Primitive elements of the Hopf algebras of tableaux

Classically, Hopf algebras are defined on the basis of modules over commutative rings. The present study seeks to extend the Hopf algebra formalism to a more general universal-algebraic setting, entropic varieties, including (pointed) sets, barycentric algebras, semilattices, and commutative monoids. The concept of a setlike (or grouplike) element may be defined, and group algebras constructed, in any such variety. In particular, group algebras within the variety of barycentric algebras consist precisely of finitely supported probability distributions on groups. For primitive elements and group quantum doubles, the natural universal-algebraic classes are entropic Jónsson-Tarski varieties (such as semilattices or commutative monoids). There, the tensor algebra on any algebra is a bialgebra, and the set of primitive elements of a Hopf algebra forms an abelian group. Coalgebra congruences on comonoids in entropic varieties are also studied.

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CLASSIFYING COALGEBRA SPLIT EXTENSIONS OF HOPF ALGEBRAS

Journal of Algebra and Its Applications ◽

10.1142/s0219498812502271 ◽

2013 ◽

Vol 12 (05) ◽

pp. 1250227

Author(s):

A. L. AGORE ◽

C. G. BONTEA ◽

G. MILITARU

Keyword(s):

Hopf Algebra ◽

Hopf Algebras ◽

Infinite Family ◽

Crossed Products ◽

Primitive Elements ◽

Characteristic P ◽

Generators And Relations ◽

Split Extensions ◽

Groups Of Automorphisms

For a given Hopf algebra A we classify all Hopf algebras E that are coalgebra split extensions of A by H4, where H4is the Sweedler's four-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras A# H4by computing explicitly two classifying objects: the cohomological "group" [Formula: see text] and CRP (H4, A) ≔ the set of types of isomorphisms of all crossed products A# H4. All crossed products A# H4are described by generators and relations and classified: they are parameterized by the set [Formula: see text] of all central primitive elements of A. Several examples are worked out in detail: in particular, over a field of characteristic p ≥ 3 an infinite family of non-isomorphic Hopf algebras of dimension 4p is constructed. The groups of automorphisms of these Hopf algebras are also described.

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Primitive Generators for Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1982-030-4 ◽

1982 ◽

Vol 34 (2) ◽

pp. 454-465

Author(s):

Stanley O. Kochman

Keyword(s):

Hopf Algebra ◽

Commutative Algebra ◽

Hopf Algebras ◽

Primitive Elements ◽

Explicit Formulas ◽

Recursive Computation

LetHbe a graded commutative algebra with a nice set of algebra generators. LetHalso be a comodule over a Hopf algebraA. In Section 2 we give conditions under which certain of these generators ofHcan be rechosen to be primitive. In addition we give explicit formulas expressing these primitive generators in terms of the original set of generators.In Section 3 we apply the theory of Section 2 to the modphomology of the Thorn spectraMO, MUandMSp.In particular we give two explicit descriptions of the image of the Hurewicz homomorphism forMO.One of these makes explicit the recursive computation of E. Brown and F. Peterson [1].In Section 4 we give a variation of the theory of Section 2 which computes primitive generators of certain Hopf algebras. This theory is applied to study the primitive elements ofH*(BU)andH*(SO;Z2).

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Combinatorial Hopf Algebras and Towers of Algebras

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3634 ◽

2008 ◽

Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽

Author(s):

Nantel Bergeron ◽

Thomas Lam ◽

Huilan Li

Keyword(s):

Hopf Algebras ◽

Primitive Elements ◽

Combinatorial Hopf Algebras ◽

Grothendieck Groups ◽

International Audience ◽

Dual Graded Graphs ◽

Tower Of Algebras

International audience Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras $\bigoplus_{n \geq 0}A_n$ can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower $\bigoplus_{n \geq 0}A_n$ gives rise to graded dual Hopf algebras then we must have $\dim (A_n)=r^nn!$ where $r = \dim (A_1)$. Bergeron et Li ont donné un ensemble d'axiomes qui garanti que les groupes de Grothendieck d'une tour d'algèbres $\bigoplus_{n \geq 0}A_n$ peuvent être dotés d'une structure d'algèbres de Hopf graduées duales. Hivert et Nzeutzhap, et indépen\-damment Lam et Shimozono, ont construit des graphes gradués duals à partir d'éléments primitifs dans des algèbres de Hopf. Dans cet article, nous appliquons la composition de ces constructions aux tours des algèbres. Nous prouvons que si une tour $\bigoplus_{n \geq 0}A_n$ donne des algèbres de Hopf graduées duales, alors nous devons avoir $\dim (A_n)=r^nn!$ où $r = \dim (A_1)$.

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