Frobenius-Perron Theory of Endofunctors

Transcription

1 March 17th, 2018 Recent Developments in Noncommutative Algebra and Related Areas, Seattle, WA

2 Throughout let k be an algebraically closed field.

3 Throughout let k be an algebraically closed field. The Frobenius-Perron dimension for an object in a finite tensor category has been essential to the classification of fusion categories and semisimple Hopf algebras.

4 Throughout let k be an algebraically closed field. The Frobenius-Perron dimension for an object in a finite tensor category has been essential to the classification of fusion categories and semisimple Hopf algebras. What if we could generalize this invariant to any k-linear category?

5 Throughout let k be an algebraically closed field. The Frobenius-Perron dimension for an object in a finite tensor category has been essential to the classification of fusion categories and semisimple Hopf algebras. What if we could generalize this invariant to any k-linear category? In joint work with Chen, Gao, Zhang, Zhang, and Zhu, we introduced a new invariant called the Frobenius-Perron dimension of a k-linear category C with chosen endofunctor σ (denoted fpd(σ)). In the case where C is a fusion category and X is an object of C, fpd(x ) gives the Frobenius-Perron dimension of X.

6 Throughout let k be an algebraically closed field. The Frobenius-Perron dimension for an object in a finite tensor category has been essential to the classification of fusion categories and semisimple Hopf algebras. What if we could generalize this invariant to any k-linear category? In joint work with Chen, Gao, Zhang, Zhang, and Zhu, we introduced a new invariant called the Frobenius-Perron dimension of a k-linear category C with chosen endofunctor σ (denoted fpd(σ)). In the case where C is a fusion category and X is an object of C, fpd(x ) gives the Frobenius-Perron dimension of X. In particular, we defined the fpd for a derived category to be fpd(σ), where Σ is the suspension functor. The fpd of an abelian category is defined via the embedding into its derived category.

7 Previous results Proposition (CGWZZZ) If A, B are abelian (respectively triangulated) categories such that A is a full abelian (respectively triangulated) subcategory of B, then fpd A fpd B.

8 Previous results Proposition (CGWZZZ) If A, B are abelian (respectively triangulated) categories such that A is a full abelian (respectively triangulated) subcategory of B, then fpd A fpd B. Theorem (CGWZZZ) fpd(d b (Mod - k[x 1,..., x n])) = n.

9 Previous results Proposition (CGWZZZ) If A, B are abelian (respectively triangulated) categories such that A is a full abelian (respectively triangulated) subcategory of B, then fpd A fpd B. Theorem (CGWZZZ) fpd(d b (Mod - k[x 1,..., x n])) = n. Theorem (CGWZZZ) Let Q be a finite connected quiver and let D b (Q) be the bounded derived category of finite dimensional left kq-modules. 1. kq is of finite representation type (A-D-E) fpd(d b (Q)) = kq is of tame representation type (à - D - Ẽ) fpd(db (Q)) = kq is of wild representation type (not A-D-E or à - D - Ẽ) fpd(d b (Q)) =.

10 Calculating the fpd for f.d. radical square zero algebras Theorem (W) Let A = kq/( 2) for some finite quiver Q. Let Q be the quiver formed from Q by removing all the loops. Then there is a one-to-one correspondence between the isomorphism classes below: {brick A - modules} {brick kq - modules annihilated by ( 2)}

11 Calculating the fpd for f.d. radical square zero algebras Theorem (W) Let A = kq/( 2) for some finite quiver Q. Let Q be the quiver formed from Q by removing all the loops. Then there is a one-to-one correspondence between the isomorphism classes below: {brick A - modules} {brick kq - modules annihilated by ( 2)} Furthermore, if M, N are brick A-modules, we have a natural isomorphism Hom A (M, N) = HomkQ (M, N).

12 Calculating the fpd for f.d. radical square zero algebras Theorem (W) Let A = kq/( 2) for some finite quiver Q. Let Q be the quiver formed from Q by removing all the loops. Then there is a one-to-one correspondence between the isomorphism classes below: {brick A - modules} {brick kq - modules annihilated by ( 2)} Furthermore, if M, N are brick A-modules, we have a natural isomorphism Hom A (M, N) = HomkQ (M, N). Proposition (W) There exists a family of finite-dimensional radical squared zero algebras A which can be used to construct arbitrarily large irrational values of fpd (A - mod).

13 Calculating the fpd for f.d. radical square zero algebras Theorem (W) Let A = kq/( 2) for some finite quiver Q. Let Q be the quiver formed from Q by removing all the loops. Then there is a one-to-one correspondence between the isomorphism classes below: {brick A - modules} {brick kq - modules annihilated by ( 2)} Furthermore, if M, N are brick A-modules, we have a natural isomorphism Hom A (M, N) = HomkQ (M, N). Proposition (W) 1 2 n a 1 l a 2 l a n l a n 1 l n n 2 n There exists a family of finite-dimensional radical squared zero algebras A which can be used to construct arbitrarily large irrational values of fpd (A - mod). a l 1 a l 2 a l 2 a l n 2 a l n Theorem (W) 2 Let Q be one of the quivers depicted with N i loops at each vertex i. Define A := kq/( 2). Then n fpd (A - mod) = max{n 1,..., N n}. a l 1 a l 3 a l 4 a l n