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On Hopf algebras related to triangular matrices. (English) Zbl 0807.16033

A class of Hopf algebras is constructed which are neither commutative nor commutative. Drinfel’d considered these as the most interesting and mysterious ones. They are not obtained by the method of Drinfel’d quantization. The construction generalizes examples given by Sweedler and Pareigis.
The invertible triangular \(n \times n\) matrices are considered as a group valued functor \(T_ n\) on the category (\(K\text{-Alg}\)) of associative, unital \(K\)-algebras. \(T_ n\) is represented by an algebra \(A_ n\) and the multiplication of \(T_ n\) induces a comultiplication on \(A_ n\) which turns it into a bigebra. The main point is the construction of the antipode \(s_ n\) of this bigebra. It is the unique anti-homomorphism \(s_ n\) which agrees on canonical generators \(x_{ij}\) of \(A_ n\) with the value of the algebra homomorphism induced by the inverse map of \(T_ n\). Also a few examples of Hopf ideals \(I\) of \(A_ n\) and its quotients \(A_ n / I\) are considered. One obtains deformations of Hopf algebras of unipotent algebraic group schemes and a Hopf algebra related to \(U_ h({\mathfrak sl}_ 2)\), the quantum universal enveloping algebra of \({\mathfrak sl}_ 2\).

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
14A22 Noncommutative algebraic geometry
16S50 Endomorphism rings; matrix rings
14L15 Group schemes

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References:

[1] M. Demazure andP. Gabriel, Introduction to Algebraic Geometry and Algebraic Groups, North Holland, Amsterdam 1980. · Zbl 0431.14015
[2] V. G.Drinfel’d, Quantum Groups. Proc. Int. Coogr. Math. Berkley 1986.
[3] B. Pareigis, A Non-commutative Non-cocommutative Hopf Algebra in ?Nature?. J. Algebra70, 336-374 (1981). · Zbl 0463.18003 · doi:10.1016/0021-8693(81)90224-6
[4] B.Pareigis, Endomorphism Bialgebras of Diagrams and of Non-Commutative Algebras and Spaces. Preprint. · Zbl 0821.16041
[5] M. E.Sweedler, Hopf Algebras. New York 1969.
[6] L. A. Takhtajan, Quantum Groups and Integrable Models. In: Integrable Systems in Quantum Field Theory and Statistical Mechanics. Adv. Stud. Pure Math.19, 435-457 (1989).
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