A diagrammatic definition of \(U_q(\mathfrak{sl}_2)\) (Q2922967)

The Temperley-Lieb category \(TL\) has objects the non-negative integers, with morphisms from \(n\) to \(m\) being formal linear combinations of Temperley-Lieb diagrams with \(n\) endpoints at the bottom and \(n\) on the top. The tensor product of \(n\) and \(m\) is \(n+m\). The author extends \(TL\) to \(TL^*\) by introducing diagrams with univalent vertices, where a vertex is the endpoint of a strand lying in the interior of the diagram. Relations called turning, confetti and cutting are imposed. He then defines a Hopf algebra \(H\) of formal linear combinations of diagrams similar to hose in \(TL^*\), but also with a vertical pole. For a diagram \(h\) in \(H\), let \(r(h)\) be the diagram in \(TL^*\) obtained by replacing the pole by a strand. Then \(r\) extends to an algebra morphism from \(H\) to the 2 by 2 matrices over the ground field \(F\). He shows that \(H\) is generated by 8 specific elements, describes the coproduct, counit and antipode on these 8 elements, and gives relations among them. He then gets a quotient \(H'\) of \(H\) by the intersection of the kernels of all tensor powers of \(r\). \(H'\) has 4 generators \(e, f, k\) and \(k^(-1)\), and he describes the coproduct, counit and and antipode on them, and gets relations among them. He defines a Hopf algebra morphism \(f\) from \(U_q(\mathfrak{sl}_2)\) to \(H'\) whose image is the words of even length in the 4 generators of \(H'\), and describes the kernel of \(f\). Finally he shows that if \(F\) is the complex numbers, and \(q\) is not a root of unity, then \(U_q(\mathfrak{sl}_2)\) is isomorphic to the algebra of words of even length in the 4 generators of \(H'\).