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Combinatorial rank of quantum groups

• f 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 = x1, x2, . . . , xn || f1 = 0, f2 = 0, . . . , fm = 0.

V.K. Kharchenko, M.L. D´ ıaz Sosa Combinatorial rank of quantum groups of infinite series

Algebras and bialgebras

◮ A = x1, x2, . . . , xn || f1 = 0, f2 = 0, . . . , fm = 0. ◮ ∆(A) → A ⊗ A;

∆(a) =

(a) a(1) ⊗ a(2). Biideal:

∆(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 = x1, x2, . . . , xn || f1 = 0, f2 = 0, . . . , fm = 0. ◮ ∆(A) → A ⊗ A;

∆(a) =

(a) a(1) ⊗ a(2). Biideal:

∆(I) ⊆ A ⊗ I + I ⊗ A. If f ∈ I, then either f (1) ∈ I or f (2) ∈ I, but not both.

◮ Example: f = x1x2; ∆(f ) = f ⊗ 1 + x1 ⊗ x2 + x2 ⊗ x1 + 1 ⊗ f ;

Biidx1x2 is either Idx1 or Idx2.

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 Idf 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 Idf 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|C1 is injective. Then φ in injective.

◮ Here C0 ⊂ C1 ⊂ C2 ⊂ . . . = C is the coradical filtration:

∆(Cn) ⊆

n

• i=1

Ci ⊗ Cn−i.

V.K. Kharchenko, M.L. D´ ıaz Sosa Combinatorial rank of quantum groups of infinite series

Combinatorial rank

◮ Every biideal has nontrivial intersection with C1, and

∆(C1) ⊆ C0 ⊗ C1 + C1 ⊗ C0.

V.K. Kharchenko, M.L. D´ ıaz Sosa Combinatorial rank of quantum groups of infinite series

Combinatorial rank

◮ Every biideal has nontrivial intersection with C1, and

∆(C1) ⊆ C0 ⊗ C1 + C1 ⊗ C0.

◮ Theorem (Taft-Wilson, 74). If C is pointed, then C1 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 C1, and

∆(C1) ⊆ C0 ⊗ C1 + C1 ⊗ C0.

◮ Theorem (Taft-Wilson, 74). If C is pointed, then C1 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.

I1 ⊂ I2 ⊂ I3 ⊂ . . . ⊂ Iκ = I, It/It−1 = I/It−1 ∩ C1(F/It−1).

V.K. Kharchenko, M.L. D´ ıaz Sosa Combinatorial rank of quantum groups of infinite series

Combinatorial ranks

κ(uq(sln+1)) = ⌊log2 n⌋ + 1, Kh., A. ´ Alvarez, Contemporary Mathematics, 376(2005), 299–308. κ(uq(so2n+1)) = ⌊log2(n − 1)⌋ + 2, Kh., M.L. D´ ıaz Sosa, Comm. in Algebra, 39(2011), 4705–4718. κ(uq(so2n)) = ⌊log2(n − 2)⌋ + 2, Kh., M.L. D´ ıaz Sosa, Journal of Algebra, 448(2016), 48–73. κ(uq(sp2n)) = ⌊log2(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: ∆([ukm]) = [ukm] ⊗ 1 + gkm ⊗ [ukm] +

m−1

• i=k

τigki[u1+i m] ⊗ [uki], 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