LAPTH Laboratoire d’Annecy-le-Vieux de Physique Th´eorique Yangian realisations from finite W-algebras E. Ragoucya,1 and P. Sorbab,2 a Theory Division, CERN, CH-1211 Gen`eve 23 b Laboratoire de Physique Th´eorique LAPTH3 LAPP, BP 110, F-74941 Annecy-le-Vieux Cedex, France. Abstract We construct an algebra homomorphism between the Yangian Y (sl(n)) and the finite W-algebras W(sl(np),n.sl(p)) for any p. We show how this result can be applied to determine properties of the finite dimensional representations of such W-algebras. hep-th/9803243 CERN-TH/98-104 LAPTH-672/98 March 1998
[email protected] On leave of absence from LAPTH.
[email protected]. 3URA 14-36 du CNRS, associ´ee `a l’Universit´edeSavoie. Contents 1 Introduction 2 2 Finite W algebras: notation and classification. 3 2.1 Classical W(G, S)algebras.............................. 3 2.2 Quantum W(G, S)algebras............................. 4 2.3Miurarepresentations................................. 5 2.4 Example: W(sl(np),n.sl(p)))algebras....................... 5 3 Yangians Y (G) 7 3.1Definition....................................... 7 3.2 Evaluation representations of Y (sl(n))....................... 8 4 Yangians and classical W-algebra 8 5 Yangians and quantum W(sl(np),n.sl(p)) algebras 9 6 Representations of W(sl(2n),n.sl(2)) algebras 11 7 Conclusion 13 A The soldering procedure 14 B Classical W(sl(np),n.sl(p)) algebras 16 B.1Generalities...................................... 16 B.2 The generic case n 6=2................................ 18 B.3 The particular case of Y (sl(2))..........................