Is A 1 of type B 2 ? Noncommutative algebraic geometry Shanghai - - PowerPoint PPT Presentation

Is A1 of type B2?

Noncommutative algebraic geometry Shanghai 2011

St´ ephane Launois (University of Kent)

Plan

• Construction of quantum analogues of the first Weyl algebra.
• Quantum Dixmier conjecture.
• Automorphisms of quantum algebras.

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The first Weyl algebra A1(C) A1(C) is the C-algebra generated by x and ∂ with ∂x − x∂ = 1.

• A1(C) is a Noetherian domain.
• GKdim(A1(C)) = 2.
• A1(C) is a simple algebra.
• Z(A1(C)) = C.
• U(A1(C)) = C∗.
• Frac(A1(C)) = D1(C) the Weyl skew-field.
• Conjecture (Dixmier):

Every endomorphism of A1(C) is an automorphism.

3

Question Can we quantise the first Weyl algebra? That is, can we find a family of algebras having the above prop- erties (at least generically)? Remark: The quantum Weyl algebra is generated by x and y subject to yx − qxy = 1. It is not simple when q is not a root of unity.

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Objective of the first section Construct a family (Rq)q∈C∗ of C-algebras such that R1 = A1(C) and, if q is not a root of unity:

• Rq is a Noetherian domain.
• GKdim(Rq) = 2.
• Rq is a simple algebra.
• Z(Rq) = C.
• U(Rq) = C∗.
• Frac(Rq) = Frac(Cq[x±1, y±1]) the quantum Weyl skew-field.

(Cq[x±1, y±1] is the C-algebra generated by x±1 and y±1 with yx = qxy.)

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Plan Dixmier Let n be a finite dimensional nilpotent Lie algebra over

C.

Then P ∈ Prim(U(n)) if and only if U(n)

P

is isomorphic to An(C) for a certain n. Idea: Study the simple factor algebras of Gelfand-Kirillov dimen- sion 2 of the positive part U+

q (g) of the quantised enveloping

algebra of a finite dimensional complex simple Lie algebra g.

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Basics on U+

q (g)

We now assume that q ∈ C∗ is not a root of unity. U+

q (g) is the C-algebra generated by n = rk(g) indeterminates

Ei subject to the quantum Serre relations:

1−aij

• k=0

(−1)k

• 1 − aij

k

• i

E1−aij−k

i

EjEk

i = 0 (i = j)

This algebra can be presented as an Ore extension over C and so this is a Noetherian domain. Goodearl-Letzter Prime ideals of U+

q (g) are completely prime.

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Case where g is of type A2 U+

q (A2) is the C-algebra generated by two indeterminates E1 and

E2 subject to the following relations: E2

1E2 − (q2 + q−2)E1E2E1 + E2E2 1 = 0

E2

2E1 − (q2 + q−2)E2E1E2 + E1E2 2 = 0

Normal elements of U+

q (A2).

Set E3 := E1E2 − q2E2E1 and E3 := E1E2 − q−2E2E1. Then E3 and E3 are normal in U+

q (A2).

Centre of U+

q (A2). (Alev-Dumas)

Z(U+

q (A2)) is a polynomial algebra in one variable Z(U+ q (A2)) =

C[Ω], where Ω denotes the quantum Casimir, that is:

Ω = E3E3.

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Prime spectrum of U+

q (A2). (Malliavin) << E1, E2 − β >> << E1, E2 >>

• << E1 − α, E2 >>

< E1 >

• < E2 >
• << E3 >>
• << Ω − γ >>

<< E3 >>

• < 0 >

with αβγ = 0.

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Properties of the simple factors algebras in U+

q (A2)

Kirkman-Small For all γ ∈ C∗, Aγ := U+

q (A2)

<Ω−γ> is a Noetherian

domain of Gelfand-Kirillov dimension 2. Further

• Aγ is a simple algebra.
• Z(Aγ) = C.
• Frac(Aγ) ≃ Frac(Cq2[x±1, y±1]).
• U(Aγ) C∗.

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Case where g is of type B2 U+

q (B2) is the C-algebra generated by two indeterminates e1 and

e2 subject to the quantum Serre relations: e2

1e2 − (q2 + q−2)e1e2e1 + e2e2 1 = 0

and e3

2e1 − [3]qe2 2e1e2 + [3]qe2e1e2 2 − e1e3 2 = 0

where [3]q = q2 + 1 + q−2. We set e3 = e1e2 − q2e2e1 z = e2e3 − q2e3e2 e3 = e1e2 − q−2e2e1 Note that e1e3 = q−2e3e1 and z is central. The monomials (ziej

3ek 1el 2)(i,j,k,l)∈N4 form a PBW-basis of U+ q (B2).

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Centre of U+

q (B2)

We set z′ = (1 − q−4)(1 − q−2)e3e1e2 + q−4(1 − q−2)e2

3

+(1 − q−4)ze1 Caldero The centre Z(U+

q (B2)) of U+ q (B2) is a polynomial ring

in two variables. More precisely, we have Z(U+

q (B2)) = C[z, z′].

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The prime and primitive spectra of R := U+

q (B2)

The torus H := (C∗)2 acts by automorphisms on R via : (h1, h2).ei = hiei ∀i ∈ {1, 2}. We denote by H-Spec(R) the set of those prime ideals in R which are H-invariant. Gorelik R has exactly 8 H-primes. < e1, e2 >

• < e1 >
• < e2 >
• < e3 >
• < e3 >
• < z >
• < z′ >
• < 0 >

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Stratification Theorem (Goodearl-Letzter) If J ∈ H-Spec(R), then we set SpecJ(R) := {P ∈ Spec(R) |

• h∈H

h.P = J}.

• 1. Spec(R) =
• J∈H-Spec(R)

SpecJ(R)

• 2. For all J ∈ H-Spec(R), SpecJ(R) is homeomorphic to the

prime spectrum of a (commutative) Laurent polynomial ring

• ver C.
• 3. The primitive ideals of R are precisely the primes maximal in

their H-strata.

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H-strata coming from U+

q (A2)

Note that U+

q (B2)

z

≃ U+

q (A2). So the H-strata corresponding to

those H-primes that contain z are:

e1, e2 − β

• e1, e2
• e1 − α, e2
• e1
• e2
• e3
• z, z′ − γ

e3

• z

where α, β, γ ∈ C∗

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H-strata not coming from U+

q (A2)

It remains to deal with two H-strata, those corresponding to 0 and z′. z − α, z′ − β z − α, z′ I z′ , where α, β ∈ C∗. Sketch of proof. Spec0(R) ≃ Spec(R[z±1, z′±1]). Next, stratifi- cation theory of Goodearl and Letzter shows that Spec0(R) ≃ Spec

• Z
• R[z±1, z′±1]
• , that is:

Spec0(R) ≃ Spec(C[z±1, z′±1]). Finally, be careful with the localisations!

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Picture Spec(R[z±1, z′±1]) P[z±1, z′±1] = Q Spec0(R)

≃

• Spec(C[z±1, z′±1])
• ≃
• P
• Q ∩ C[z±1, z′±1]
• 17

The 8 H-strata of Spec(R) e1, e2 − β

• e1, e2
• e1 − α, e2
• e1
• e2
• e3
• z, z′ − β

z − α, z′ − β z − α, z′ e3

• z
• I

z′

• where α, β ∈ C∗.

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Simple factor algebras of Gelfand-Kirillov dimension 2 For all (α, β) ∈ C2 \ {(0, 0)}, we set Rα,β := R z − α, z′ − β. Rα,β is the C-algebra generated by three indeterminates f1, f2, f3 with f1f3 = q−2f3f1, f2f3 = q2f3f2 + α, f2f1 = q−2f1f2 − q−2f3 and f2

3 + c1f3f1f2 + αc2f1 + βc3 = 0,

where c1 = q4 − 1, c2 = q2(q2 + 1) and c3 =

q6 1−q2.

Observe that, for α = 1, β = 0 and q = 1, we retrieve A1(C).

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Properties of Rα,β

• Rα,β is a Noetherian domain.
• GKdim(Rα,β) = 2.
• Rα,β is a simple algebra.
• F := (fi

1fj 2)i,j∈N ∪ (f3fi 1fj 2)i,j∈N is a linear basis of Rα,β.

• U(Rα,β) = C∗ iff α = 0 and β = 0.
• Z(Rα,β) = C.
• Frac(Rα,β) = Frac(Cq2[x±1, y±1]) the quantum Weyl skew-field.

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Quantum Dixmier Conjecture (1) Dixmier Conjecture Every endomorphism of A1(C) is an auto- morphism. R1,0 is the C-algebra generated by three indeterminates f1, f2, f3 with f1f3 = q−2f3f1, f2f3 = q2f3f2 + 1, f2f1 = q−2f1f2 − q−2f3 and f2

3 + (q4 − 1)f3f1f2 + q2(q2 + 1)f1 = 0,

Observe that, for q = 1, we retrieve A1(C).

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Quantum Dixmier Conjecture (2) Theorem Every endomorphism of R1,0 is an automorphism. Proof: Let σ an endomorphism of R1,0. First, use the fact that U(R1,0) = {λhk|λ ∈ C∗, k ∈ Z}, where h := (q2 − 1)f3f2 + 1. This leads to σ(h) = λh or σ(h) = λh−1. Then one concludes using the fact that {r ∈ R1,0|hrh−1 = q2r} = C[h±1]f2 and {r ∈ R1,0|hrh−1 = q−2r} = C[h±1]f3

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Automorphism group of U+

q (A2)

Recall that U+

q (A2) is the C-algebra generated by E1 and E2

with E2

1E2 − (q2 + q−2)E1E2E1 + E2E2 1 = 0

E2

2E1 − (q2 + q−2)E2E1E2 + E1E2 2 = 0

(C∗)2 acts by automorphism on U+

q (A2) via

(h1, h2) · (Ei) = hiEi ∀i ∈ {1, 2}. Denote by w the diagram automorphism defined by w(E1) = E2 and w(E2) = E1. Alev-Dumas Aut(U+

q (A2)) ≃ (C∗)2 ⋊ w.

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Conjecture (Andruskiewitsch-Dumas) Aut(U+

q (g)) ≃ (C∗)rk(g) ⋊ Autdiagr(g)

True when g is of type: A2 (Alev-Dumas); B2 (L); A3 (L-Lopes). It is open otherwise.

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Automorphism group of U+

q (B2): preliminary

Andruskiewitsch-Dumas The sub-group Autz(R) of Aut(R) of those automorphisms that fix < z > is isomorphic to the torus (C∗)2. Sketch of proof. Recall that R z ≃ U+

q (A2)

So every automorphism σ ∈ Autz(R) induces an automorphism

• σ of U+

q (A2). Andruskiewitsch and Dumas have proved that the

homomorphism φ : Autz(R) → Aut(U+

q (A2)) sending σ to

σ is injective. Moreover, using the description of Aut(U+

q (A2)), they prove that

Im(φ) ≃ (C∗)2.

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Automorphism group of U+

q (B2) (1)

Let σ be an automorphism of R. We want to prove that σ(< z >) =< z > . Let (βi)i∈N be any family of pairewise distinct nonzero complex

• numbers. Then

z =

• i∈N

P0,βi, where Pα,β := z − α, z′ − β. Hence we get σ (z) =

• i∈N

σ(P0,βi).

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Automorphism group of U+

q (B2) (2)

Since P0,βi is an height 2 maximal ideal of R, so is σ(P0,βi). Thus, there exist (γi, δi) = (0, 0) such that σ(P0,βi) = Pγi,δi. Now, comparing the groups of invertible elements of U+

q (B2)

P0,βi = R0,βi and U+

q (B2)

Pγi,δi = Rγi,δi leads to either γi = 0, or δi = 0. Naturally, we can choose the family of scalars (βi)i∈N such that either γi = 0 for all i ∈ N, or δi = 0 for all i ∈ N.

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Automorphism group of U+

q (B2) (3)

Hence, either σ (z) =

• i∈N

σ(P0,βi) =

• i∈N

Pγi,0,

• r

σ (z) =

• i∈N

σ(P0,βi) =

• i∈N

P0,δi, that is, either σ (z) = z′ or σ (z) = z. To conclude, it remains to show that σ (z) = z′ is impossible. In order to do this, we use a graded argument.

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Automorphism group of U+

q (B2) (4)

Let σ be an automorphism of U+

q (B2). Considering the N-grading

• f U+

q (B2) obtained by assigning degree 1 to e1 and e2. As there

are enough q-commutation relations, one can prove: Proposition If x = 0 is homogeneous of degree d then σ(x) = yd + y>d, where yd = 0 is homogeneous of degree d and y>d is a sum of elements of degree > d. Now, recall that z = e2e3 − q2e3e2 and z′ = •e3e1e2 + •e2

3 + •ze1.

Hence deg(z) = 3 and deg(z′) = 4. So we cannot have σ (z) = z′. Hence σ(< z >) =< z >, and we conclude that Aut(U+

q (B2)) ≃ (C∗)2.

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Convention R denote an irreducible root system of rank n. Π = {α1, . . . , αn} is a basis of R.

g denotes the corresponding simple Lie algebra, and W is its

Weyl group. (aij) denotes the Cartan matrix. q ∈ C∗ is not a root of unity.

30

Positive roots Recall that W is the subgroup of O(E) generated by the hyper- plane reflections sα with α ∈ Π. Every element w ∈ W has a reduced decomposition: w = si1 ◦ · · · ◦ sit. This integer t is called the length of w. We set l(w) = t. Observe that 0 ≤ l(w) ≤ N = |R+|. We set β1 = αi1, β2 = si1(αi2), . . . , βt = si1 ◦ · · · ◦ sit−1(αit). It is well known that these are distinct positive roots and that the set {β1, ..., βt} does not depend on the chosen reduced expression

• f w.

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Quantised enveloping algebra Uq(g) Uq(g) is the C-algebra generated by indeterminates E1, . . . , En, F1, . . . , Fn and K±1

1 , . . . , K±1 n

with the following relations KiKj = KjKi KiEjK−1

i

= qaij

i

Ej and KiFjK−1

i

= q−aij

i

Fj EiFj − FjEi = δij Ki − K−1

i

qi − q−1

i

and the quantum Serre relations

1−aij

• k=0

(−1)k

• 1 − aij

k

• i

E1−aij−k

i

EjEk

i = 0 (i = j)

and

1−aij

• k=0

(−1)k

• 1 − aij

k

• i

F 1−aij−k

i

FjF k

i = 0 (i = j).

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Quantised enveloping algebra Uq(g): first properties Uq(g) is a Hopf algebra. Uq(g) is ZR-graded with deg(Ei) = αi, deg(Fi) = −αi and deg(K±1

i

) = 0. We denote by U+

q

the positive part of Uq(g), that is the subal- gebra of Uq(g) generated by E1, ..., En.

33

Braid group action There is a natural action of the relative braid group on Uq(g) via the so-called Lusztig automorphisms. We denote by Ti the Lusztig automorphism associated to αi ∈ Π. Moreover we set: Xβ1 = Ei1, Xβ2 = Ti1(Ei2), . . . , Xβt = Ti1 ◦ · · · ◦ Tit−1(Eit). De Concini-Kac-Procesi, Levendorskii-Soibelman

• 1. Xβi ∈ U+

q .

• 2. Xβi is homogeneous of degree βi.

34

De Concini-Kac-Procesi algebra We denote by Uq[w] the subalgebra of U+

q

generated by the root- vectors Xβ1, Xβ2, ..., Xβt.

• 3. Uq[w] does only depend on w.

4. The monomials Xa1

β1Xa2 β2 · · · Xat βt with a1, . . . , at ∈ N0 form a

basis of Uq[w].

• 5. XβjXβi = q−(βi,βj)XβiXβj +
• λaXai+1

βi+1 · · · Xaj−1 βj−1 (i < j),

where the sum runs through all a such that ai+1βi+1 + · · · + aj−1βj−1 = βi + βj.

• 6. Let w0 be the longest element in W. Then Uq[w0] = U+

q .

35

Conjecture Let w = si1 ◦ · · · ◦ sit. We set Supp(w) := {i1, . . . , it} and d(w) := |Supp(w)|. We can regroup the elements of Supp(w) according to the length

• f the corresponding simple roots:

Supp(w) := Supp(w)1 ⊔ · · · ⊔ Supp(w)k Let di := |Supp(w)i|. Moreover we denote by S(w) the subgroup of Sd1 × · · · × Sdk of those u such that w = su(i1) ◦ · · · ◦ su(it). Conjecture: Aut(Uq[w]) ≃ (C∗)|Supp(w)| ⋊ S(w). (We need to assume that for all i ∈ Supp(w), there exists j ∈ Supp(w) such that aij = 0.)

36

An example Assume that g is of type Am+p−1. Identify W with Sm+p. Set w = (1 2 · · · m + p)m =

• 1

2 · · · p p + 1 · · · m + p m + 1 m + 2 · · · m + p 1 · · · m

• A reduced expression for w is given by

w = (sm · · · s1)(sm+1 · · · s2) · · · (sm+p−1 · · · sp). Then the root-vectors Xβ can be arrange in a matrix as follows:

      

Xβ1 Xβm+1 · · · Xβm(p−1)+1 Xβ2 Xβm+2 · · · Xβm(p−1)+2 . . . . . . . . . Xβm Xβ2m · · · Xβmp

      

37

An example: quantum matrices Recall that g is of type Am+p−1 and w = (1 2 · · · m + p)m. For any 2×2 submatrix

• a

b c d

• , we have the following relations

ab = qba, cd = qdc ac = qca, bd = qdb bc = cb, ad − da = (q − q−1)cb. Hence Uq[w] is isomorphic to the algebra Oq(Mm,p). L-Lenagan Assume m = p. Then Aut(Oq(Mm,p)) ≃ (C∗)m+p−1.

38