Weyl algebra S p g q7 S p g q Deformation of Weyl algebra Problems - PowerPoint PPT Presentation

C OMPLETED H OPF ALGEBROIDS D OCTORAL PRESENTATION Martina Stoji´ c Faculty of Science Department of Mathematics October 20, 2017

I NTRODUCTION T HE CATEGORY indproVect H OPF ALGEBROID AND SCALAR EXTENSION H EISENBERG DOUBLES E XAMPLES 1. I NTRODUCTION Weyl algebra S p g ˚ q7 S p g q Deformation of Weyl algebra Problems with Weyl algebra deformations Yetter-Drinfeld module algebra and Hopf algebroid Idea for the solution – the thesis 2. T HE CATEGORY indproVect Requirements, intuition and strategy Categories indVect and proVect Dual subcategories of Grothendieck’s categories The category indproVect Tensor products, formal sums and formal basis Commutation of the tensor product and coequalizers 2 / 39

I NTRODUCTION T HE CATEGORY indproVect H OPF ALGEBROID AND SCALAR EXTENSION H EISENBERG DOUBLES E XAMPLES 3. I NTERNAL H OPF ALGEBROID AND SCALAR EXTENSION Hopf algebroids, motivation and definition Internal bialgebroid of Gabriella B¨ ohm Definition of internal Hopf algebroid Scalar extensions of Lu, Brzezi´ nski and Militaru Internal scalar extension theorem 4. H EISENBERG DOUBLES OF FILTERED H OPF ALGEBRAS AND GENERALIZATIONS Canonical elements and representations Theorem about Yetter-Drinfeld module algebra Theorem with canonical elements for A in indVectFin Theorem with anihilators for A in indVect and H in proVect 3 / 39

I NTRODUCTION T HE CATEGORY indproVect H OPF ALGEBROID AND SCALAR EXTENSION H EISENBERG DOUBLES E XAMPLES 5. E XAMPLES Heisenberg double U p g q ˚ 7 U p g q Noncommutative phase space U p g q7 ˆ S p g ˚ q Minimal scalar extension U p g q min 7 U p g q Reduced Heisenberg double U p g q ˝ 7 U p g q Minimal algebra O min p G q7 U p g q of differential operators Algebra O p Aut p g qq7 U p g q Heisenberg double U q p sl 2 q ˚ 7 U q p sl 2 q when q is a root of unity 4 / 39

I NTRODUCTION T HE CATEGORY indproVect H OPF ALGEBROID AND SCALAR EXTENSION H EISENBERG DOUBLES E XAMPLES I NTRODUCTION S p g ˚ q7 S p g q Ż S p g ˚ q op b ? ˆ b S p g ˚ q Ž S p g q7 S p g ˚ q S p g q7 S p g ˚ q § S p g q deformation U p g q7 ˆ S p g ˚ q § U p g q 1. I NTRODUCTION Weyl algebra S p g ˚ q7 S p g q Deformation of Weyl algebra Problems with Weyl algebra deformations Yetter-Drinfeld module algebra and Hopf algebroid Idea for the solution – the thesis 5 / 39

I NTRODUCTION T HE CATEGORY indproVect H OPF ALGEBROID AND SCALAR EXTENSION H EISENBERG DOUBLES E XAMPLES Weyl algebra S p g ˚ q7 S p g q Weyl algebra L – x x 1 , . . . , x n , B 1 , . . . , B n y I , where the ideal I is generated by B α x β ´ x β B α ´ δ αβ , α, β P t 1 , . . . , n u � ř K – ring Diff p R n q – I “ 0 p I p x qB I | K P N n ( 0 , p I polynomials – smash product S p g ˚ q7 S p g q , for g “ T 0 V – V vector space S p g ˚ q – k r V ˚ s – k r x 1 , . . . , x n s , S p g q “ U p g q – k rB 1 , . . . , B n s Smash product S p g ˚ q7 S p g q is S p g ˚ q b S p g q with multiplication § f 7 D ¨ g 7 E “ ř f p D p 1 q Ż g q7 D p 2 q E , where D Ż f “ Df § coproduct ∆ p D q “ ř D p 1 q b D p 2 q , D P U p g q , is defined with ∆ p D qp f b g q “ D p f ¨ g q “ ř D p 1 q f ¨ D p 2 q g (Leibniz rule) Dual Hopf algebras S p g ˚ q – k r V ˚ s and S p g q “ U p g q § product dual to coproduct, unit to counit, etc. 6 / 39

I NTRODUCTION T HE CATEGORY indproVect H OPF ALGEBROID AND SCALAR EXTENSION H EISENBERG DOUBLES E XAMPLES Deformation of Weyl algebra Deformation ù noncommutative coordinates § first Diff p R n q op – p S p g ˚ q7 S p g qq op – S p g q7 S p g ˚ q (geometry: algebra of diff. operators that act to the left Ž ) x n s , S p g ˚ q – k r ˆ B 1 , . . . , ˆ § now S p g q – k r ˆ x 1 , . . . , ˆ B n s § deformation: g becomes a noncommutative Lie algebra, generated by ˆ x 1 , . . . , ˆ x n , mod the ideal J generated by σ C σ r ˆ x α , ˆ x β s ´ ř αβ ˆ x σ , α, β P t 1 , . . . , n u § S p g q deforms as algebra to U p g q , S p g ˚ q deforms as coalgebra to a Hopf algebra dual to U p g q Meljanac, ˇ Skoda, Stoji´ c, Lie algebra type noncommutative phase spaces are Hopf algebroids , Lett. Math. Phys. 107:3, 475–503 (2017) § U p g q7 ˆ S p g ˚ q Why is it OK? Comparison with t -deformations. 7 / 39

I NTRODUCTION T HE CATEGORY indproVect H OPF ALGEBROID AND SCALAR EXTENSION H EISENBERG DOUBLES E XAMPLES Problems with Weyl algebra deformations Problem with infinite dimensionality of S p g q and U p g q : § coproduct ∆: S p g ˚ q Ñ S p g ˚ q b S p g ˚ q deforms to coproduct S p g ˚ q Ñ S p g ˚ q ˆ b S p g ˚ q with completion One possible solution: § coproduct ˆ ∆: ˆ S p g ˚ q Ñ ˆ b ˆ S p g ˚ q ˆ S p g ˚ q and smash product U p g q7 ˆ S p g ˚ q defined out of the action ˆ S p g ˚ q Ž U p g q Problem: combining b and ˆ b § we need to work with the ’action’ of the deformed differential operators ˆ S p g ˚ q § U p g q there are no axioms for § there is no definition of a ’completed’ Hopf algebroid Solution ad hoc in the LMP article: § ’action’ without axioms... infinite sums... coordinates... § unstable definition of a ’completed’ Hopf algebroid... 8 / 39

I NTRODUCTION T HE CATEGORY indproVect H OPF ALGEBROID AND SCALAR EXTENSION H EISENBERG DOUBLES E XAMPLES Yetter-Drinfeld module algebra and Hopf algebroid Algebra of formal diff. operators around the unit of a Lie group: Diff ω p G , e q – J 8 p G , e q7 U p g L q – U p g L q ˚ 7 U p g L q Diff ω p G , e q – J 8 p G , e q co 7 U p g R q – U p g R q ˚ 7 U p g R q Noncommutative phase space is the opposite algebra: S p g ˚ q co 7 U p g R qq op – Diff ω p G , e q op U p g L q7 ˆ S p g ˚ q – p ˆ ˆ S p g ˚ q – J 8 p G , e q – U p g L q ˚ ’Completed’ Heisenberg double U p g q ˚ 7 U p g q ? Corrolary. (Lu) § If A is a finite-dimensional Hopf algebra, then the Heisenberg double A ˚ 7 A is a Hopf algebroid over A . Theorem. (Brzezi´ nski, Militaru) Scalar extension. § If A is a braided-commutative Yetter-Drinfeld module algebra over H , then the smash product H 7 A is a Hopf algebroid over A . 9 / 39

I NTRODUCTION T HE CATEGORY indproVect H OPF ALGEBROID AND SCALAR EXTENSION H EISENBERG DOUBLES E XAMPLES Idea for the solution – the thesis 1. Category § new category which has vector spaces with filtrations and vector spaces with cofiltrations, and U p g q ˚ 7 U p g q § it has to have a monoidal product ˜ b which is equal to b when vector spaces are filtered and ˆ b when cofiltered § it has to admit coequalizers and they have to commute with the monoidal product for the definition of ˜ b A to be possible 2. Definition of an internal Hopf algebroid § based on the definition of internal bialgebroid of Gabi B¨ ohm 3. The scalar extension theorem § simetrical definition, antipod antiisomorphism, geometry 4. Proof that U p g q is an internal braided-commutative YD-module algebra over U p g q ˚ 5. What can be more generally known about A ˚ 7 A and H 7 A for dual infinite-dimensional H and A ? 10 / 39

I NTRODUCTION T HE CATEGORY indproVect H OPF ALGEBROID AND SCALAR EXTENSION H EISENBERG DOUBLES E XAMPLES T HE CATEGORY indproVect p indVect , b , k q p indproVect , ˜ p Vect , b , k q b , k q p proVect , ˆ b , k q 2. T HE CATEGORY indproVect Requirements, intuition and strategy Categories indVect and proVect Dual subcategories of Grothendieck’s categories The category indproVect Tensor products, formal sums and formal basis Commutation of the tensor product and coequalizers 11 / 39

I NTRODUCTION T HE CATEGORY indproVect H OPF ALGEBROID AND SCALAR EXTENSION H EISENBERG DOUBLES E XAMPLES Requirements, intuition and strategy Vector spaces with structure and tensor products 1. A , B ’filtered’ vector spaces ñ A b B “ colim n , m A n b B m H , K ’cofiltered’ vector spaces ñ H ˆ b K “ lim k , l H k b K l 2. Filtering components A n ã Ñ A are subspaces, duality ñ cofiltering components H ։ H k are quotients 3. A fin-dim-filtered, H fin-dim-cofiltered ñ A ˜ b H “ A b H ñ let’s try with A ˜ A n ˆ b H “ colim lim k A n b H k “ colim b H n n k V k ñ ’filtered-cofiltered’ vector space V “ colim lim n n 4. Hopefully this is a symmetric monoidal category. 5. Hopefully it admits coequalizers and the monoidal product b commutes with them. ˜ Let’s name these categories: indVect , proVect and indproVect . 12 / 39

I NTRODUCTION T HE CATEGORY indproVect H OPF ALGEBROID AND SCALAR EXTENSION H EISENBERG DOUBLES E XAMPLES Requirements, intuition and strategy Morphisms that respect this structure 6. Multiplication A b A Ñ A , comultiplication H Ñ H ˆ b H and action § : H b A Ñ A should be morphisms in this category. 7. Axiom of action: p H ˆ b H q b A Ñ H b A Ñ A i H ˆ b p H b A q Ñ H b A Ñ A become H ˜ b H ˜ b A Ñ H ˜ b A Ñ A . 8. When cofiltered algebra H ’acts on’ filtered vector space A , ´ ř ¯ B α 1 B α 2 x β 1 x β 2 x β n 0 a α ˆ 1 ˆ 2 ¨ ¨ ¨ ˆ B α n § ˆ 1 ˆ 2 ¨ ¨ ¨ ˆ α P N n n n the result is always a finite sum ¯ ` ˆ B α 1 1 ˆ B α 2 2 ¨ ¨ ¨ ˆ B α n x β 1 x β 2 x β n ˘ ř § ˆ 1 ˆ 2 ¨ ¨ ¨ ˆ 0 a α α P N n n n even though each summand of the infinite sum acts. ñ Infinitness is controlled by interaction of filtrations and cofiltrations. Formalization: morphisms in indproVect . 13 / 39