PBW deformation of Koszul Calabi-Yau algebras Can Zhu Shanghai - - PowerPoint PPT Presentation

Contents Calabi-Yau algebra Deformation and extension Main results

PBW deformation of Koszul Calabi-Yau algebras

Can Zhu

Shanghai Polytechnic University

Noncommutative Algebraic Geometry 2011 Shanghai Workshop 9-12, 2011

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results

Joint work with Quan Shui WU.

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results

Contents

1

Calabi-Yau algebra Calabi-Yau algebra and Calabi-Yau category Examples and properties Yonada Ext algebra

2

Deformation and extension PBW deformation Central regular extension Relations

3

Main results Central Regular Extension PBW deformation Application

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results

Notations

k is field with characteristic 0 graded algebras are generated in degree 1 all modules are left graded ones Ae := A ⊗ Ao,

∗ := Homk(

, k)

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results Calabi-Yau algebra and Calabi-Yau category Examples and properties Yonada Ext algebra

Calabi-Yau algebra

Definition (V. Ginzburg, 2007) A graded algebra A is d-Calabi-Yau (d-CY, for short)if (1) A is homologically smooth, i.e., A has a finitely generated Ae-projective resolution of finite length; (2) there exists an integer l such that Exti

Ae(A, Ae) ∼

=

• A(l),

if i = d; 0, if i = d as Ae-modules.

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results Calabi-Yau algebra and Calabi-Yau category Examples and properties Yonada Ext algebra

Calabi-Yau category

Suppose that (T , Σ) is a triangulated k-category which is Hom-finite. Let d be a nonnegative integer. Definition (M. Kontsevich, 1998) The triangulated category (T , Σ) is called a d-Calabi-Yau (d-CY, for short) category if Σd is a Serre functor, i.e., there is a bifunctorial isomorphism HomT (X, Y) ∼ = HomT (Y, ΣdX)∗.

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results Calabi-Yau algebra and Calabi-Yau category Examples and properties Yonada Ext algebra

Remarks

(1) (Yekutieli-Zhang, Jørgensen, 1997) A is a d-CY algebra ⇒ Db(qgrA) is a (d −1)-CY category (2) (Keller, 2008) A is a d-CY algebra ⇒ Db

fd(Mod A) is a d-CY category

(3) (He-Oystaeyen-Zhang, 2010) Suppose A is p-Koszul. A is a d-CY algebra ⇔ Db

fd(Mod A) is a d-CY category

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results Calabi-Yau algebra and Calabi-Yau category Examples and properties Yonada Ext algebra

Examples

(1) k[x1, x2, · · · , xn]#G for a finite subgroup G of SL(n, k) (2) 3-dim. AS-regular algebras of type diag(1, 1), (1, 1, 1) (3) Yang-Mills algebras (4) Sklyanin algebras of dimension 4 (5) Weyl algebras An (6) preprojective algebras of non-Dynkin quivers (7) quantum enveloping algebras (8) rational Cherednik algebras

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results Calabi-Yau algebra and Calabi-Yau category Examples and properties Yonada Ext algebra

Proposition (Berger-Taillefer, Iyama-Reiten) If A is a graded d-CY algebra, then (1) the global dimension of A is d; (2) the Hochschild dimension of A is d; (3) if, moreover, A is connected, then A is AS-regular; (4) d = 0 if and only if A is semisimple.

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results Calabi-Yau algebra and Calabi-Yau category Examples and properties Yonada Ext algebra

Remarks (1) For ungraded d-CY algebra, its global dimension is not necessary to be d. Eg.: Weyl algebra An (2n-CY, BUT gl.dimAn = n). (2) Finite dim. CY algebras must be 0-CY.

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results Calabi-Yau algebra and Calabi-Yau category Examples and properties Yonada Ext algebra

Yonada Ext algebras of AS-regular algebras

Theorem(Bondal-Polishchuk, Smith, Berger-Marconnet, LPWZ) A is an AS-regular algebra if and only if its Yoneda Ext algebra E(A) := Ext∗

A(k, k) is a Frobenius algebra.

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results Calabi-Yau algebra and Calabi-Yau category Examples and properties Yonada Ext algebra

Yonada Ext algebras of Calabi-Yau algebras

Theorem (Van den Bergh, 2008) A is a d-CY algebra if and only if its Yoneda Ext algebra E(A) has a cyclic A∞-structure of degree d. Here, a cyclic structure on an A∞-algebra (E, m2, m3, · · · ) is a symmetric bilinear form (., .) on E such that for any n, (mn(a1, · · · , an), an+1) = ±(a1, mn(a2, · · · , , an+1)).

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results PBW deformation Central regular extension Relations

Definitions

Deformation A = Tk(V)/ R where R = {r1, · · · , rm}. U = Tk(V)/ P where P = {r1 + l1, · · · , rm + lm} with |li| < |ri|. U is called a deformation of A. ∃ A ։ gr U PBW deformation U is called a Poincaré-Birkhoff-Witt (PBW) deformation of A if A → gr U is an isomorphism as graded algebras.

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results PBW deformation Central regular extension Relations

Examples

(1) universal enveloping algebras U(g) (2) Weyl algebras (3) Sridharan enveloping algebras (4) symplectic reflection algebras

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results PBW deformation Central regular extension Relations

PBW-theorem for Koszul algebras

Let A be a quadratic algebra. Then ∃α : R − → V, β : R − → k

• s. t. P = {r − α(r) − β(r) | r ∈ R}.

Theorem (Braverman-Gaitsgory, 1996) Let A be a Koszul algebra. Then U is a PBW deformation if and

• nly if the following are satisfied:

(1) Im(α ⊗ id − id ⊗α) ⊂ R (on R ⊗ V V ⊗ R); (2) α(α ⊗ id − id ⊗α) = −(β ⊗ id − id ⊗β); (3) β(α ⊗ id − id ⊗α) = 0.

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results PBW deformation Central regular extension Relations

Remarks

(1) Similar results due to Polishchuk-Positselski, Berger-Ginzburg. (2) Universal enveloping algebras U(g) α(xy − yx) = [x, y], β = 0 PBW-theorem = “ Jacobi identity of Lie algebra g”

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results PBW deformation Central regular extension Relations

PBW deformation of 3-Calabi-Yau algebras

Let U be a PBW deformation of a 3-CY algebra A. Theorem (Berger-Taillefer, 2007) U is a 3-CY algebra if the following equivalent conditions are satisfied (1) U is derived from a potential, (2) id ⊗α − α ⊗ id = 0.

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results PBW deformation Central regular extension Relations

Central regular extension

Definition Let A and D be two graded algebras. If there is a central regular element t such that A ∼ = D/t, then D is called a central regular extension of A.

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results PBW deformation Central regular extension Relations

Rees algebra

Suppose that U has an increasing filtration FU = {FnU}n∈Z. Rees algebra Rees(U) =

n∈Z FnU tn ⊂ U[t, t−1]

Fact Rees(U) is a central regular extension of A.

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results PBW deformation Central regular extension Relations

Relations

Let U be a deformation of A and H(U) be the central extension associated to U (homogenization). Theorem (Cassidy-Shelton, 2007) U is a PBW deformation ⇔ H(U) is a central regular extension. Theorem If U is a PBW deformation, then Rees(U) ∼ = H(U).

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results Central Regular Extension PBW deformation Application

Central Regular Extension

Theorem Suppose that A = T(V)/R is a Koszul algebra. Let D be a central regular extension of A. Then (1) if D is a CY algebra, then so is A; (2) if A is a d-CY algebra, then, D is (d + 1)-CY if and only if

d−2

• i=0

(−1)i id⊗i ⊗α ⊗ id⊗(d−2−i)(x) = 0, for any x ∈ V ⊗i ⊗ R ⊗ V ⊗(d−2−i).

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results Central Regular Extension PBW deformation Application

PBW deformation

Suppose that A is a Noetherian k-algebras which is a domain and U is a PBW deformation of A. Theorem If A is a Koszul d-CY algebra (d ≥ 2), then the following are equivalent: (1) U is a d-CY algebra; (2) Rees(U) is a (d + 1)-CY algebra; (3)

d−2

• i=0

(−1)i id⊗i ⊗α ⊗ id⊗(d−2−i) = 0.

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results Central Regular Extension PBW deformation Application

PBW deformation

Remark (1) independent of β; (2) Chevalley-Eilenberg resolution.

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results Central Regular Extension PBW deformation Application

Application

Let g be a Lie algebra over k, f ∈ Z 2(g, k) be a 2-cocycle. Sridharan enveloping algebra Uf(g) = Tk(g)/x ⊗ y − y ⊗ x − [x, y] − f(x, y) Corollary (He-Oystaeyen-Zhang, 2010) The following statements are equivalent. (1) dim g = d and tr(ad(x)) = 0 for all x ∈ g . (2) U(g) is a d-CY algebra. (3) Uf(g) is a d-CY algebra.

Can Zhu PBW deformation of Koszul Calabi-Yau algebras

Contents Calabi-Yau algebra Deformation and extension Main results Central Regular Extension PBW deformation Application

Thank You!

Can Zhu PBW deformation of Koszul Calabi-Yau algebras