Representations of Quantizations
نویسنده
چکیده
1.1. Algebra case. Let A be a filtered quantization of a Z-graded finitely generated Poisson algebra A. By A -mod we denote the category of finitely generated A-modules. A basic tool to study such modules is to reduce them to finitely generated A-modules that can be studied by means of Commutative algebra/ Algebraic geometry. Given an A-module M , one introduces the notion of a good filtration M = ∪ i∈ZM6i: this is a complete and separated A-module filtration on M such that grM is a finitely generated A-module. Note that if A is Z>0-filtered, then any good filtration on M is bounded from below. A good filtration exists if and only if the module is finitely generated. That a module with a good filtration is finitely generated is an exercise. Let us produce a good filtration on a finitely generated module. Choose generators m1, . . . ,mk and integers d1, . . . , dk. Set M6n := ∑k j=1A6n−dkmk. This is a good filtration. Indeed, we have an epimorphism A⊕k M defined by the generators. We equip each summand A with the original filtration shifted by di. Then the filtration on M is the induced filtration on the quotient, which easily shows that the filtration is good. In fact, any good filtration on M has this form. The construction also implies that there are many good filtrations. However, despite this fact any two of them are “not very far from one another”.
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