Representation rings of quantum groups.
Let \({\mathcal O}_q\) be the complex coordinate algebra of one of the quantum groups \(\text{GL}_q(n)\), \(\text{SL}_q(n)\), \(O_q(n)\), \(\text{Sp}_q(n)\), where the complex number \(q\) is not a root of one. Moreover in the three last cases \(q\) is transcendental. Then the algebra \({\mathcal O}_q\) is cosemisimple. An element \(z\in{\mathcal O}_q\) is cocommutative if \(\tau\Delta(z)=\Delta(z)\) where \(\tau\) is an automorphism of \({\mathcal O}_q^{\otimes 2}\) such that \(\tau(a\otimes b)=b\otimes a\). The set of all cocommutative elements forms a subalgebra \({\mathcal O}_q^{coc}\) in \({\mathcal O}_q\). There is given an explicit form of generators of \({\mathcal O}_q^{coc}\). Let \({\mathcal A}(G_q)\) be a complex algebra generated by elements \(u^i_j\), \(1\leqslant i,j\leqslant n\), subject to the defining relations \(R(u\otimes 1)(1\otimes u)=(1\otimes u)(u\otimes 1)R\) where \(R\) is the \(R\)-matrix of the vector representation of \(U_q(\mathfrak g)\) for the simple Lie algebra \(\mathfrak g\) corresponding to \(G_q\), and \(u=(u^i_j)\) is a square matrix of size \(n\). There is also given an explicit form of generators of the algebra \({\mathcal A}(G_q)^{coc}\) in the case \(G_1=\text{SL}_q(n)\), \(\text{Sp}_q(m)\), \(O_q(n)\) where \(m\) is even. Using quantum traces an algebra isomorphism between \({\mathcal O}_q^{coc}\) and its classical counterpart is established.
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