{"id":516,"date":"2017-09-11T23:14:40","date_gmt":"2017-09-11T21:14:40","guid":{"rendered":"http:\/\/bela.phy.hr\/quantixlie\/?p=516"},"modified":"2017-09-11T23:14:40","modified_gmt":"2017-09-11T21:14:40","slug":"conformal-embeddings-of-affine-vertex-algebras-in-minimal-w-algebras-ii-decompositions","status":"publish","type":"post","link":"http:\/\/bela.phy.hr\/quantixlie\/conformal-embeddings-of-affine-vertex-algebras-in-minimal-w-algebras-ii-decompositions\/","title":{"rendered":"Conformal embeddings of affine vertex algebras in minimal W-algebras II: decompositions"},"content":{"rendered":"

Authors:\u00a0Dra\u017een\u00a0Adamovi\u0107<\/span>,\u00a0<\/span><\/span>Victor\u00a0G.\u00a0Kac,\u00a0Pierluigi\u00a0M\u00f6seneder Frajria, Paolo\u00a0Papi, Ozren\u00a0Per\u0161e<\/p>\n

Japanese Journal of Mathematics<\/a>,\u00a0<\/span>Issue\u00a02<\/a>,\u00a0pp 261\u2013315.<\/span><\/p>\n

https:\/\/doi.org\/10.1007\/s11537-017-1621-x<\/a><\/p>\n

Abstract: We present methods for computing the explicit decomposition of the minimal simple affine\u00a0W-algebra \"{W_k(\mathfrak{g}, \theta)}\"\u00a0as a module for its maximal affine subalgebra \"{\mathscr{V}_k(\mathfrak{g}^{\natural})}\"\u00a0at a conformal level\u00a0k, that is, whenever the Virasoro vectors of \"{W_k(\mathfrak{g}, \theta)}\"\u00a0and \"{\mathscr{V}_k(\mathfrak{g}^\natural)}\"\u00a0coincide. 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}}\"\u00a0is a semisimple Lie algebra, we show that, for a suitable conformal level k, \"{W_k(\mathfrak{g}, \theta)}\"\u00a0is isomorphic to an extension of \"{\mathscr{V}_k(\mathfrak{g}^{\natural})}\"\u00a0by its simple module. We are able to prove that in certain cases \"{W_k(\mathfrak{g}, \theta)}\"\u00a0is a simple current extension of \"{\mathscr{V}_k(\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 \"{W_{k}(\mathit{sl}(4), \theta)}\" at\u00a0k\u00a0=\u00a0\u22128\/3. We prove, as conjectured in [3], that \"{W_{k}(\mathit{sl}(4), \theta)}\"\u00a0is isomorphic to the vertex algebra \"{\mathscr{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 \"{V_k (\mathit{sl}(n))}\"\u00a0at certain admissible levels and for \"{V_k (\mathit{sl}(m \vert n)), m\ne n, m,n\geq 1}\"\u00a0at arbitrary levels.<\/p>\n","protected":false},"excerpt":{"rendered":"

Authors:\u00a0Dra\u017een\u00a0Adamovi\u0107,\u00a0Victor\u00a0G.\u00a0Kac,\u00a0Pierluigi\u00a0M\u00f6seneder Frajria, Paolo\u00a0Papi, Ozren\u00a0Per\u0161e Japanese Journal of Mathematics,\u00a0September 2017,\u00a0Volume 12,\u00a0Issue\u00a02,\u00a0pp 261\u2013315. https:\/\/doi.org\/10.1007\/s11537-017-1621-x Abstract: We present methods for computing the explicit decomposition of the minimal simple affine\u00a0W-algebra \u00a0as a module for its maximal affine subalgebra \u00a0at a conformal level\u00a0k, that is, whenever the Virasoro vectors of \u00a0and \u00a0coincide. A particular emphasis […]<\/a><\/p>\n<\/div>","protected":false},"author":5,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[18,14],"tags":[],"_links":{"self":[{"href":"http:\/\/bela.phy.hr\/quantixlie\/wp-json\/wp\/v2\/posts\/516"}],"collection":[{"href":"http:\/\/bela.phy.hr\/quantixlie\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/bela.phy.hr\/quantixlie\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/bela.phy.hr\/quantixlie\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"http:\/\/bela.phy.hr\/quantixlie\/wp-json\/wp\/v2\/comments?post=516"}],"version-history":[{"count":14,"href":"http:\/\/bela.phy.hr\/quantixlie\/wp-json\/wp\/v2\/posts\/516\/revisions"}],"predecessor-version":[{"id":531,"href":"http:\/\/bela.phy.hr\/quantixlie\/wp-json\/wp\/v2\/posts\/516\/revisions\/531"}],"wp:attachment":[{"href":"http:\/\/bela.phy.hr\/quantixlie\/wp-json\/wp\/v2\/media?parent=516"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/bela.phy.hr\/quantixlie\/wp-json\/wp\/v2\/categories?post=516"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/bela.phy.hr\/quantixlie\/wp-json\/wp\/v2\/tags?post=516"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}