Quantizations of Hopf algebras of decorated planar trees and connection with quantum groups.
type: Item
instance of: scholarly article
title: Quantizations of Hopf algebras of decorated planar trees and connection with quantum groups.
zbMATH Open document ID: 1042.16023
DOI: 10.1016/S0007-4497(03)00028-9
published in: Bulletin des Sciences Mathématiques
publication date: 2003-09-15T00:00:00Z
review text: 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 t…
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1981488
zbMATH Keywords: rooted trees
zbMATH Keywords: Hopf algebras
zbMATH Keywords: quantum groups
author: Loïc Foissy
reviewed by: Earl J. Taft
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1981488