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Vertex operator algebra analogue of embedding of $B_4$ into $F_4$
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Vertex operator algebra analogue of
embedding of B4 into F4
Ozren Pers?e
Abstract
Let LB(?
5
2 , 0) (resp. LF (?
5
2 , 0)) be the simple vertex operator
algebra associated to affine Lie algebra of type B
(1)
4 (resp. F
(1)
4 ) with
the lowest admissible half-integer level ?52 . We show that LB(?
5
2 , 0)
is a vertex subalgebra of LF (?
5
2 , 0) with the same conformal vector,
and that LF (?
5
2 , 0) is isomorphic to the extension of LB(?
5
2 , 0) by its
only irreducible module other than itself. We also study the repre-
sentation theory of LF (?
5
2 , 0), and determine the decompositions of
irreducible weak LF (?
5
2 , 0)-modules from the category O into direct
sums of irreducible weak LB(?
5
2 , 0)-modules.
1 Introduction
The embedding of B4 into F4 has been studied on the level of Lie groups
and Lie algebras both in mathematics and physics (cf. [Ba, GKRS, PR]). It
is particularly interesting that simple Lie algebra of type F4 contains three
copies of simple Lie algebra of type B4 as Lie subalgebras.
Let gF be the simple Lie algebra of type F4, and gB its Lie subalgebra of
type B4. The decomposition of gF into a direct sum of irreducible gB-modules
is:
gF ～= gB ⊕ VB(ω?4), (1.1)
12000 Mathematics Subject Classification. Primary 17B69; Secondary 17B67.
2Partially supported by the Ministry of Science, Education and Sports of the Republic
of Croatia, grant 0037125.
1
where VB(ω?4) is the irreducible highest weight gB-module whose highest
weight is the fundamental weight ω?4 for gB.
We want to study the analogue of relation (1.1) in the case of vertex
operator algebras associated to affine Lie algebras. Let g?B be the affine Lie
algebra of type B
(1)
4 , and g?F the affine Lie algebra of type F
(1)
4 . For k ∈ C,
denote by LB(k, 0) and LF (k, 0) simple vertex operator algebras associated
to g?B and g?F of level k, respectively, with conformal vectors obtained from
Segal-Sugawara construct
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