Jordan pairs and Hopf algebras
The well-known Tits-Kantor-Koecher (TKK) construction relates Jordan systems and Lie algebras. More specifically, given a Jordan pair \(V=(V^+,V^-)\), \(L=\text{TKK}(V)\) is a \(3\)-graded Lie algebra: \(L=L_{-1}\oplus L_0\oplus L_1\), and conversely, provided the characteristic is not \(2\), the \(\pm 1\)-components of any \(3\)-graded Lie algebra form a Jordan pair. In the paper under review, the role of the Lie algebra is replaced by a certain kind of Hopf algebra, whose primitive elements form a \(3\)-graded Lie algebra and which also contains divided power sequences that capture the quadratic properties of the Jordan pair. Thus, after reviewing the basic properties of Hopf algebras, Jordan pairs and the TKK construction, one of the main results is proved: If \(A\) is a \(\mathbb{Z}\)-graded Hopf algebra, whose primitive elements \(P(A)\) satisfy \(P(A)=P_{-1}\oplus P_0\oplus P_1\), where \(P_i=P(A)\cap A_i\), and such that for any \(x\in P_{\pm 1}\) there is a `homogeneous divided power sequence' \((x^{(0)}=1,x^{(1)}=x,x^{(2)},\ldots)\), where \(x^{(n)}\in A_{\pm n}\) and \(\Delta(x^{(n)})=\sum_{i+j=n}x^{(i)}\otimes x^{(j)}\), then \((P_{-1},P_1)\) is a Jordan pair with \(Q_xy=x^{(2)}y-xyx+yx^{(2)}\). Next a concept of divided power representation of a Jordan pair \(V\) into a unital associative algebra is introduced. A related universal associative algebra \(\mathcal{U}(V)\) is shown to exist, and this is proved to be a \(\mathbb{Z}\)-graded cocommutative Hopf algebra on which \(V\) is embedded, and that verifies, under some restrictions, the conditions in the result above, with \(P_{\pm 1}=V_{\pm 1}\). In characteristic zero, \(\mathcal{U}(V)\) is shown to be isomorphic to the universal enveloping algebra of the universal cover of \(\text{TKK}(V)\). To complete the work, a detailed appendix is given with the treatment of homogeneous maps that is used throughout the paper.
scientific article; zbMATH DE number 1539381