Page - 147 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Warning. Here algebrameans amultiplication a ·bwithout any rule of calculations. So the product a ·b · c is meaningless. Throughout thispaper, the smoothmanifoldswedealwithare connectedandparacompact. Ina smooth manifoldMallgeometrical objectsweare interested inare smoothaswell. Thevector space of smoothvectorfields in amanifoldM isdenotedbyX(M). It is a leftmodule of the associative commutativealgebraC∞(M). Considera realvectorbundle E→M. Therealvector spaceof sectionsofE isdenotedbyΓ(E). Definition3. Areal algebroidovera smoothmanifoldMisa realvectorbundlewhosevector spaceof sections is a real algebra. SothevectorspaceofsectionsofarealalgebroidE isendowedwithaR-bilinearmap Γ(E)×Γ(E) (s,s∗)→ s ·s∗ ∈Γ(E) Tosimplify themultiplicationof twosections isdenoted s ·s∗. Definition4. Atwo-sidedmoduleof analgebroidE is avectorbundle V→M whosevector spaceof sections is a two-sidedmoduleof thealgebraΓ(E). Let sbesectionE andletvbeasectionofV. Both leftaction sonvandtherightactionof sonv aredenotedby s ·vandv ·s. Definition5. AnanchoredvectorbundleoverMisapair (E,b) formedbyarealvectorbundleE andavectorbundlehomomorphism E e→ b(e)∈TM. Thehomomorphismb is called theanchormap. 2.2.AnomalyFunctionsofAlgebroidsandofModules LetV beatwo-sidedmoduleofanalgebroid (E,b). Definition6. Ananomaly function of an algebroidE is a 3-linearmapAE ofΓ(E)3 inΓ(E)whose values AE(s1,s2,s3)belongto spanR[(si·sj)·sk,si·(sj·sk); i, j,k∈ [1,2,3]].Ananomaly functionofanE-moduleV is a3-linearmapAEV ofΓ(E)2×Γ(V) inΓ(V)whosevaluesAEV(s,s∗,v)belongtospanR[(s·s∗)·v,s·(s∗·v)∀s, s∗ ∈Γ(E),∀v∈Γ(V)]. In thispaperweare interestedinsomeanomalyfunctionswhichhavestronggeometrical impacts. Theyaredefinedbelow. 147