197:, p.8). Note that Montgomery describes a slight variant of this Hopf algebra using the opposite coproduct, i.e. the coproduct described above composed with the tensor flip on
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211:. This Hopf algebra is isomorphic to the Hopf algebra described here by the Hopf algebra homomorphism
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is a certain 4-dimensional quotient of it that is neither commutative nor cocommutative.
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Armour, Aaron; Chen, Hui-Xiang; Zhang, Yinhuo (2006), "Structure theorems of H
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387:; Zhang, Yinhuo (2001), "The Brauer group of Sweedler's Hopf algebra H
362:, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York,
27:, p. 89–90) introduced an example of an infinite-dimensional
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The following infinite dimensional Hopf algebra was introduced by
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Example of a non-commutative and non-cocommutative Hopf algebra
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Sweedler's 4-dimensional Hopf algebra is a quotient of the
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394:Proceedings of the American Mathematical Society
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159:is the quotient of this by the relations
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152:Sweedler's 4-dimensional Hopf algebra
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305:10.1016/j.jalgebra.2005.10.020
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408:10.1090/S0002-9939-00-05628-8
68:The coproduct Δ is given by
259:{\displaystyle x\mapsto gx}
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356:Sweedler, Moss E. (1969),
328:Montgomery, Susan (1993),
230:{\displaystyle g\mapsto g}
136:The counit ε is given by
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33:Sweedler's Hopf algebra
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272:Pareigis Hopf algebra
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181:so it has a basis 1,
288:-Azumaya algebras",
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21:Moss E. Sweedler
385:Van Oystaeyen, Fred
291:Journal of Algebra
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278:References
45:Definition
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314:0021-8693
248:↦
222:↦
445:Category
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166:= 0,
144:)=0, ε(
72:Δ(g) =
23: (
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170:= 1,
84:) = 1⊗
31:, and
148:) = 1
114:) = –
423:ISSN
364:ISBN
336:ISBN
310:ISSN
237:and
129:) =
80:, Δ(
61:and
25:1969
413:hdl
403:doi
399:129
391:",
300:doi
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174:= –
447::
431:MR
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421:,
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346:MR
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318:MR
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191:xg
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176:xg
172:gx
140:ε(
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