A Primer of Hopf Algebras

• Page 9, §2.1: Here, Cartier claims that “by invariant theory, Ωp for p > 2n is decomposable as a product of forms of degree≤ 2n−1”. I don’t know what results from invariant theory yield this; however, I think the Amitsur-Levitzki theorem yields that Ωp = 0 for p > 2n (and, even stronger, the antisymmetrization of A1A2...Ap (and not only of Tr (A1A2...Ap)) is 0 for p > 2n). • Page 9, §2.1: Here, Cartier claims that “It follows that the algebra T · (U (n)) = ⊕

[1]  Jean-Louis Loday.  On the algebra of quasi-shuffles , 2005 .

[2]  Jean Alexandre Dieudonné,et al.  Introduction to the Theory of Formal Groups , 2020 .

[3]  Harish-Chandra.  Lie Algebras and the Tannaka Duality Theorem , 1950 .

[4]  E. Cartan,et al.  La théorie des groupes finis et continus et l'Analysis situs , 1952 .

[5]  Daniel Krob,et al.  Noncommutative Symmetric Functions II: Transformations of Alphabets , 1997, Int. J. Algebra Comput..

[6]  R. J. Wilson.  Analysis situs , 1985 .

[7]  H. Samelson,et al.  Topology of Lie groups , 1952 .

[8]  Alain Connes,et al.  Hopf Algebras, Cyclic Cohomology and the Transverse Index Theorem , 1998 .

[9]  C. Chevalley.  Theory of Lie Groups , 1946 .

[10]  Pierre Cartier,et al.  On the structure of free baxter algebras , 1972 .

[11]  L. H. Loomis.  An Introduction to Abstract Harmonic Analysis , 1953 .

[12]  H. Weyl,et al.  Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. I , 1926 .

[13]  L. Pontrjagin.  Über die topologische Struktur der Lieschen Gruppen , 1940 .

[14]  Nicolas Bourbaki,et al.  Groupes et algèbres de Lie , 1971 .

[15]  Maria O. Ronco.  A Milnor–Moore theorem for dendriform Hopf algebras , 2001 .

[16]  A. Connes,et al.  Modular Hecke Algebras and their Hopf Symmetry , 2003, math/0301089.

[17]  Maria O. Ronco.  Eulerian idempotents and Milnor–Moore theorem for certain non-cocommutative Hopf algebras , 2002 .

[18]  F. Murnaghan,et al.  LINEAR ALGEBRAIC GROUPS , 2005 .

[19]  C. Reutenauer,et al.  Duality between Quasi-Symmetrical Functions and the Solomon Descent Algebra , 1995 .

[20]  M. Lothaire.  Algebraic Combinatorics on Words , 2002 .

[21]  H. Weyl.  The Classical Groups , 1939 .

[22]  Jean-Louis Loday,et al.  Hopf Algebra of the Planar Binary Trees , 1998 .

[23]  Matilde Marcolli,et al.  Renormalization, the Riemann–Hilbert Correspondence, and Motivic Galois Theory , 2004, hep-th/0411114.

[24]  J. Bertin,et al.  Propriétés générales des schémas en groupes , 1970 .

[25]  Charles Ehresmann,et al.  Sur la topologie de certains espaces homogènes , 1934 .

[26]  H. Hopf,et al.  Ein Satz über die Wirkungsräume geschlossener Liescher Gruppen , 1940 .

[27]  Pierre Cartier,et al.  Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents , 2001 .

[28]  A. Borel,et al.  Sur L'Homologie et la Cohomologie des Groupes de Lie Compacts Connexes , 1954 .

[29]  Pierre Cartier,et al.  Jacobiennes généralisées, monodromie unipotente et intégrales itérées , 1987 .

[30]  Une histoire du théorème de Poincare-Birkhoff-Witt , 2004 .

[31]  François Goichot.  Un théoréme de Milnor-Moore pour les algèbres de Leibniz , 2001 .

[32]  Michael E. Hoffman,et al.  Quasi-Shuffle Products , 1999 .

[33]  P. D. Val.  The Theory and Applications of Harmonic Integrals , 1941, Nature.

[34]  John Milnor,et al.  On the Structure of Hopf Algebras , 1965 .

[35]  E. Cartan,et al.  La géométrie des groupes simples , 1927 .

[36]  H. Samelson.  Beiträge zur Topologie der Gruppen-Mannigfaltigkeiten , 1941 .

[37]  N. Bourbaki.  Espaces vectoriels topologiques , 1955 .

[38]  W. V. Hodge,et al.  The Theory and Applications of Harmonic Integrals , 1941 .

[39]  A. Borel.  Topology of Lie groups and characteristic classes , 1955 .

[40]  M. Geck.  An Introduction to Algebraic Geometry and Algebraic Groups , 2004 .

[41]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[42]  Frédéric Chapoton,et al.  Dialgebras and Related Operads , 2001 .

[43]  Georges de Rham.  Sur l’analysis situs des variétés à $n$ dimensions , 1931 .

[44]  F. Patras.  Decent Algebra of a Graduated Bigebra , 1994 .

[45]  John E. Roberts,et al.  Endomorphisms of C*-algebras, cross products and duality for compact groups , 1989 .

[46]  John E. Roberts,et al.  A new duality theory for compact groups , 1989 .

[47]  M. Lothaire,et al.  Algebraic Combinatorics on Words: Index of Notation , 2002 .

[48]  E. Cartan.  Sur les invariants intégraux de certains espaces homogènes clos et les propriétés topologiques de ces espaces , 1930 .

[49]  J. Leray,et al.  L'anneau spectral et l'anneau filtré d'homologie d'un espace localement compact et d'une application continue , 1950 .

[50]  Nicholas M. Katz,et al.  The Grothendieck Festschrift , 1990 .

[51]  P. Deligné.  Le Groupe Fondamental de la Droite Projective Moins Trois Points , 1989 .

[52]  Chris D. Godsil,et al.  ALGEBRAIC COMBINATORICS , 2013 .

[53]  H. Hopf,et al.  Über die Topologie der Gruppen-Mannigfaltigkeiten und ihrer Verallgemeinerungen , 1941 .

[54]  Israel M. Gelfand,et al.  Noncommutative Symmetrical Functions , 1995 .