• Title, Summary, Keyword: Hopf algebra

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TOWARDS UNIQUENESS OF MPR, THE MALVENUTO-POITIER-REUTENAUER HOPF ALGEBRA OF PERMUTATIONS

  • Hazewinkel, Michiel
    • Honam Mathematical Journal
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    • v.29 no.2
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    • pp.119-192
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    • 2007
  • A very important Hopf algebra is the graded Hopf algebra Symm of symmetric functions. It can be characterized as the unique graded positive selfdual Hopf algebra with orthonormal graded distinguished basis and just one primitive element from the distinguished basis. This result is due to Andrei Zelevinsky. A noncommutative graded Hopf algebra of this type cannot exist. But there is a most important positive graded Hopf algebra with distinguished basis that is noncommutative and that is twisted selfdual, the Malvenuto-Poirier-Reutenauer Hopf algebra of permutations. Thus the question arises whether there is a corresponding uniqueness theorem for MPR. This prepreprint records initial investigations in this direction and proves that uniquenees holds up to and including the degree 4 which has rank 24.

HOPF STRUCTURE FOR POISSON ENVELOPING ALGEBRAS

  • Min, Kangju;Oh, Sei-Qwon
    • Journal of the Chungcheong Mathematical Society
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    • v.13 no.2
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    • pp.29-39
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    • 2001
  • For a Poisson Hopf algebra A, we find a natural Hopf structure in the Poisson enveloping algebra U(A) of A. As an application, we show that the Poisson enveloping algebra U(S(L)), where S(L) is the symmetric algebra of a Lie algebra L, has a Hopf structure such that a sub-Hopf algebra of U(S(L)) is Hopf-isomorphic to the universal enveloping algebra of L.

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A REMARK ON THE CONJUGATION IN THE STEENROD ALGEBRA

  • TURGAY, NESET DENIZ
    • Communications of the Korean Mathematical Society
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    • v.30 no.3
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    • pp.269-276
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    • 2015
  • We investigate the Hopf algebra conjugation, ${\chi}$, of the mod 2 Steenrod algebra, $\mathcal{A}_2$, in terms of the Hopf algebra conjugation, ${\chi}^{\prime}$, of the mod 2 Leibniz-Hopf algebra. We also investigate the fixed points of $\mathcal{A}_2$ under ${\chi}$ and their relationship to the invariants under ${\chi}^{\prime}$.

Department of Mathematics, Dongeui University

  • Yoon, Suk-Bong
    • Bulletin of the Korean Mathematical Society
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    • v.38 no.3
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    • pp.527-541
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    • 2001
  • We find the necessary and sufficient conditions for the smash product algebra structure and the crossed coproduct coalgebra structure with th dual cocycle $\alpha$ to afford a Hopf algebra (A equation,※See Full-text). If B and H are finite algebra and Hopf algebra, respectively, then the linear dual (※See Full-text) is also a Hopf algebra. We show that the weak coaction admissible mapping system characterizes the new Hopf algebras (※See Full-text).

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ON HOPF ALGEBRAS IN ENTROPIC JÓNSSON-TARSKI VARIETIES

  • ROMANOWSKA, ANNA B.;SMITH, JONATHAN D.H.
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.5
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    • pp.1587-1606
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    • 2015
  • Comonoid, bi-algebra, and Hopf algebra structures are studied within the universal-algebraic context of entropic varieties. Attention focuses on the behavior of setlike and primitive elements. It is shown that entropic $J{\acute{o}}nsson$-Tarski varieties provide a natural universal-algebraic setting for primitive elements and group quantum couples (generalizations of the group quantum double). Here, the set of primitive elements of a Hopf algebra forms a Lie algebra, and the tensor algebra on any algebra is a bi-algebra. If the tensor algebra is a Hopf algebra, then the underlying $J{\acute{o}}nsson$-Tarski monoid of the generating algebra is cancellative. The problem of determining when the $J{\acute{o}}nsson$-Tarski monoid forms a group is open.

DUALITY OF CO-POISSON HOPF ALGEBRAS

  • Oh, Sei-Qwon;Park, Hyung-Min
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.1
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    • pp.17-21
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    • 2011
  • Let A be a co-Poisson Hopf algebra with Poisson co-bracket $\delta$. Here it is shown that the Hopf dual $A^{\circ}$ is a Poisson Hopf algebra with Poisson bracket {f, g}(x) = < $\delta(x)$, $f\;{\otimes}\;g$ > for any f, g $\in$ $A^{\circ}$ and x $\in$ A if A is an almost normalizing extension over the ground field. Moreover we get, as a corollary, the fact that the Hopf dual of the universal enveloping algebra U(g) for a finite dimensional Lie bialgebra g is a Poisson Hopf algebra.

POISSON HOPF STRUCTURE INDUCED BY THE UNIVERSAL ENVELOPING ALGEBRA OF A GRADED LIE ALGEBRA

  • Oh, Sei-Qwon;Park, Miran
    • Journal of the Chungcheong Mathematical Society
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    • v.23 no.1
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    • pp.177-184
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    • 2010
  • Let G be an abelian group, $\alpha$ an antisymmetric bicharacter on G and g a (G, $\alpha$)-Lie algebra. Here we give a complete proof for that the associated graded algebra determined by a natural filtration in the universal enveloping algebra U(g) is a (G, $\alpha$)-Poisson Hopf algebra.

INDUCED HOPF CORING STRUCTURES

  • Saramago, Rui Miguel
    • Journal of the Korean Mathematical Society
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    • v.48 no.3
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    • pp.627-639
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    • 2011
  • Hopf corings are dened in this work as coring objects in the category of algebras over a commutative ring R. Using the Dieudonn$\'{e}$ equivalences from [7] and [19], one can associate coring structures built from the Hopf algebra $F_p[x_0,x_1,{\ldots}]$, p a prime, with Hopf ring structures with same underlying connected Hopf algebra. We have that $F_p[x_0,x_1,{\ldots}]$ coring structures classify thus Hopf ring structures for a given Hopf algebra. These methods are applied to dene new ring products in the Hopf algebras underlying known Hopf rings that come from connective Morava ${\kappa}$-theory.

ACTIONS OF FINITE-DIMENSIONAL SEMISIMPLE HOPF ALGEBRAS AND INVARIANT ALGEBRAS

  • Min, Kang-Ju;Park, Jun-Seok
    • Communications of the Korean Mathematical Society
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    • v.13 no.2
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    • pp.225-232
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    • 1998
  • Let H be a finite dimensional Hopf algebra over a field k, and A be an H-module algebra over k which the H-action on A is D-continuous. We show that $Q_{max}(A)$, the maximal ring or quotients of A, is an H-module algebra. This is used to prove that if H is a finite dimensional semisimple Hopf algebra and A is a semiprime right(left) Goldie algebra than $A#H$ is a semiprime right(left) Goldie algebra. Assume that Asi a semiprime H-module algebra Then $A^H$ is left Artinian if and only if A is left Artinian.

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