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Is A 1 of type B 2 ? 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


  1. Is A 1 of type B 2 ? Noncommutative algebraic geometry Shanghai 2011 St´ ephane Launois (University of Kent)

  2. Plan • Construction of quantum analogues of the first Weyl algebra. • Quantum Dixmier conjecture. • Automorphisms of quantum algebras. 2

  3. The first Weyl algebra A 1 ( C ) A 1 ( C ) is the C -algebra generated by x and ∂ with ∂x − x∂ = 1. • A 1 ( C ) is a Noetherian domain. • GKdim( A 1 ( C )) = 2. • A 1 ( C ) is a simple algebra. • Z ( A 1 ( C )) = C . • U ( A 1 ( C )) = C ∗ . • Frac( A 1 ( C )) = D 1 ( C ) the Weyl skew-field. Every endomorphism of A 1 ( C ) is an • Conjecture (Dixmier): automorphism. 3

  4. 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. 4

  5. Objective of the first section Construct a family ( R q ) q ∈ C ∗ of C -algebras such that R 1 = A 1 ( C ) and, if q is not a root of unity: • R q is a Noetherian domain. • GKdim( R q ) = 2. • R q is a simple algebra. • Z ( R q ) = C . • U ( R q ) = C ∗ . • Frac( R q ) = Frac( C q [ x ± 1 , y ± 1 ]) the quantum Weyl skew-field. ( C q [ x ± 1 , y ± 1 ] is the C -algebra generated by x ± 1 and y ± 1 with yx = qxy .) 5

  6. Plan Dixmier Let n be a finite dimensional nilpotent Lie algebra over Then P ∈ Prim( U ( n )) if and only if U ( n ) C . is isomorphic to P A n ( 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 . 6

  7. 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 E i subject to the quantum Serre relations: � � 1 − a ij � 1 − a ij E 1 − a ij − k ( − 1) k E j E k i = 0 ( i � = j ) i k i k =0 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. 7

  8. Case where g is of type A 2 U + q ( A 2 ) is the C -algebra generated by two indeterminates E 1 and E 2 subject to the following relations: 1 E 2 − ( q 2 + q − 2 ) E 1 E 2 E 1 + E 2 E 2 E 2 1 = 0 2 E 1 − ( q 2 + q − 2 ) E 2 E 1 E 2 + E 1 E 2 E 2 2 = 0 Normal elements of U + q ( A 2 ) . Set E 3 := E 1 E 2 − q 2 E 2 E 1 and E 3 := E 1 E 2 − q − 2 E 2 E 1 . Then E 3 and E 3 are normal in U + q ( A 2 ). Centre of U + q ( A 2 ) . (Alev-Dumas) Z ( U + q ( A 2 )) is a polynomial algebra in one variable Z ( U + q ( A 2 )) = C [Ω], where Ω denotes the quantum Casimir, that is: Ω = E 3 E 3 . 8

  9. Prime spectrum of U + q ( A 2 ) . (Malliavin) << E 1 , E 2 − β >> << E 1 , E 2 >> << E 1 − α, E 2 >> � ������������������������������� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � < E 1 > < E 2 > � �������������������������������������������������������� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � << Ω − γ >> << E 3 >> << E 3 >> � � ������������������������������� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � < 0 > with αβγ � = 0. 9

  10. Properties of the simple factors algebras in U + q ( A 2 ) Kirkman-Small For all γ ∈ C ∗ , A γ := U + q ( A 2 ) < Ω − γ> is a Noetherian domain of Gelfand-Kirillov dimension 2. Further • A γ is a simple algebra. • Z ( A γ ) = C . • Frac( A γ ) ≃ Frac( C q 2 [ x ± 1 , y ± 1 ]). • U ( A γ ) � C ∗ . 10

  11. Case where g is of type B 2 U + q ( B 2 ) is the C -algebra generated by two indeterminates e 1 and e 2 subject to the quantum Serre relations: 1 e 2 − ( q 2 + q − 2 ) e 1 e 2 e 1 + e 2 e 2 e 2 1 = 0 and e 3 2 e 1 − [3] q e 2 2 e 1 e 2 + [3] q e 2 e 1 e 2 2 − e 1 e 3 2 = 0 where [3] q = q 2 + 1 + q − 2 . We set e 1 e 2 − q 2 e 2 e 1 = e 3 e 2 e 3 − q 2 e 3 e 2 = z e 1 e 2 − q − 2 e 2 e 1 = e 3 Note that e 1 e 3 = q − 2 e 3 e 1 and z is central. The monomials ( z i e j 2 ) ( i,j,k,l ) ∈ N 4 form a PBW-basis of U + 3 e k 1 e l q ( B 2 ). 11

  12. Centre of U + q ( B 2 ) We set z ′ (1 − q − 4 )(1 − q − 2 ) e 3 e 1 e 2 + q − 4 (1 − q − 2 ) e 2 = 3 +(1 − q − 4 ) ze 1 Caldero The centre Z ( U + q ( B 2 )) of U + q ( B 2 ) is a polynomial ring in two variables. More precisely, we have Z ( U + q ( B 2 )) = C [ z, z ′ ] . 12

  13. The prime and primitive spectra of R := U + q ( B 2 ) The torus H := ( C ∗ ) 2 acts by automorphisms on R via : ( h 1 , h 2 ) .e i = h i e i ∀ 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. < e 1 , e 2 > � ������������� � � � � � � � � � � � � < e 1 > < e 2 > � ������������������������������� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � < e 3 > < e 3 > � �������������������������������� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � < z ′ > � < z > � � � ������������� � � � � � � � � � � � < 0 > 13

  14. Stratification Theorem (Goodearl-Letzter) If J ∈ H -Spec( R ), then we set � Spec J ( R ) := { P ∈ Spec( R ) | h.P = J } . h ∈H � 1. Spec( R ) = Spec J ( R ) J ∈H - Spec( R ) 2. For all J ∈ H -Spec( R ), Spec J ( R ) is homeomorphic to the prime spectrum of a (commutative) Laurent polynomial ring over C . 3. The primitive ideals of R are precisely the primes maximal in their H -strata. 14

  15. H -strata coming from U + q ( A 2 ) Note that U + q ( B 2 ) ≃ U + q ( A 2 ). So the H -strata corresponding to � z � those H -primes that contain z are: �� e 1 , e 2 − β �� �� e 1 , e 2 �� �� e 1 − α, e 2 �� � � � ����������������� � ������������������ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � e 1 � � e 2 � � ������������������������������������������������ � � � � � � � ������������������ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� z, z ′ − γ �� � � �� e 3 �� �� e 3 �� � � �������������������������������� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � z � where α, β, γ ∈ C ∗ 15

  16. H -strata not coming from U + q ( A 2 ) It remains to deal with two H -strata, those corresponding to � 0 � and � z ′ � . �� z − α, z ′ − β �� �� z − α, z ′ �� , � z ′ � I � 0 � where α, β ∈ C ∗ . Sketch of proof. Spec � 0 � ( R ) ≃ Spec( R [ z ± 1 , z ′± 1 ]). Next, stratifi- cation theory of Goodearl and Letzter shows that Spec � 0 � ( R ) ≃ � � �� R [ z ± 1 , z ′± 1 ] Spec , that is: Z Spec � 0 � ( R ) ≃ Spec( C [ z ± 1 , z ′± 1 ]) . Finally, be careful with the localisations! 16

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