Classification of bicovariant differential calculi on quantum groups of type A, B, C and D

Abstract

Under the assumptions that q is not a root of unity and that the differentials du) of the matrix entries span the left module of first order forms, we classify bicovariant differential calculi on quantum groups A nu B n ,C n and D n . We prove that apart one dimensional differential calculi and from finitely many values of q, there are precisely In such calculi on the quantum group A n -ι = SL q (n) for n Ξ> 3. All these calculi have the dimension n 2 . For the quantum groups B n , C n and D n we show that except for finitely many q there exist precisely two iV 2 -dimensional bicovariant calculi for N ^ 3, where N = In + 1 for B n and N = 2n for C n ,D n . The structure of these calculi is explicitly described and the corresponding ad-invariant right ideals of ker ε are determined. In the limit q-*l two of the 2n calculi for A n -\ and one of the two calculi for B n , C n and D n contain the ordinary classical differential calculus on the corresponding Lie group as a quotient.

Levi-Civita Connections on the Quantum Groups SL q ( N ), O q ( N ) and Sp q ( N )

“…Woronowicz has developed a general theory of such calculi which in many aspects can be considered as a non-commutative version of the classical Lie group theory. Bicovariant differential calculi on the quantum matrix groups SL q (N ), O q (N ) and Sp q (N ) have been classified (under natural assumptions) in two recent papers [18] and [19]. An outcome of this classification is that except for finitely many values of q there are precisely 2N such calculi on SL q (N ) for N ≥ 3 and two on O q (N ) and Sp q (N ) for N ≥ 4.…”

Classification of bicovariant differential calculi on the quantum groups SLq(n+1) and Spq(2n)

“…Here (a −1 ) ij are the entries of the inverse Cartan matrix. From the normalization g i (K j ) = (a −1 ) ij we get g i (K αj ) = δ i j which implies the simple commutation rules (16) below between E i , F i and g j . Now the Hopf dual U 0 := (CT ) • is the (commutative and cocommutative) Hopf algebra generated by the functionals f µ and g i with relations f µ f µ ′ = f µµ ′ , g i g j = g j g i , f µ g i = g i f µ and Hopf algebra structure given by…”

Differential Hopf Algebras on Quantum Groups of Type A

“…First recall the defining parameters (for n τ,z , r τ and r ± see before (15); for s ± see before (38)). We use the same notations as in [22,23].…”