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Combinatorial rank of quantum groups of infinite series V.K. Kharchenko, M.L. D´ ıaz Sosa FES-C UNAM MEXICO Groups, Rings, and the Yang–Baxter equation 18–24 of June 2017, Spa, Belgium V.K. Kharchenko, M.L. D´ ıaz Sosa Combinatorial rank of quantum groups of infinite series

Algebras and bialgebras ◮ A = � x 1 , x 2 , . . . , x n || f 1 = 0 , f 2 = 0 , . . . , f m = 0 � . V.K. Kharchenko, M.L. D´ ıaz Sosa Combinatorial rank of quantum groups of infinite series

Algebras and bialgebras ◮ A = � x 1 , x 2 , . . . , x n || f 1 = 0 , f 2 = 0 , . . . , f m = 0 � . ( a ) a (1) ⊗ a (2) . Biideal: ◮ ∆( A ) → A ⊗ A ; ∆( a ) = � ∆( I ) ⊆ A ⊗ I + I ⊗ A . If f ∈ I , then either f (1) ∈ I or f (2) ∈ I , but not both. V.K. Kharchenko, M.L. D´ ıaz Sosa Combinatorial rank of quantum groups of infinite series

Algebras and bialgebras ◮ A = � x 1 , x 2 , . . . , x n || f 1 = 0 , f 2 = 0 , . . . , f m = 0 � . ( a ) a (1) ⊗ a (2) . Biideal: ◮ ∆( A ) → A ⊗ A ; ∆( a ) = � ∆( I ) ⊆ A ⊗ I + I ⊗ A . If f ∈ I , then either f (1) ∈ I or f (2) ∈ I , but not both. ◮ Example: f = x 1 x 2 ; ∆( f ) = f ⊗ 1 + x 1 ⊗ x 2 + x 2 ⊗ x 1 + 1 ⊗ f ; Biid � x 1 x 2 � is either Id � x 1 � or Id � x 2 � . V.K. Kharchenko, M.L. D´ ıaz Sosa Combinatorial rank of quantum groups of infinite series

Primitive elements and coradical filtration ◮ If ∆( f ) = a ⊗ f + f ⊗ b , then Id � f � is a biideal. The combinatorial representation exists if the defining relations are skew-primitive. It is not true that every biideal is generated by skew-primitives. V.K. Kharchenko, M.L. D´ ıaz Sosa Combinatorial rank of quantum groups of infinite series

Primitive elements and coradical filtration ◮ If ∆( f ) = a ⊗ f + f ⊗ b , then Id � f � is a biideal. The combinatorial representation exists if the defining relations are skew-primitive. It is not true that every biideal is generated by skew-primitives. ◮ Theorem (Heyneman–Radford, 74). Let C and D be coalgebras y φ : C → D be a morphism of coalgebras such that the restriction φ | C 1 is injective. Then φ in injective. ◮ Here C 0 ⊂ C 1 ⊂ C 2 ⊂ . . . = C is the coradical filtration: n � ∆( C n ) ⊆ C i ⊗ C n − i . i =1 V.K. Kharchenko, M.L. D´ ıaz Sosa Combinatorial rank of quantum groups of infinite series

Combinatorial rank ◮ Every biideal has nontrivial intersection with C 1 , and ∆( C 1 ) ⊆ C 0 ⊗ C 1 + C 1 ⊗ C 0 . V.K. Kharchenko, M.L. D´ ıaz Sosa Combinatorial rank of quantum groups of infinite series

Combinatorial rank ◮ Every biideal has nontrivial intersection with C 1 , and ∆( C 1 ) ⊆ C 0 ⊗ C 1 + C 1 ⊗ C 0 . ◮ Theorem (Taft-Wilson, 74). If C is pointed, then C 1 is spanned by 1 and by skew-primitive elements . Corollary . Every nonzero biideal I of a pointed bialgebra A has a nonzero skew-primitive element. A = � X || f 1 1 , . . . , f 1 m | f 2 1 , . . . , f 2 m | . . . | f κ 1 , . . . , f κ m � . V.K. Kharchenko, M.L. D´ ıaz Sosa Combinatorial rank of quantum groups of infinite series

Combinatorial rank ◮ Every biideal has nontrivial intersection with C 1 , and ∆( C 1 ) ⊆ C 0 ⊗ C 1 + C 1 ⊗ C 0 . ◮ Theorem (Taft-Wilson, 74). If C is pointed, then C 1 is spanned by 1 and by skew-primitive elements . Corollary . Every nonzero biideal I of a pointed bialgebra A has a nonzero skew-primitive element. A = � X || f 1 1 , . . . , f 1 m | f 2 1 , . . . , f 2 m | . . . | f κ 1 , . . . , f κ m � . ◮ The number κ is the combinatorial rank of A . I 1 ⊂ I 2 ⊂ I 3 ⊂ . . . ⊂ I κ = I , I t / I t − 1 = I / I t − 1 ∩ C 1 ( F / I t − 1 ) . V.K. Kharchenko, M.L. D´ ıaz Sosa Combinatorial rank of quantum groups of infinite series

Combinatorial ranks κ ( u q ( s l n +1 )) = ⌊ log 2 n ⌋ + 1 , Kh., A. ´ Alvarez, Contemporary Mathematics, 376(2005), 299–308. κ ( u q ( s o 2 n +1 )) = ⌊ log 2 ( n − 1) ⌋ + 2 , Kh., M.L. D´ ıaz Sosa, Comm. in Algebra, 39(2011), 4705–4718. κ ( u q ( s o 2 n )) = ⌊ log 2 ( n − 2) ⌋ + 2 , Kh., M.L. D´ ıaz Sosa, Journal of Algebra, 448(2016), 48–73. κ ( u q ( s p 2 n )) = ⌊ log 2 ( n − 1) ⌋ + 2 , Kh., M.L. D´ ıaz Sosa, to appear. V.K. Kharchenko, M.L. D´ ıaz Sosa Combinatorial rank of quantum groups of infinite series

Coproduct formula In the proof, we use an explicit formula for the coproduct: m − 1 � ∆([ u km ]) = [ u km ] ⊗ 1 + g km ⊗ [ u km ] + τ i g ki [ u 1+ i m ] ⊗ [ u ki ] , i = k Kh, Israel Journal of Mathematics, 208(2015), 13–43. V.K. Kharchenko, M.L. D´ ıaz Sosa Combinatorial rank of quantum groups of infinite series

BOOK For more details and related results, see the book “Quantum Lie Theory” Lecture Notes in Mathematics, v. 2150, Springer, 2015. THANK YOU V.K. Kharchenko, M.L. D´ ıaz Sosa Combinatorial rank of quantum groups of infinite series