Poisson and Hochschild cohomology and the semiclassical limit

Transcription

1 Poisson and Hochschild cohomology and the semiclassical limit Matthew Towers University of Kent Matthew Towers (University of Kent) arxiv / 15

2 Motivation A quantum algebra q 1 A, {, } Poisson algebra HH(A) Hochschild cohomology?? HP(A, {, }) Poisson cohomology Matthew Towers (University of Kent) arxiv / 15

3 Poisson algebras A Poisson algebra is a commutative algebra A with a Lie bracket {, }, such that {a, } : A A is a derivation for all a A. Matthew Towers (University of Kent) arxiv / 15

4 Poisson algebras A Poisson algebra is a commutative algebra A with a Lie bracket {, }, such that {a, } : A A is a derivation for all a A. Example A = k[x, y], {x, y} = xy Matthew Towers (University of Kent) arxiv / 15

5 Poisson algebras A Poisson algebra is a commutative algebra A with a Lie bracket {, }, such that {a, } : A A is a derivation for all a A. Example A = k[x, y], {x, y} = xy A left Poisson module over A is a module M for the associative algebra A and for the Lie algebra (A, {, }) such that Matthew Towers (University of Kent) arxiv / 15

6 Poisson algebras A Poisson algebra is a commutative algebra A with a Lie bracket {, }, such that {a, } : A A is a derivation for all a A. Example A = k[x, y], {x, y} = xy A left Poisson module over A is a module M for the associative algebra A and for the Lie algebra (A, {, }) such that {a, bm} = {a, b}m + b{a, m} Matthew Towers (University of Kent) arxiv / 15

7 Poisson algebras A Poisson algebra is a commutative algebra A with a Lie bracket {, }, such that {a, } : A A is a derivation for all a A. Example A = k[x, y], {x, y} = xy A left Poisson module over A is a module M for the associative algebra A and for the Lie algebra (A, {, }) such that {a, bm} = {a, b}m + b{a, m} {ab, m} = a{b, m} + b{a, m} Matthew Towers (University of Kent) arxiv / 15

8 Poisson algebras A Poisson algebra is a commutative algebra A with a Lie bracket {, }, such that {a, } : A A is a derivation for all a A. Example A = k[x, y], {x, y} = xy A left Poisson module over A is a module M for the associative algebra A and for the Lie algebra (A, {, }) such that Example {a, bm} = {a, b}m + b{a, m} {ab, m} = a{b, m} + b{a, m} A is a Poisson module over itself. Matthew Towers (University of Kent) arxiv / 15

9 Poisson algebras The Poisson enveloping algebra P(A) is an associative algebra such that P(A) mod = A PoissonMod Matthew Towers (University of Kent) arxiv / 15

10 Poisson algebras The Poisson enveloping algebra P(A) is an associative algebra such that P(A) mod = A PoissonMod Example A = k[x, y], {x, y} = xy. Then P(A) is generated by x, y, Ω(x), Ω(y) subject to xy = yx xω(x) = Ω(x)x yω(y) = Ω(y)y Ω(y)x xω(y) = xy Ω(x)y yω(x) = xy Ω(x)Ω(y) Ω(y)Ω(x) = xω(y) + yω(x) Matthew Towers (University of Kent) arxiv / 15

11 Poisson algebras If A is a polynomial Poisson algebra, we define the Poisson cohomology by HP (A) := Ext P(A) (A, A) Matthew Towers (University of Kent) arxiv / 15

12 Poisson algebras If A is a polynomial Poisson algebra, we define the Poisson cohomology by HP (A) := Ext P(A) (A, A) Otherwise, this may not be equivalent to the standard definition. Matthew Towers (University of Kent) arxiv / 15

13 Graded Poisson algebras Definition A graded Poisson algebra is a graded commutative algebra B with a bilinear bracket {, } such that {a, b} = ( 1) a b +1 {b, a} {a, bc} = {a, b}c + ( 1) a b b{a, c} ( 1) a c {a, {b, c}} + ( 1) a b {b, {c, a}} + ( 1) b c {c, {a, b}} = 0 Matthew Towers (University of Kent) arxiv / 15

14 Graded Poisson algebras Definition A graded Poisson algebra is a graded commutative algebra B with a bilinear bracket {, } such that {a, b} = ( 1) a b +1 {b, a} {a, bc} = {a, b}c + ( 1) a b b{a, c} ( 1) a c {a, {b, c}} + ( 1) a b {b, {c, a}} + ( 1) b c {c, {a, b}} = 0 A quadratic Poisson bracket on A = k[x 1,..., x n ] is one such that {x i, x j } is a quadratic polynomial for all i, j. It is determined by a map β : Λ 2 ( x 1,..., x n ) S 2 ( x 1,..., x n ). Matthew Towers (University of Kent) arxiv / 15

15 Graded Poisson algebras Definition A graded Poisson algebra is a graded commutative algebra B with a bilinear bracket {, } such that {a, b} = ( 1) a b +1 {b, a} {a, bc} = {a, b}c + ( 1) a b b{a, c} ( 1) a c {a, {b, c}} + ( 1) a b {b, {c, a}} + ( 1) b c {c, {a, b}} = 0 A quadratic Poisson bracket on A = k[x 1,..., x n ] is one such that {x i, x j } is a quadratic polynomial for all i, j. It is determined by a map β : Λ 2 ( x 1,..., x n ) S 2 ( x 1,..., x n ). If B = Λ(U) is an exterior algebra, a quadratic graded Poisson bracket is determined by a map γ : S 2 (U) Λ 2 (U). Matthew Towers (University of Kent) arxiv / 15

16 Homological aspects Theorem If A = k[v ] is a quadratic Poisson algebra with Poisson bracket determined by β : S 2 (V ) Λ 2 (V ) then P(A) is Koszul and P(A)! = Ext P(A) (k, k) is isomorphic to the graded Poisson enveloping algebra P gr (B) of the exterior algebra B on V with graded Poisson bracket determined by β : Λ 2 (V ) S 2 (V ). Matthew Towers (University of Kent) arxiv / 15

17 Semiclassical limits Let A be a torsion free R = k[q, q 1 ]-algebra such that A := A/(q 1)A is commutative. Matthew Towers (University of Kent) arxiv / 15

18 Semiclassical limits Let A be a torsion free R = k[q, q 1 ]-algebra such that A := A/(q 1)A is commutative. If x, y A then xy yx = (q 1)β(x, y) for unique β(x, y) A. Matthew Towers (University of Kent) arxiv / 15

19 Semiclassical limits Let A be a torsion free R = k[q, q 1 ]-algebra such that A := A/(q 1)A is commutative. If x, y A then xy yx = (q 1)β(x, y) for unique β(x, y) A. Write x for the image of x in A. Then { x, ȳ} := β(x, y) is a Poisson bracket on A. This Poisson algebra is called the semiclassical limit of A. Matthew Towers (University of Kent) arxiv / 15

20 Example semiclassical limit Example A = R x, y (xy qyx) Then xy yx = (q 1)yx, so the semiclassical limit is A = k[ x, ȳ] with Poisson bracket { x, ȳ} = xȳ Matthew Towers (University of Kent) arxiv / 15

21 Observation Let B = k q [x, y] = k(q) x,y xy qyx. Then the enveloping algebra Be = B B op is a deformation of P(A). Matthew Towers (University of Kent) arxiv / 15

22 Observation Let B = k q [x, y] = k(q) x,y xy qyx. Then the enveloping algebra Be = B B op is a deformation of P(A). Write X = x 1 Y = y 1 O(x) = (x 1 1 x)/(q 1) O(y) = (y 1 1 y)/(q 1) (elements of B e ). Matthew Towers (University of Kent) arxiv / 15

23 Observation Let B = k q [x, y] = k(q) x,y xy qyx. Then the enveloping algebra Be = B B op is a deformation of P(A). Write X = x 1 Y = y 1 O(x) = (x 1 1 x)/(q 1) O(y) = (y 1 1 y)/(q 1) (elements of B e ). Then: XY = qyx O(x)X = XO(x) O(y)Y = YO(y) O(y)X XO(y) = XY O(x)Y YO(x) = XY qo(x)o(y) O(y)O(x) = XO(y) + YO(x) Matthew Towers (University of Kent) arxiv / 15

24 Observation Let B = k q [x, y] = k(q) x,y xy qyx. Then the enveloping algebra Be = B B op is a deformation of P(A). Write X = x 1 Y = y 1 O(x) = (x 1 1 x)/(q 1) O(y) = (y 1 1 y)/(q 1) (elements of B e ). Then: XY = qyx O(x)X = XO(x) O(y)Y = YO(y) O(y)X XO(y) = XY O(x)Y YO(x) = XY qo(x)o(y) O(y)O(x) = XO(y) + YO(x) Compare with the relations for P(A) given earlier: xy = yx Ω(x)x = xω(x) Ω(y)y = yω(y) Ω(y)x xω(y) = xy Ω(x)y yω(x) = xy Ω(x)Ω(y) Ω(y)Ω(x) = xω(y) + yω(x) Matthew Towers (University of Kent) arxiv / 15

25 Deformation theorem Theorem Let B be a Z 0 -graded k(q)-algebra generated by elements x 1,..., x n of degree one, with a PBW basis of polynomial type, having a R-form B with a semiclassical limit A polynomial on the images of the x i. Then there is an R-subalgebra S of B e such that S R k(q) = B e S R k = P(A) Matthew Towers (University of Kent) arxiv / 15

26 Deformation theorem Theorem Let B be a Z 0 -graded k(q)-algebra generated by elements x 1,..., x n of degree one, with a PBW basis of polynomial type, having a R-form B with a semiclassical limit A polynomial on the images of the x i. Then there is an R-subalgebra S of B e such that S R k(q) = B e S R k = P(A) In this situation we say B e is a q-deformation of P(A), and S is an R-form of B e. Matthew Towers (University of Kent) arxiv / 15

27 Deformation theorem Theorem Let B be a Z 0 -graded k(q)-algebra generated by elements x 1,..., x n of degree one, with a PBW basis of polynomial type, having a R-form B with a semiclassical limit A polynomial on the images of the x i. Then there is an R-subalgebra S of B e such that S R k(q) = B e S R k = P(A) In this situation we say B e is a q-deformation of P(A), and S is an R-form of B e. Algebras B satisfying these conditions will be called P-algebras. They include quantum affine spaces and coordinate rings of quantum matrices. Matthew Towers (University of Kent) arxiv / 15

28 Cohomology Suppose M is a Λ-module, and is a free resolution of M. P 1 P 0 M 0 Matthew Towers (University of Kent) arxiv / 15

29 Cohomology Suppose M is a Λ-module, and P 1 P 0 M 0 is a free resolution of M. Then hom Λ (P, M) is a cocomplex, and Ext Λ (M, M) = H (hom Λ (P, M)). Matthew Towers (University of Kent) arxiv / 15

30 Cohomology Suppose M is a Λ-module, and P 1 P 0 M 0 is a free resolution of M. Then hom Λ (P, M) is a cocomplex, and Ext Λ (M, M) = H (hom Λ (P, M)). HH(B) is the special case Λ = B e = B B op, M = B. Matthew Towers (University of Kent) arxiv / 15

31 Cohomology Suppose M is a Λ-module, and P 1 P 0 M 0 is a free resolution of M. Then hom Λ (P, M) is a cocomplex, and Ext Λ (M, M) = H (hom Λ (P, M)). HH(B) is the special case Λ = B e = B B op, M = B. HP(A) is the special case Λ = P(A), M = A. Matthew Towers (University of Kent) arxiv / 15

32 Koszul algebras A Koszul module over a Z 0 -graded connected algebra Λ is one admitting a graded free resolution Λ[ 2] n 2 Λ[ 1] n 1 Λ n 0 M 0 Matthew Towers (University of Kent) arxiv / 15

33 Koszul algebras A Koszul module over a Z 0 -graded connected algebra Λ is one admitting a graded free resolution Λ[ 2] n 2 Λ[ 1] n 1 Λ n 0 M 0 Λ is called Koszul if the trivial Λ-module is Koszul. Matthew Towers (University of Kent) arxiv / 15

34 Koszul algebras A Koszul module over a Z 0 -graded connected algebra Λ is one admitting a graded free resolution Λ[ 2] n 2 Λ[ 1] n 1 Λ n 0 M 0 Λ is called Koszul if the trivial Λ-module is Koszul. Such an algebra has the form Λ = T (V )/(R), R V V, and Ext Λ (k, k) = T (V )/(R ) = Λ!. Matthew Towers (University of Kent) arxiv / 15

35 Cohomology: Poisson vs. Hochschild Recall that our goal was to compare HH(B) = Ext B e (B, B) and HP(A) = Ext P(A) (A, A) where B is a P-algebra with semiclassical limit A. Matthew Towers (University of Kent) arxiv / 15

36 Cohomology: Poisson vs. Hochschild Recall that our goal was to compare HH(B) = Ext B e (B, B) and HP(A) = Ext P(A) (A, A) where B is a P-algebra with semiclassical limit A. P-algebras B are Koszul: B B! is a DGA whose homology is HH(B). Matthew Towers (University of Kent) arxiv / 15

37 Cohomology: Poisson vs. Hochschild Recall that our goal was to compare HH(B) = Ext B e (B, B) and HP(A) = Ext P(A) (A, A) where B is a P-algebra with semiclassical limit A. P-algebras B are Koszul: B B! is a DGA whose homology is HH(B). Equally, P(A) is a Koszul algebra and A is a Koszul P(A)-module, and A A! is a DGA whose homology is HP(A). Matthew Towers (University of Kent) arxiv / 15

38 Deformation theorem II Theorem If B is a P-algebra with semiclassical limit A then B B! has a R-sub-DGA C such that as DGAs. C R k(q) = B B! C R k = A A! Matthew Towers (University of Kent) arxiv / 15

39 Deformation theorem II Theorem If B is a P-algebra with semiclassical limit A then B B! has a R-sub-DGA C such that as DGAs. Corollary C R k(q) = B B! C R k = A A! If H (C) has no (q 1)-torsion then HH (B) is a q-deformation of HP (A). Matthew Towers (University of Kent) arxiv / 15