{"id":286,"date":"2016-06-06T09:50:58","date_gmt":"2016-06-06T07:50:58","guid":{"rendered":"http:\/\/bela.phy.hr\/czi\/?p=286"},"modified":"2016-10-25T16:19:02","modified_gmt":"2016-10-25T14:19:02","slug":"hrvatski-whittaker-modules-for-the-affine-lie-algebra-a_1-1","status":"publish","type":"post","link":"http:\/\/bela.phy.hr\/quantixlie\/hrvatski-whittaker-modules-for-the-affine-lie-algebra-a_1-1\/","title":{"rendered":"Whittaker modules for the affine Lie algebra \"A_1 ^{(1)}\""},"content":{"rendered":"


\nWe prove the irreducibility of the universal non-degenerate Whittaker modules for the affine Lie algebra \"\widehat{sl_2}\" of type\u00a0\"A_1 ^{(1)}\" with noncritical level. These modules can become simple Whittaker modules over \"\widetilde{sl_2}=\widehat{sl_2}+\mathbb{C}d\" with the same Whittaker function and central charge. We have to modulo a central character for sl2sl2 to obtain simple degenerate Whittaker \"\widehat{sl_2}\"-modules with noncritical level. In the case of critical level the universal Whittaker module is reducible. We prove that the quotient of universal Whittaker \"\widehat{sl_2}\"-module by a submodule generated by a scalar action of central elements of the vertex algebra \"V_{\widehat{sl_2}}\" is simple as \"\widehat{sl_2}\"-module. We also explicitly describe the simple quotients of universal Whittaker modules at the critical level for \"\widetilde{sl_2}\". Quite surprisingly, with the same Whittaker function some simple degenerate \"\widetilde{sl_2}\" Whittaker modules at the critical level have semisimple action of d and others have free action of d . At last, by using vertex algebraic techniques we present a Wakimoto type construction of a family of simple generalized Whittaker modules for\u00a0\"\widehat{sl_2}\" at the critical level. This family includes all classical Whittaker modules at critical level. We also have Wakimoto type realization for degenerate Whittaker modules for\u00a0\"\widehat{sl_2}\" at noncritical level.<\/p>\n

D. Adamovi\u0107, R. Lu, K. Zhao, Whittaker modules for the affine Lie algebra \"A_1 ^{(1)}\",<\/a>\u00a0<\/strong> Advances in Mathematics 289 (2016) 438-479.<\/p>","protected":false},"excerpt":{"rendered":"

We prove the irreducibility of the universal non-degenerate Whittaker modules for the affine Lie algebra of type\u00a0 with noncritical level. These modules can become simple Whittaker modules over with the same Whittaker function and central charge. We have to modulo a central character for sl2sl2 to obtain simple degenerate Whittaker […]<\/a><\/p>\n<\/div>","protected":false},"author":1,"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\/286"}],"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\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/bela.phy.hr\/quantixlie\/wp-json\/wp\/v2\/comments?post=286"}],"version-history":[{"count":26,"href":"http:\/\/bela.phy.hr\/quantixlie\/wp-json\/wp\/v2\/posts\/286\/revisions"}],"predecessor-version":[{"id":439,"href":"http:\/\/bela.phy.hr\/quantixlie\/wp-json\/wp\/v2\/posts\/286\/revisions\/439"}],"wp:attachment":[{"href":"http:\/\/bela.phy.hr\/quantixlie\/wp-json\/wp\/v2\/media?parent=286"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/bela.phy.hr\/quantixlie\/wp-json\/wp\/v2\/categories?post=286"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/bela.phy.hr\/quantixlie\/wp-json\/wp\/v2\/tags?post=286"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}