Page - 148 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Definition7. LetE beanalgebroidand let s,s∗,s∗∗∈Γ(E). (1) Theassociatoranomaly functionofE isdefinedby Ass(s,s∗,s∗∗)=(s ·s∗) ·s∗∗−s ·(s∗ ·s∗∗). (2) TheKoszul-Vinberganomaly functionofE isdefinedby KV(s,s∗,s∗∗)=Ass(s,s∗,s∗∗)−Ass(s∗,s,s∗∗). (3) The Jacobi anomaly functionsofE aredefinedby J(s,s∗,s∗∗)=(s ·s∗) ·s∗∗+(s∗ ·s∗∗) ·s+(s∗∗ ·s) ·s∗. Definition8. Letvbea sectionof a two-sidedE-moduleV. (1) Theassociatoranomaly functionof a leftmoduleV isdefinedas Ass(s,s∗,v)=(s ·s∗) ·v−s ·(s∗ ·v). (2) TheKVanomaly functionsof a twosidedmoduleV aredefinedas KV(s,s∗,v)=Ass(s,s∗,v)−Ass(s∗,s,v), KV(s,v,s∗)=(s ·v) ·s∗−s ·(v ·s∗)−(v ·s) ·s∗+v ·(s ·s∗). Definition9. Wekeepthenotationusedabove. Lets,s∗besectionsofE, letvbeasectionofV and f ∈C∞(M). (1) TheLeibnizanomaly functionof ananchoredalgebroidE isdefinedby L(s, f,s∗)= s ·(fs∗)−df(b(s))s∗− fs ·s∗. (2) TheLeibnizanomaly functionof theE-moduleV isdefinedby L(s, f,v)= s ·(fv)−df(b(s))v− fs ·v. A category of algebroids and modules of algebroids is defined by its anomaly functions. Theanomalyfunctionsarealsousedfor introducingtheoriesofhomologyofalgebroids. Somecategoriesofanchoredalgebroidsplay important roles in thedifferentialgeometry. Definition10. (A1):ALiealgebroid is ananchoredalgebroid (E,b) satisfying the identities s ·s∗=0, L(s, f,s∗)=0. (B1):AKValgebroid is ananchoredalgebroid (E,b) satisfying the identities KV(s,s∗,s∗∗)=0, L(s, f,s∗)=0. (B2):AvectorbundleV is amoduleofLie algebroid (E,b) if it satisfies the identities L(s, f,v)=0, (s ·s∗) ·v−s ·(s∗·v)+s∗·(s ·v)=0. 148