator algebra is always a simple point, the cohomological variety can have as large a dimension as possible. In this paper, we study R(V) as a commutative algebra only and do not use the property of its Poisson structure, which is expected to provide more refined invariants. The goal of this work is to study the cohomological supports of modules for vertex algebras as the cohomological support varieties for finite groups and restricted Lie algebras.
We study the affine analogue FTp(sl2) of the triplet algebra. We show that FTp(sl2) is quasi-lisse and the associated variety is the nilpotent cone of sl2. We realize FTp(sl2) as the global sections ...of a sheaf of vertex algebras in the spirit of Feigin–Tipunin and thereby construct infinitely many simple modules and, in particular solve a conjecture by Semikhatov and Tipunin. We introduce the Kazama–Suzuki dual superalgebra sWp(sl2|1) of FTp(sl2) and their singlet type subalgebras sMp(sl2|1) and Mp(sl2) and show their correspondence of categories. For p=1, we show the logarithmic Kazhdan–Lusztig correspondence for these (super)algebras and, in particular, show that the quantum group corresponding to sMp(sl2|1) is the unrolled restricted quantum supergroup u−1H(sl2|1) as suggested by Semikhatov and Tipunin.
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