函数恒等式理论在泊松超代数上的应用(英文)

函数恒等式理论在泊松超代数上的应用(英文)IntroductionThe theory of algebraic identities has been widely applied in diffe

() 函数恒等式理论在泊松超代数上的应用英文 Introduction Thetheoryofalgebraicidentitieshasbeenwidelyappliedin differentmathematicalfieldssuchasalgebraicgeometry, representationtheory,andmathematicalphysics.Inthispaper,we willstudytheapplicationofthetheoryofalgebraicidentitieson Poissonsuperalgebras. Poissonsuperalgebrasareoneoftheimportantmathematical structuresthathavefoundapplicationindifferentareasofphysics, includingquantummechanicsandintegrablesystems.The Poissonbracketisafundamentalobjectthatunderliesthe structureofPoissonsuperalgebrasandencodesthealgebraic propertiesofthesestructures.ThestudyofPoissonsuperalgebras providesaframeworkforunderstandingthegeometricstructure ofclassicalandquantummechanicalsystemsandhasbeenwidely usedinthetheoryofintegrablesystems. Inthispaper,wewillfirstgiveabriefintroductiontoPoisson superalgebrasandthendiscussthetheoryofalgebraicidentities. Thenwewillstudytheapplicationofthetheoryofalgebraic identitiesonPoissonsuperalgebras,particularlyinthecontextof Hamiltonianoperators. PoissonSuperalgebras APoissonsuperalgebraisagradedalgebra,denotedbyA= ⨁n≥-1An,endowedwithaLiebracketoperation{.,.}:AnxAm→ An+m,suchthatthefollowingaxiomshold: -Antisymmetry:{a,b}=-(-1)pq{b,a}fora,b∈Apandp+q=1 mod2. -Leibnizrule:{a,bc}={a,b}c+(-1)pa{a,c}bfora∈Apandb,c ∈A. -Jacobiidentity:{a,{b,c}}+{b,{c,a}}+{c,{a,b}}=0fora,b,c∈ A. Inaddition,wedefinethePoissonbracketonthedualspace A*ofAby{f,g}(a)=-(-1)pq{(df)(a),(dg)(a)},wheref,garelinear functionalsonAandp+q=1mod2. ThePoissonbracketisafundamentaltoolthatunderliesthe