: ▁classical ▁W - al gebras , ▁quantum ▁W - al gebras , ▁and ▁finite ▁W - al gebras . ▁ ▁Class ical ▁W - al gebras ▁Perform ing ▁classical ▁Dr inf eld - S ok ol ov ▁reduction ▁on ▁a ▁Lie ▁algebra ▁provides ▁the ▁Po isson ▁bra cket ▁on ▁this ▁algebra . ▁ ▁Quant um ▁W - al gebras ▁▁ ▁defines ▁a ▁( quant um ) ▁W - algebra ▁to ▁be ▁a ▁mer omorphic ▁conform al ▁field ▁theory ▁( rough ly ▁a ▁vertex ▁operator ▁algebra ) ▁together ▁with ▁a ▁distinguished ▁set ▁of ▁gener ators ▁satisfying ▁various ▁properties . ▁ ▁They ▁can ▁be ▁constructed ▁from ▁a ▁Lie ▁( super ) algebra ▁by ▁quantum ▁Dr inf eld – S ok ol ov ▁reduction . ▁Another ▁approach ▁is ▁to ▁look ▁for ▁other ▁conser ved ▁curr ents ▁besides ▁the ▁Str ess – energy ▁tensor ▁in ▁a ▁similar ▁manner ▁to ▁how ▁the ▁Vir as oro ▁algebra ▁can ▁be ▁read ▁off ▁from ▁the ▁expansion ▁of ▁the ▁stress ▁tensor . ▁ ▁Fin ite ▁W - al gebras ▁▁ ▁comp ares ▁several ▁different ▁definitions ▁of ▁finite ▁W - al gebras , ▁which ▁are ▁certain ▁associ ative ▁al gebras ▁associated ▁to ▁nil pot ent ▁elements ▁of ▁sem is im ple ▁Lie ▁al gebras . ▁ ▁The ▁original ▁definition , ▁provided ▁by ▁Alexander ▁Prem et , ▁starts ▁with ▁a ▁pair ▁ ▁consisting ▁of ▁a ▁re duct ive ▁Lie ▁algebra ▁ ▁over ▁the ▁complex ▁numbers ▁and ▁a ▁nil pot ent ▁element ▁e . ▁By ▁the ▁Jacob son - Mor oz ov ▁theorem ,