Hopf Algebra of the Planar Binary Trees

✍ Scribed by Jean-Louis Loday; Marı́a O. Ronco

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 416 KB
• Volume: 139
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1998.1759

✦ Synopsis

Let k be a field and let S n be the symmetric group. The group algebra k[S n ] contains the Solomon descent algebra, which is of dimension 2 n&1 . In [MR] Malvenuto and Reutenauer construct a graded Hopf algebra structure on

so that the sum of the Solomon descent algebras

There is a basis

In between S n and Q n there is an intermediate set Y n , which is made of planar binary trees with n vertices. For instance, for n=3 the number of elements in S 3 , Y 3 , and Q 3 is 6, 5 and 4 respectively. The projection from S n to Q n is the composite of two maps