The $q$-boson-fermion realizations of quantum suprealgebra $U_q(gl(21))$

We show that our construction of realizations for Lie algebras and quantum algebras can be generalized to quantum superalgebras, too. We study an example of quantum superalgebra $U_q(gl(2/1))$ and give the boson-fermion realization with respect to one pair

Theq–boson–fermionrealizationsofthequantumsuperalgebraUq(gl(2/1))

ˇBurd´C. k

DepartmentofMathematicsandDopplerInstitute,FNSPE,CzechTechnicalUniversity,

Trojanova13,CZ-12000Prague2,CzechRepublic

O.Navr´atil

DepartmentofMathematics,FTS,CzechTechnicalUniversity,

NaFlorenci25,CZ-11000Prague1,CzechRepublic

Abstract

WeshowthatourconstructionofrealizationsforLiealgebrasandquantum

algebrascanbegeneralizedtoquantumsuperalgebras,too.WestudyanexampleofquantumsuperalgebraUq(gl(2/1))andgivetheboson–fermionrealizationwithrespecttoonepairofq–bosonoperatorsand2pairsoffermions.

arXiv:math/0307178v1 [math.QA] 12 Jul 20031IntroductionBoson–fermionrealizationsofagivensetofoperatorsviaBose–Fermioncreationandan-nihilationoperatorsareamongthemaintoolsofsolvingvariousquantumproblems.TheoriginislinkedwiththeSchwinger[1],Dyson[2]andHolstein–Primako [3]realizationswhicharedi erentbosonrealizationsofthealgebrasl(2).GeneralizationsoftheDysonrealizationtotheLiealgebrasl(n)werederivedin[4].Inourpaper[5]weformulatedthemethodstartingfromtheVermamodulesforobtainingbosonrealizationsandin[6]weobtainedexplicitlyabraidclassofrealizationswhichgeneralizedtheresultsfrom[7,8].LatertheideawasextendedtotheLiesuperalgebra,andtheDysontypeboson–fermionrealizationswereexplicitlygivenin[9],generalizingtheresultstosl(2/1)([10],[11]).Todaytheseboson–fermionrealizationsbecomeastandardtechniqueinquantummany–bodyphysicsandwecanalso ndseveralotherapplicationsinall eldsofquantum

physics.

Quantumgroupsandquantumsupergroupsorq–deformedLiealgebrasandsuperal-gebrasimplysomespeci cdeformationsoftheclassicalLiealgebrasandsuperalgebras.Fromamathematicalpointofview,thosearenoncommutativeassociativeHopfalgebrasandsuperalgebras.ThestructureandrepresentationtheoryofquantumgroupswereextensivelydevelopedbyJimbo[12]andDrinfeld[13].The rst”quantum”versionofHolstein–Primako wasworkedoutforUq(sl(2))[14]andthenforUq((sl(3))[15].TheSchwingertyperealizationwaswrittenin[16]and[17].Theserealizationsfoundimme-diateapplications[18–23].

Inourpapers[24,25,26]westudiedtheDysonrealizationsoftheseriesalgebrasUq(sl(2)),Uq(gl(n)),Uq(Bn),Uq(Cn)andUq(Dn).Thereissomespecialcase[25]for

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