187 research outputs found
Invariant subalgebras of affine vertex algebras
Given a finite-dimensional complex Lie algebra g equipped with a
nondegenerate, symmetric, invariant bilinear form B, let V_k(g,B) denote the
universal affine vertex algebra associated to g and B at level k. For any
reductive group G of automorphisms of V_k(g,B), we show that the invariant
subalgebra V_k(g,B)^G is strongly finitely generated for generic values of k.
This implies the existence of a new family of deformable W-algebras W(g,B,G)_k
which exist for all but finitely many values of k.Comment: Final version, proof of main result simplified. arXiv admin note:
substantial text overlap with arXiv:1006.562
Cosets of affine vertex algebras inside larger structures
Given a finite-dimensional reductive Lie algebra equipped with
a nondegenerate, invariant, symmetric bilinear form , let
denote the universal affine vertex algebra associated to
and at level . Let be a vertex
(super)algebra admitting a homomorphism . Under some technical conditions on , we
characterize the coset for
generic values of . We establish the strong finite generation of this coset
in full generality in the following cases: , , and . Here and are
finite-dimensional Lie (super)algebras containing , equipped with
nondegenerate, invariant, (super)symmetric bilinear forms and which
extend , is fixed, and is a free field
algebra admitting a homomorphism .
Our approach is essentially constructive and leads to minimal strong finite
generating sets for many interesting examples. As an application, we give a new
proof of the rationality of the simple superconformal algebra with
for all positive integers .Comment: Some errors corrected, final versio
Orbifolds of symplectic fermion algebras
We present a systematic study of the orbifolds of the rank symplectic
fermion algebra , which has full automorphism group .
First, we show that and are
-algebras of type and
, respectively. Using these results, we find
minimal strong finite generating sets for and
for all . We compute the characters of the
irreducible representations of and
appearing inside , and
we express these characters using partial theta functions. Finally, we give a
complete solution to the Hilbert problem for ; we show that for
any reductive group of automorphisms, is strongly
finitely generated.Comment: Exposition streamlined, some new results added in Section 5,
references added. arXiv admin note: text overlap with arXiv:1205.446
A commutant realization of Odake's algebra
The bc\beta\gamma-system W of rank 3 has an action of the affine vertex
algebra V_0(sl_2), and the commutant vertex algebra C =Com(V_0(sl_2), W)
contains copies of V_{-3/2}(sl_2) and Odake's algebra O. Odake's algebra is an
extension of the N=2 superconformal algebra with c=9, and is generated by eight
fields which close nonlinearly under operator product expansions. Our main
result is that V_{-3/2}(sl_2) and O form a Howe pair (i.e., a pair of mutual
commutants) inside C. More generally, any finite-dimensional representation of
a Lie algebra g gives rise to a similar Howe pair, and this example corresponds
to the adjoint representation of sl_2.Comment: Minor corrections, discussion of Odake's algebra in Section 2
expanded, final versio
Cosets of the -algebra
Let be the universal
-algebra associated to with its subregular
nilpotent element, and let be its simple quotient. There is a Heisenberg subalgebra
, and we denote by the coset
,
and by its simple quotient. We show that for
where is an integer greater than and is coprime to ,
is isomorphic to a rational, regular -algebra
. In particular,
is a simple current
extension of the tensor product of with a rank one lattice vertex operator algebra, and hence is
rational.Comment: 14 pages, to appear in conference proceedings for AMS Special Session
on Vertex Algebras and Geometr
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