COMPLETED HOPF ALGEBROIDS

Transcription

1 COMPLETED HOPF ALGEBROIDS DOCTORAL PRESENTATION Martina Stojić Faculty of Science Department of Mathematics October 20, 2017

2 2 / INTRODUCTION Weyl algebra Spg q7spgq Deformation of Weyl algebra Problems with Weyl algebra deformations Yetter-Drinfeld module algebra and Hopf algebroid Idea for the solution the thesis 2. THE 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

3 3. INTERNAL HOPF ALGEBROID AND SCALAR EXTENSION Hopf algebroids, motivation and definition Internal bialgebroid of Gabriella Böhm Definition of internal Hopf algebroid Scalar extensions of Lu, Brzeziński and Militaru Internal scalar extension theorem 4. HEISENBERG DOUBLES OF FILTERED HOPF 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

4 4 / EXAMPLES Heisenberg double Upgq 7Upgq Noncommutative phase space Upgq7Ŝpg q Minimal scalar extension Upgq min 7Upgq Reduced Heisenberg double Upgq 7Upgq Minimal algebra O min pgq7upgq of differential operators Algebra OpAutpgqq7Upgq Heisenberg double U q psl 2 q 7U q psl 2 q when q is a root of unity

5 5 / 39 INTRODUCTION Spg q7spgq Ż Spg q op Spg q Ž Spgq7Spg q b? ˆb Spgq7Spg q Spgq deformation Upgq7Ŝpg q Upgq 1. INTRODUCTION Weyl algebra Spg q7spgq Deformation of Weyl algebra Problems with Weyl algebra deformations Yetter-Drinfeld module algebra and Hopf algebroid Idea for the solution the thesis

6 6 / 39 Weyl algebra Spg q7spgq Weyl algebra xx 1,..., x n, B 1,..., B n y L I, where the ideal I is generated by B α x β x β B α δ αβ, α, β P t1,..., nu ring DiffpR n q ř K I 0 p IpxqB I K P N n 0, p I polynomials ( smash product Spg q7spgq, for g T 0 V V vector space Spg q krv s krx 1,..., x n s, Spgq Upgq krb 1,..., B n s Smash product Spg q7spgq is Spg q b Spgq with multiplication f 7D g7e ř f pd p1q Ż gq7d p2q E, where D Ż f Df coproduct pdq ř D p1q b D p2q, D P Upgq, is defined with pdqpf b gq Dpf gq ř D p1q f D p2q g (Leibniz rule) Dual Hopf algebras Spg q krv s and Spgq Upgq product dual to coproduct, unit to counit, etc.

7 7 / 39 Deformation of Weyl algebra Deformation ù noncommutative coordinates first DiffpR n q op pspg q7spgqq op Spgq7Spg q (geometry: algebra of diff. operators that act to the left Ž) now Spgq krˆx 1,..., ˆx n s, Spg q krˆb 1,..., ˆB n s deformation: g becomes a noncommutative Lie algebra, generated by ˆx 1,..., ˆx n, mod the ideal J generated by rˆx α, ˆx β s řσ Cσ αβ ˆx σ, α, β P t1,..., nu Spgq deforms as algebra to Upgq, Spg q deforms as coalgebra to a Hopf algebra dual to Upgq Meljanac, Škoda, Stojić, Lie algebra type noncommutative phase spaces are Hopf algebroids, Lett. Math. Phys. 107:3, (2017) Upgq7Ŝpg q Why is it OK? Comparison with t-deformations.

8 8 / 39 Problems with Weyl algebra deformations Problem with infinite dimensionality of Spgq and Upgq: coproduct : Spg q Ñ Spg q b Spg q deforms to coproduct Spg q Ñ Spg q ˆb Spg q with completion One possible solution: coproduct ˆ : Ŝpg q Ñ Ŝpg q ˆb Ŝpg q and smash product Upgq7Ŝpg q defined out of the action Ŝpg q Ž Upgq Problem: combining b and ˆb we need to work with the action of the deformed differential operators Ŝpg q Upgq 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...

9 Yetter-Drinfeld module algebra and Hopf algebroid Algebra of formal diff. operators around the unit of a Lie group: Diff ω pg, eq J 8 pg, eq7upg L q Upg L q 7Upg L q Diff ω pg, eq J 8 pg, eq co 7Upg R q Upg R q 7Upg R q Noncommutative phase space is the opposite algebra: Upg L q7ŝpg q pŝpg q co 7Upg R qq op Diff ω pg, eq op Ŝpg q J 8 pg, eq Upg L q Completed Heisenberg double Upgq 7Upgq? Corrolary. (Lu) If A is a finite-dimensional Hopf algebra, then the Heisenberg double A 7A is a Hopf algebroid over A. Theorem. (Brzeziński, Militaru) Scalar extension. If A is a braided-commutative Yetter-Drinfeld module algebra over H, then the smash product H7A is a Hopf algebroid over A. 9 / 39

10 10 / 39 Idea for the solution the thesis 1. Category new category which has vector spaces with filtrations and vector spaces with cofiltrations, and Upgq 7Upgq it has to have a monoidal product b which is equal to b when vector spaces are filtered and ˆb 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 Böhm 3. The scalar extension theorem simetrical definition, antipod antiisomorphism, geometry 4. Proof that Upgq is an internal braided-commutative YD-module algebra over Upgq 5. What can be more generally known about A 7A and H7A for dual infinite-dimensional H and A?

11 THE CATEGORY indprovect pindvect, b, kq pvect, b, kq pindprovect, b, k q pprovect, ˆb, kq 2. THE 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

12 12 / 39 Requirements, intuition and strategy Vector spaces with structure and tensor products 1. A, B filtered vector spaces ñ A b B colim A n b B m n,m H, K cofiltered vector spaces ñ H ˆb K lim H k b K l k,l 2. Filtering components A n ãñ 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 b H colim lim A n b H k colim A n ˆb H n k n ñ filtered-cofiltered vector space V colim lim Vn k n k 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.

13 13 / 39 Requirements, intuition and strategy Morphisms that respect this structure 6. Multiplication A b A Ñ A, comultiplication H Ñ H ˆb H and action : H b A Ñ A should be morphisms in this category. 7. Axiom of action: ph ˆb Hq b A Ñ H b A Ñ A i H ˆb ph b Aq 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 αpn n a α ˆB α 1 ˆB α ˆB n αn ˆx β 1 1 ˆx β 2 βn 2 ˆx n the result is always a finite sum ř αpn n 0 a α`ˆb α 1 ˆB α ˆB n αn ˆx β 1 1 ˆx β 2 2 βn ˆx n even though each summand of the infinite sum acts. ñ Infinitness is controlled by interaction of filtrations and cofiltrations. Formalization: morphisms in indprovect.

14 14 / 39 Objects of categories indvect and provect Definition. A functor V : I Ñ V is an ℵ 0 -filtration if I is a small directed category of cofinality of at most ℵ 0 and all connecting morphisms are monomorphisms. Definition. A functor V : I op Ñ V is an ℵ 0 -cofiltration if I is a small directed category of cofinality of at most ℵ 0 and all connecting morphisms are epimorphisms. natural generalizations of standard notions of filtration and decreasing filtration (equivalently, cofiltration) objects of categories Ind s ℵ 0 V and Pro s ℵ 0 V respectively Why ℵ 0? Why monomorphisms and epimorphisms? Definition. Filtered (resp. cofiltered) vector space is a vector space V together with an ℵ 0 -filtration (resp. ℵ 0 -cofiltration) V in Vect such that V colim V (resp. V lim V ) in Vect.

15 15 / 39 Morphisms of categories indvect and provect Definition. Morphism of filtered vector spaces, or a filtered map, from V colim V to W colim W is a linear map f : V Ñ W such that p@i P IqpDj P JqpDf ji : V i Ñ W j qpf ι V i ι W j f ji q. Definition. Morphism of cofiltered vector spaces, or a cofiltered map, from V lim V to W lim W is a linear map f : V Ñ W such that p@j P JqpDi P IqpDf ji : V i Ñ W j qpf ji π V i π W j f q. f V W V W f f ji V i W j V i W j f ji

16 16 / 39 Dual subcategories of Grothendieck categories Theorems. The category indvect is equivalent to the category of strict ind-objects of cofinality of at most ℵ 0 in the category Vect, indvect Ind s ℵ 0 Vect. The category provect is equivalent to the category of strict pro-objects of cofinality of at most ℵ 0 in the category Vect, provect Pro s ℵ 0 Vect. The categories indvect and provect are dual to each other. ø These theorems are not theorems if: we don t have monomorphisms A n ãñ A in filtrations and epimorphisms H H k in cofiltrations. the cofinality is not at most ℵ0. But maybe it still works without ℵ 0 -assumption if we take VectFin instead of Vect.

17 17 / 39 The category indprovect Definition. Filtered-cofiltered vector space is a vector space V together with an ℵ 0 -filtration V in provect such that V colim V in Vect. ø Well def.: Proposition. mono in provect ñ mono in Vect. Definition. Morphism of filtered-cofiltered vector spaces, or a filtered-cofiltered map, is a linear map f : V Ñ W such that p@i P IqpDj P JqpDf ji : V i Ñ W j in provectqpf ι V i ι W j f ji q. Theorem. (Subcategory of Grothendieck s category.) The category indprovect is equivalent to the category of strict ind-pro-objects of cofinality of at most ℵ 0 in the category Vect, indprovect Ind s ℵ 0 Pro s ℵ 0 Vect. ø Later, deeper reasons: Theorem. provect has colimits and Theorem. Filtered colimit in provect is colimit in Vect.

18 18 / 39 Tensor products, formal sums and formal basis Tensor product is easily defined by lifting it to indvect from filtrations: V b W colim V b W to provect from cofiltrations: V ˆb W lim V b W to indprovect from filtrations of cofiltrations: V b W colim V ˆb W. Advantage over abstract ind-pro-objects: concrete categories. Formal sums in provect. Formal basis in provect. Propositions. Categories pindvectfin, b, kq and pprovectfin, ˆb, kq are dual. Morphisms in provect = ones that distribute over formal sums. If td α u α is a filtered basis of V in indvectfin ù dual functionals te α u α comprise a formal basis of V in provectfin. Examples. Upgq 7Upgq, Heisenberg doubles of Hopf algebras filtered by finite-dimensional components,...

19 19 / 39 Commutation of the tensor product and coequalizers Proposition 1. (Coequalizers in provect.) The category provect admits coequalizers. The coequalizers in pprovect, ˆb, kq commute with the monoidal product. Proposition 2. (Complete subspaces and quotients in provect.) Vector subspace is complete if and only if it contains values of all formal sums. Quotient by a complete subspace is a cofiltered vector space and the quotient map is a cofiltered map. Proposition 3. (Coproduct in provect.) The category provect has coproducts. Description of coproduct in provect. Sketch of the proof. Existence: use equivalence with the category Pro s ℵ 0 Vect. Description: use the notion of formal sum, Proposition 2. about formal sums and completions, and Proposition 3. for description of cofiltration on quotients.

20 20 / 39 Theorem 4. (Filtered colimit in provect.) The category provect has colimits. Description of filtered colimit in provect. Sketch of the proof. Complex proof.... Theorem 5. (Existence of coequalizers in indprovect.) The category indprovect admits coequalizers. Sketch of the proof. Use: existence of colimit in provect, quotient maps by complete subspaces in provect, completeness of kernels of maps in provect,... Theorem 6. (Coequalizers commute with b in indprovect.) Coequalizers in pindprovect, b, kq commute with the monoidal product. Sketch of the proof. Complex proof.... Uses: properties of formal sums, description of filtered colimit in provect,... This makes the definition of b A, for internal monoid A, possible.

21 21 / 39 INTERNAL HOPF ALGEBROID AND SCALAR EXTENSION G ˆM G G M FunpG ˆM Gq FunpGq b FunpMq FunpGq H b A H H b A H H A 3. INTERNAL HOPF ALGEBROID AND SCALAR EXTENSION Hopf algebroids, motivation and definition Internal bialgebroid of Gabriella Böhm Definition of internal Hopf algebroid Scalar extensions of Lu, Brzeziński and Militaru Internal scalar extension theorem

22 22 / 39 Hopf algebroids, motivation and definition Functions on a group G = (commutative) Hopf algebra H, functions on a groupoid G = (commutative) Hopf algebroid H. General Hopf algebras and Hopf algebroids = functions on spaces with noncommutative coordinates (with the structure of a group or a groupoid ) = quantum group, quantum groupoid. It is more complicated: Coproduct : H Ñ H b H becomes coproduct : H Ñ H b A H. If A is noncommutative, H b A H is not an algebra hence there is a problem with the definition of multiplicativity of coproduct... Takeuchi product. Unit η : k Ñ H becomes left unit α: A Ñ H and right unit β : A Ñ H. Much more complexity in axioms: left bialgebroid, right bialgebroid and antipode. Lu, Day & Street,..., Böhm.

23 23 / 39 Internal bialgebroid of Gabriella Böhm Modern definition of a Hopf algebroid: Gabriella Böhm, Handbook of Algebra. bialgebra H over k (left) bialgebroid H over A µ: H b H Ñ H µ: H b H Ñ H η : k Ñ H α: A Ñ H, β : A op Ñ H : H Ñ H b H : H Ñ H b A H ɛ: H Ñ k ɛ: H Ñ A Hopf algebra over k Hopf algebroid H over A ph, µ, η,, ɛq H L ph, µ, α L, β L, L, ɛ L q together with H R ph, µ, α R, β R, R, ɛ R q S : H op Ñ H S : H op Ñ H In a symmetric monoidal categoy with coequalizers that commute with the monoidal product Gabriella Böhm defines an internal bialgebroid. Complication: Takeuchi product she replaces it with actions ρ and λ. No elements only diagrams.

24 24 / 39 Definition of internal Hopf algebroid First work out Gabi s definition of internal left bialgebroid and internal right bialgebroid, with actions ρ and λ. ph b L Hq b ph b Hq ρ ph b L Hq H b ph b Hq bid idbπ H b ph b L Hq λ ph b L Hq b ph b Hq ρ ph b L Hq Definition. Internal Hopf algebroid ph L, H R, Sq in a symmetric monoidal category which admits coequalizers that commute with the monoidal product. These properties of coproducts L and R were needed for it to be well defined. Proposition. L is an R-bimodule map, with R H R. R too, L H L.

25 25 / 39 Scalar extensions of Lu, Brzeziński and Militaru Theorem. (Lu) Quantum transformation groupoid. If A is a braided-commutative module algebra over Drinfeld double DpHq of a finite-dimensional Hopf algebra H, then H7A is a Hopf algebroid over A. About the proof. Lu s definition of Hopf algebroid. Finite dimensionality. Uses canonical elements: ta s u basis of A, tx s u dual basis of A, βpaq ř t x ts 1 px s q b a s aa t. Theorem. (Brzeziński, Militaru) Scalar extension. If A is a braided-commutative YD-module algebra over Hopf algebra H, then H7A is a Hopf algebroid over A. About the proof. Lu s definition of Hopf algebroid. Omission in the proof: it is not proved antipode is an antihomomorphism. It is not clear to me whether the proof can be completed without the additional assumption of bijectivity of antipode S : A op Ñ A.

26 Internal scalar extension theorem Theorem. (Internal scalar extension in indprovect.) If A is a braided-commutative YD-module algebra over Hopf algebra H with bijective antipode, then H7A is a Hopf algebroid over A. Sketch of the proof. Geometry: for Upg R q J 8 pg, eq co : H, H7Upg R q Upg L q7h Diff ω pg, eq. H L L7H is a pretty left bialgebroid over L, Upg L q7h, H R H7R is a pretty right bialgebroid over R, H7Upg R q, isomorphism of algebras Φ: H L Ñ H R, formula is extracted from the geometrical example, antipode S : H Ñ H is an antihomomorphism (complete the proof of B-M: hard, use isomorphism Φ: easy). Abstract Sweedler notation. The proof works for any symmetric monoidal category with the needed coequalizers property. 26 / 39

27 HEISENBERG DOUBLES OF FILTERED HOPF ALGEBRAS AND GENERALIZATIONS Here we study pairings A b H Ñ k which are non-degenerate in variable in H, hence by which H ãñ A. The question is: When is A over H a braided-commutative YD-module algebra in indprovect, and hence H7A a Hopf algebroid over A? Is Upgq over Upgq a braided-commutative YD-module algebra in indprovect? 4. HEISENBERG DOUBLES OF FILTERED HOPF 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 27 / 39

28 Canonical elements and representations Action of H b A on A (from the right). S 1 : H b A Ñ HompA, Aq S 1 p ř λ h λ b a λ q: b ÞÑ ř λ xb, h λya λ Canonical element K is such that Ŝ 1 K id HompA,Aq. K : HompA, Aq Ñ A ˆb A Kpφq ř α e α7φpx α q Action H7A on A (from the right). T 1 : H b A Ñ HompA, Aq T 1 p ř λ h λ b a λ q: b ÞÑ ř λ pb đ h λqa λ Canonical element L is such that ˆT 1 L id HompA,Aq. L: HompA, Aq Ñ A ˆb A Lpφq ř α,β eβs 1 pe α q7x α φpx β q For Lpφq, Hopf algebra A has to have a bijective antipode S. 28 / 39

29 Theorem about Yetter-Drinfeld module algebra Theorem 1. (About Yetter-Drinfeld module algebra.) Let A and H be in Hopf pairing in indprovect which is non-degenerated in variable in H. Assume T 2 is injective, T 2 : H b H b A Ñ HompA b A, Aq T 2 p ř λ h λ b h 1 λ b a λq: b b b 1 ÞÑ ř λ pb đ h λqpb 1 đ h 1 λ qa λ Then A is over H a braided-commutative YD-module algebra (with action defined from pairing) if and only if there exists a morphism ρ: A Ñ H b A such that x đ ρpaq ax. Sketch of the proof. Then T 1 is also injective. By acting with the left and the right side of axiom equation on an element of A b A, or A, we prove the equation. To prove that ρ is a coaction, we use T 2, and to prove the YD-condition and that A is an algebra, we use T 1. For example, YD-condition is in H b A, ř hp2q pa đ h p1q q r 1s b pa đ h p1q q r0s ř a r 1s h p1q b pa r0s đ h p2q q. 29 / 39

30 30 / 39 Theorem with canonical elements for A in indvectfin Kpφq ř α e α b φpx α q Lpφq ř α,β e βs 1 pe α q b x α φpx β q For φ a : x ÞÑ ax, we know x ˆđ Lpφ a q ax. Put Lupaq : Lpφ a q. Is T 2 injective and when is LupAq Ď A 7A? Theorem 2. (Heisenberg double of A from indvectfin.) Let Hopf algebra A in indvectfin have a bijective antipode. Then A is over A a braided-commutative YD-module algebra (with action defined from pairing) if and only if the adjoint orbits of A are finite-dimensional. Sketch of the proof. Ŝ 1 K id and ˆT 1 L id. Propositions. S 0 injective ñ S 1, S 2 injective. Similarly, Ŝ 1, Ŝ 2 are injective. Ŝ 1 L: φ a ÞÑ ad S 1 paq, for φ a : x ÞÑ ax. It follows: Ŝ 1 bijective. L, M bijective, hence T 1, T 2 injective. From ad, adjoint orbits.

31 31 / 39 Theorem with canonical elements for A in indvectfin Theorem 3. (For A in indvectfin and H in provect.) Let A with a bijective antipode in indvectfin and H in provect be in Hopf pairing in indprovect which is non-degenerate in variable in H. Then A is over H a braided-commutative YD-module algebra (with action defined from pairing) if and only if LupAq Ď H7A. Consequence. If LupAq Ď A 7A, then there exists the smallest Hopf subalgebra A min Ď A which has all functionals needed for LupAq Ď A min 7A. The theorem is then true for all H such that A min Ď H. For Upgq, we found this minimal subalgebra explicitly. Examples Upgq min 7Upgq, Upgq 7Upgq, Upgq 7Upgq. For U q psl 2 q, we didn t, but maybe it is possible with better knowledge of q-binomial coefficients. U q psl 2 q 7U q psl 2 q

32 32 / 39 Theorem with anihilators for A in indvect, H in provect Theorem 4. (For A in indvect and H in provect.) Let A in indvect and H in provect be in Hopf pairing in indprovect which is non-degenerate in variable in H. Assume that A satisfies A paq a b 1 P A n 1 b A for a P A n and A 0 k. Then A is over H a braided-commutative YD-module algebra (with action induced by pairing) if and only if there exists a morphism ρ: A Ñ H7A such that x đ ρpaq ax. Sketch of the proof. Prove T 2 is injective, but without canonical elements. For t P H ˆb H ˆb A n, t 0, let pk, lq be minimal such that there exists d P A k ˆb A l for which S 2 ptqpdq 0. A k ˆb A l ˆb H ˆb H ˆb A n pt 2 q A m A k ˆb A l ˆb H{ AnihpA k q ˆb H{ AnihpA l q ˆb A n pt 2 S 2 q A m

33 33 / 39 EXAMPLES Upgq min 7Upgq Upgq 7Upgq Upgq 7Upgq O min pgq7upgq Upgq7Ŝpg q U q psl 2 q 7U q psl 2 q 5. EXAMPLES Heisenberg double Upgq 7Upgq Noncommutative phase space Upgq7Ŝpg q Minimal scalar extension Upgq min 7Upgq Reduced Heisenberg double Upgq 7Upgq Minimal algebra O min pgq7upgq of differential operators Algebra OpAutpgqq7Upgq Heisenberg double U q psl 2 q 7U q psl 2 q when q is a root of unity

34 Heisenberg double Upgq 7Upgq Proposition. Adjoint orbits are finite-dimensional. By Theorem 2 & The Internal Scalar Extension Theorem: Upgq 7Upgq is a Hopf algebroid over Upgq in indprovect. It is the algebra of formal differential operators around the unit of a Lie group G: Upg R q 7Upg R q J 8 pg, eq co 7Upg R q Diff ω pg, eq H7Upg R q Upg L q7h Diff ω pg, eq Let X 1,..., X n be a basis of g L, let Y 1,..., Y n in g R be such that py α q e px α q e. Then, for L Upg L q and R Upg R q, we have α L px α q X α α R py α q Y α β L px α q Y α ` řβ Cβ αβ β R py α q X α ř β Oβ α7y β SpX α q Y α, Spf q Sf 34 / 39

35 35 / 39 Noncommutative phase space Upgq7Ŝpg q It is the algebra opposite to algebra Diff ω pg, eq: Upg L q7ŝpg q pŝpg q co 7Upg R qq op Diff ω pg, eq op Upg L q7ŝpg q Ŝpg q7upg R q Let ˆx 1,..., ˆx n be a basis of g L, let ŷ 1,..., ŷ n be in g R such that pŷ α q e pˆx α q e. These are noncommutative coordinates, and Ŝpg q krrˆb 1,..., ˆB n ss has a coproduct dual to product on Upg L q L. With Upg R q R, α L pˆx α q ˆx α α R pŷ α q ŷ α β L pˆx α q ŷ α ř β ˆx β7oα β β R pŷ α q ˆx α řβ Cβ αβ Spŷ α q ˆx α SpˆB α q ˆB α ˆB α ˆx β δ αβ, ŷ α ˆx β ˆx β ˆx α ˆB α1 α s ˆB α1 ˆB αs ` def. Matrix O exp C, where Cβ α ř σ Cα βσˆb σ.

36 Examples Upgq min 7Upgq and Upgq 7Upgq Proposition. Minimal Hopf subalgebra Upgq min Ď Upgq for which LupUpgqq Ď Upgq min 7Upgq is generated with Ū α β, U α β, α, β P t1,..., nu, which satisfy: ř σ U α σ Ū σ β δα β ř σ Ū α σ U σ β pu α β q ř σ U α σ b U σ β, pū α β q ř k Ū σ β b Ū α σ ɛpu α β q δα β ɛpū α β q SpU α β q Ū α β, SpŪ α β q U α β xx γ, U α β y Cα γβ řσ,τ Cα στ U σ β U τ γ ř ρ U α ρ C ρ βγ, ř σ,τ Cα στ Ū σ β Ū τ γ ř ρ Ū α ρ C ρ βγ Generators were found by computing LupX 1 q,..., LupX n q: LupX α q ř β Ū β α7x β... Ū exp C n exp C 1, p C σ q α β Cα βσ e X σ. It follows: Upgq min 7Upgq ãñ Upgq 7Upgq ãñ Upgq 7Upgq are HA. 36 / 39

37 37 / 39 Algebras O min pgq7upgq and OpAutpgqq7Upgq All formulas for components of matrices U, Ū are true for components of matrices O Ad, Ō O 1. Ad Ad G GLpgq OpGq OpGLpgqq Autpgq OpAutpgqq Proposition. Algebras O min pgq7upgq and OpAutpgqq7Upgq are Hopf algebroids over Upgq. Sketch of the proof. Algebraicly using generators and relations. Upgq 7Upgq OpGq7Upgq LupX α q ř β Ōβ α7x β Upgq min 7Upgq O min pgq7upgq OpAutpgqq7Upgq

38 38 / 39 Example U q psl 2 q 7U q psl 2 q when q is a root of unity Proposition. Adjoint orbits are finite-dimensional. Sketch of the proof. We compute ad 1 K pe n F m K r q q 2n`2m E n F m K r ad 1 E pe n F m K r q q 2 2n`2m pq 2r 1qE n`1 F m K r 1` ` q 1 2n`2m`2r pq 2m 1qE n F m 1 K r ` pq q 1 q 2 ` q 1 2n`2m`2r pq 2m 1qE n F m 1 K r 2 pq q 1 q 2 ad 1 E pe n K r q q 2 2n pq 2r 1qE n`1 K r 1. By playing combinatorially with exponents, and using q d 1, we get ad 1 E N F M K R pe n F m K r q 0 when M ą pn ` 1qe or N ą me ` pn ` 1qe 2. Here e is minimal such that q 2e 1. Remark. If q is not a root of unity, this is not true. Maybe it is true more generally, for U q psl n q.

39 Thank you!