Quantizations of Hopf algebras of decorated planar trees and connection with quantum groups.
The author has previously introduced and studied the Hopf algebra \(H^D_{P,R}\) of planar labeled rooted trees [Bull. Sci. Math. 126, No. 3, 193-239 (2002; Zbl 1013.16026) and ibid. No. 4, 249-288 (2002; Zbl 1013.16027)]. \(H^D_{P,R}\) is self-dual, and has a non-degenerate pairing with itself. There is a functor \(H_{P,R}\) from the category of vector spaces into the category of graded Hopf algebras, such that \(H_{P,R}(V)=H^D_{P,R}\) if the cardinality of \(D\) equals the dimension of \(V\). In the paper under review, this functor is generalized to a functor \(H_{P,R}\) from braided vector spaces to braided graded Hopf algebras. On spaces with a trivial braiding, one recovers the \(H_{P,R}\) on vector spaces. If \(V\) is a braided vector space, \(H_{P,R}(V)\) has a basis of forests, the coproduct is described by admissible cuts and the action of the braid groups. (Non-degenerate) pairings of braided spaces yield (non-degenerate) pairings of the corresponding Hopf algebras. Yetter-Drinfeld modules \(V\) over a Hopf algebra \(C\) are examples. In this case, \(H_{P,R}(V)\) is a Hopf algebra in the category of Yetter-Drinfeld modules over \(C\). The Nichols algebra of \(V\) is a sub-quotient of \(H_{P,R}(V)\). When \(V\) is a diagonal symmetric braided space, it is a Yetter-Drinfeld module over the Hopf algebra \(C\) of Laurent series in the variables \(X_i\), \(i\) in \(D\). Then one can form the Drinfeld double of the crossed product \(H_{P,R}(V)\#C\). This Hopf algebra contains a copy of \(H_{P,R}(V)\), a copy of \(H_{P,R}(V)^{\text{coop}}\) and a copy of \(C\). Finally, when the base field is \(k(q)\), \(q\) an indeterminate, and \(A=(a_{ij})\) for \(i,j\) in \(D\) is a finite Cartan matrix, let \(V\) be a vector space with basis \(\{v_i\mid i\) in \(D\}\) and braiding \(v_i\otimes v_j\to q^{a_{ij}}v_j\otimes v_i\). If \(L\) is the Lie algebra associated to \(A\), then the author shows that the quantum universal enveloping algebra \(U_q(L)\) is a sub-quotient of the Drinfeld double of \(H_{P,R}(V)\).
scientific article; zbMATH DE number 1981488
scientific article