2016/04/30 by Drazen Adamovic, Victor G. Kac, Pierluigi Moseneder Frajria +2
Mathematics · Physics and Astronomy · #math.RT #math-ph #math.MP #math.QA
published as Japanese Journal of Mathematics, 12, (2017), n. 2, 261-315 · 48 pages, latex file. Final version, to appear in Japanese Mathematical Journal
arxiv created 2017/04/12 · arxiv updated 2018/02/09
We present methods for computing the explicit decomposition of the minimal simple affine W-algebra Wk(\mathfrak g, θ) at a conformal level k as a module for its maximal affine subalgebra \mathcal Vk(\mathfrak g\natural). A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when \mathfrak g\natural is a semisimple Lie algebra, we show that, for a suitable conformal level k, Wk(\mathfrak g, θ) is isomorphic to an extension of \mathcal Vk(\mathfrak g\natural) by its simple module. We are able to prove that in certain cases Wk(\mathfrak g, θ) is a simple current extension of \mathcal Vk(\mathfrak g\natural). In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple W-algebra Wk(sl(4), θ) at k=-8/3. We prove, as conjectured in arXiv:1407.1527, that Wk(sl(4), θ) is isomorphic to the vertex algebra \mathcal R(3), and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra Vk (sl(n)) at certain admissible levels and for Vk (sl(m | n)), m≠ n, m,n≥ 1 at arbitrary levels.