Enveloping algebras of double Poisson-Ore extensions

Transcription

1 Enveloping algebras of double Poisson-Ore extensions Xingting Wang Department of Mathematics, Temple University, Philadelphia Special Session on Topics Related to the Interplay of Noncommutative Algebra and Geometry, Denton Joint with Jiafeng Lü, Sei-Qwon Oh, Xiaolan Yu and Guangbin Zhuang September 9, 2017

2 Motivations We want to understand the behavior of certain noncommutative algebras arising as universal enveloping algebras of Poisson algebras. Representation theory: irreducible representations, Dixmier-Moeglin equivalence, etc. Homological properties: Artin-Schelter regularity, Calabi-Yau property, rigid dualizing complex, etc. Ring-theoretic properties: prime spectrum, noetherianess, domain, PBW deformation, etc.

3 A special family of Poisson algebras We will be focusing on universal enveloping algebras of certain Poisson algebras, which can be obtained by iterated double Poisson-Ore extensions of the polynomial algebra with one variable.

4 Outlines Part I: Basic definitions and notations. Part II: An example. Part III: General results.

5 Part I Basic definitions and notations. k denotes a base field. R denotes a (commutative) Poisson algebra. PMod R denotes the category of (left) Poisson modules over R. R e denotes the universal enveloping algebra of R such that Mod R e PMod R.

6 Part I Definition (in honor of Siméon Denis Poisson) A Poisson algebra is a commutative algebra R with a Lie bracket {, } such that {ab, c} = a{b, c} + {a, c}b for all a, b, c R. A graded Poisson algebra can be defined in a similar way.

7 Part I Definition A left Poisson R-module M is a left R-module ( denoting the R-module structure on M) together with a bilinear map such that {, } M : R M M {{a, b} R, m} M = {a, {b, m} M } M {b, {a, m} M } M, {ab, m} M = a {b, m} M + b {a, m} M, {a, b m} M = {a, b} R m + b {a, m} M, for any a, b R and m M.

8 Part I There are three equivalent ways to define the universal enveloping algebra of a Poisson algebra. explicit generators and relations. smash product. Lie-Rinehart algebra. Definition (Oh, 1999) Let V = R R with two inclusions m, h : R V. R e is the tensor algebra T V modulo the following relations: m rs = m r m s, m 1 = 1, h {r,s} = h r h s h s h r = [h r, h s ], h rs = m r h s + m s h r, m {r,s} = h r m s m s h r = [h r, m s ], for all r, s R. We still write m, h : R R e.

9 Part I Let R be a Poisson k-algebra with Poisson bracket {, } R and let R[y 1, y 2 ] be the commutative polynomial R-algebra. Set Q = {q 11, q 12 } k, w = {w 1, w 2, w 0 } R, α11 (a) α α : R M 2 2 (R), α(a) = 12 (a), α 21 (a) α 22 (a) ν1 (a) ν : R M 2 1 (R), ν(a) =, ν 2 (a) y1 y = M 2 1 (R[y 1, y 2 ]). y 2

10 Part I Theorem (Lou-Oh-Wang, 2017) We have R[y 1, y 2 ] is a Poisson algebra with Poisson bracket {a, b} = {a, b} R, {y 2, y 1 } = q 11 y q 12 y 1 y 2 + w 1 y 1 + w 2 y 2 + w 0, {y, a} = α(a)y + ν(a) for all a, b R if and only if the DE-data {Q, α, ν, w} satisfies the following conditions (i)-(v). (i) α(ab) = aα(b) + bα(a). (ii) ν(ab) = aν(b) + bν(a). (iii) α({a, b}) = {α(a), b} + {a, α(b)} + [α(a), α(b)]. (iv) ν({a, b}) = {ν(a), b} + {a, ν(b)} + α(a)ν(b) α(b)ν(a). (v) {y 2, {y 1, a}} + {y 1, {a, y 2 }} + {a, {y 2, y 1 }} = 0.

11 Part I Definition (LOW, 2017) The Poisson algebra R[y 1, y 2 ] with Poisson bracket given above is called a double Poisson-Ore extension with DE-data {Q, α, ν, w} and is denoted by R[y 1, y 2 ; α, ν] p.

12 Part II We work out an explicit example. Suppose k = k and Char k = 0. Let R = k[x, y, z] be the polynomial algebra with three variables, where the Poisson bracket of R is given by {x, y} = xy + az, {y, z} = zy + bx, {z, x} = xz + cy for any a, b, c k. Then R = k[x][y, z, α, ν] p with DE-data {Q, α, ν, w} given by Q = {q 11 = 0, q 12 = 1}, w = {w 1 = 0, w 2 = 0, w 0 = bx}, x a α : R M 2 2 (R), α(x) =, α(ab) = aα(b) + bα(a) c x ν : R M 2 1 (R), ν = 0.

13 Part II The universal enveloping algebra of R is subject to R e = k x 1, x 2, x 3, y 1, y 2, y 3 [x i, x j ] = 0, [y i, x i ] = 0, [y i, y j ] = x i y j + x j y i + a k y k, [y i, x j ] = x i x j + a k x k, where {i, j, k} is a permutation of {1, 2, 3} and a 1 = a, a 2 = b, a 3 = c.

14 Part II Theorem (Chansuriya-Sasom-Seankarun, 2011) There are precisely five d-dimensional simple Poisson modules over R for every d 1. When d = 1, the 1-dimensional simple Poisson modules are corresponding to the five maximal Poisson ideals of R. J 1 = (x, y, z) J 2 = (x ac, y ab, z + bc) J 3 = (x ac, y + ab, z bc) J 4 = (x + ac, y ab, z bc) J 5 = (x + ac, y + ab, z + bc) Corollary There are precisely five d-dimensional simple modules over R e for every d 1.

15 Part III The right double Ore extension [Zhang-Zhang, 2008] of an associative algebra A is adding two generators y 1, y 2 to A, subject to y1 y1 a = σ(a) + η(a), for all a A, where y 2 y 2 σ11 σ σ = 12 : A M σ 21 σ 2 (A) is an algebra map and 22 η1 η = : A A 2 is a σ-derivation satisfying η 2 η(ab) = σ(a)η(b) + η(a)b, for all a, b A; y 2 y 1 = p 12 y 1 y 2 + p 11 y τ 1y 1 + τ 2 y 2 + τ 0, where P := {p 12, p 11 } is a set of elements of k and τ := {τ 0, τ 1, τ 2 } is a set of elements of A;

16 6 compatible conditions: Part III σ 21 σ 11 + p 11 σ 22 σ 11 = p 11 σ p 2 11σ 12 σ 11 + p 12 σ 11 σ 21 + p 11 p 12 σ 12 σ 21, σ 21 σ 12 + p 12 σ 22 σ 11 = p 11 σ 11 σ 12 + p 11 p 12 σ 12 σ 11 + p 12 σ 11 σ 22 + p 2 12σ 12 σ 21, σ 22 σ 12 = p 11 σ p 12 σ 12 σ 22, σ 20 σ 11 + σ 21 σ 10 + σ 22 σ 11 τ 1 = p 11 (σ 10 σ 11 + σ 11 σ 10 + τ 1 σ 12 σ 11 ) +p 12 (σ 10 σ 21 + σ 11 σ 20 +τ 1 σ 12 σ 21 ) + τ 1 σ 11 + τ 2 σ 21, σ 20 σ 12 + σ 22 σ 10 + σ 22 σ 11 τ 2 = p 11 (σ 10 σ 12 + σ 12 σ 10 + τ 2 σ 12 σ 11 ) +p 12 (σ 10 σ 22 + σ 12 σ 20 + τ 2 σ 12 σ 21 ) + τ 1 σ 12 + τ 2 σ 22, σ 20 σ 10 + σ 22 σ 11 τ 0 = p 11 (σ τ 0 σ 12 σ 11 ) + p 12 (σ 10 σ 20 +τ 0 σ 12 σ 21 )+τ 1 σ 10 + τ 2 σ 20 + τ 0 Id, where σ i0 = η i for i = 1, 2. The data {P, σ, η, τ} collected are called the DE-data. Denote A P [y 1, y 2 ; σ, η, τ].

17 Theorem (Lü-Oh-W-Yu, 2017) Part III For any iterated double Poisson-Ore extension A of R, the universal enveloping algebra A e is an iterated double Ore extension of R e of double length. Proof. Step 1. Let B be any Poisson algebra. There is a natural filtration F i 0 on the Poisson universal enveloping algebra B e such that F 0 = m B, F 1 = m B + h B, F i = F i 1, for i 2 where m, h : B B B B e. We know gr(b e ) = S B (Ω B ), which is the B-symmetric algebra over the Kähler differential module Ω B. We prove that gr(a e ) = gr(r e )[x 1, x 2 ][y 1, y 2 ], where x 1, x 2, y 1, y 2 are certain elements of A e.

18 Part III Continued. Step 2. Consider the double Ore-extension of R e R e [x 1, x 2 ; σ 1, δ 1 ][y 1, y 2 ; σ 2, δ 2 ] where the DE-data {P 1, σ 1, δ 1, τ 1 } of R e [x 1, x 2 ; σ 1, δ 1 ] is P 1 = (0, 1), τ 1 = (0, 0, 0), σ 1 a 0 (a) =, σ 1 α11 (a) + da α (da) = 12 (a), 0 a α 21 (a) α 22 (a) + da δ 1 0 (a) =, δ 1 ν1 (a) (da) = 0 ν 2 (a) for all a R.

19 Part III Continued. and DE-data {P 2, σ 2, δ 2, τ 2 } is P 2 = (0, 1), τ 2 = (2q 11 x 1 + q 12 x 2 + w 1, q 12 x 1 + w 2, x 1 dw 1 + x 2 dw 2 + dw 0 ), σ 2 a 0 (a) =, σ 2 α11 (a) + da α (da) = 12 (a), 0 a α 21 (a) α 22 (a) + da σ 2 x1 0 (x 1 ) =, σ 2 x2 0 (x 0 x 2 ) =, 1 0 x 2 δ 2 α11 (a)x (a) = 1 + α 12 (a)x 2 + ν 1 (a), α 21 (a)x 1 + α 22 (a)x 2 + ν 2 (a) δ 2 x1 dα (da) = 11 (a) + x 2 dα 12 (a) + dν 1 (a), x 1 dα 21 (a) + x 2 dα 22 (a) + dν 2 (a) ( ) δ 2 0 (x 1 ) =, q 11 x1 2 + q 12x 1 x 2 + w 1 x 1 + w 2 x 2 + w 0 ( δ 2 (q11 x (x 2 ) = q 12x 1 x 2 + w 1 x 1 + w 2 x 2 + w 0 ) 0 ), for all a R.

20 Part III Continued. Step 3. We prove via the isomorphism A e = R e [x 1, x 2 ; σ 1, δ 1 ][y 1, y 2 ; σ 2, δ 2 ] gr(a e ) = gr(r e )[x 1, x 2 ][y 1, y 2 ].

21 Part III Theorem (ZZ, 2008&LOWY, 2017) Let R be a connected graded Poisson algebra, and A be a graded iterated double Poisson-Ore extension of R. If R e is Artin-Schelter regular, then A e is also Artin-Schelter regular. Corollary The Poisson universal enveloping algebras of the semiclassical limits of Artin-Schelter regular algebras of dimension 4 that are obtained from double Ore extensions of quantum planes or Jordan planes are Artin-Schelter regular of dimension 8. See [Goodearl-Launois, 2011] for definitions of these semiclassical limits.

22 References P. Chansuriya, N. Sasom, and S. Seankarun, Finite-dimensional simple Poisson modules over certain Poisson algebras, Proceeding of the International Conference on Mathematics and Sciences (ICOMSc). Q. Lou, S.-Q. Oh, and S.-Q. Wang, Double Poisson extensions, preprint, arxiv: J.-F. Lü, S.-Q. Oh, X.-T. Wang, and X.-L. Yu, Universal enveloping algebras of double Poisson Ore-extensions, preprint, arxiv: K.R. Goodearl and S. Launois, The Dixmier-Moeglin equivalence and a Gel fand-kirillov problem for Poisson Polynomial algebras, Bull. Soc. Math. France, (1)139 (2011), J.J. Zhang and J. Zhang, Double Ore extensions, J. Pure Appl. Algebra, (12)212 (2008), S.-Q. Oh, Poisson enveloping algebras, Comm. Algebra, 27 (1999),