Integrals for Hopf Algebras

Sweedler, Moss E.

doi:10.2307/1970672

Cited by 214 publications

(255 citation statements)

References 5 publications

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“…This is a bi-ideal, and denoting the cosets of Z, A, and B = ZA by z, a, and b = za, respectively, the factor algebra H1 has a basis 1, z, a, b, with multiplication table [6]. A left integral u in H satisfies hu = e(h)u for all h in H, and a right integral v in H satisfies vh = e(h)v for all h in H, see [1,7]. The left integrals and the right integrals are always one-dimensional subspaces of H (see [1, p. …”

Section: Further Remarksmentioning

confidence: 99%

The Order of the Antipode of Finite-dimensional Hopf Algebra

Taft

1971

Proc. Natl. Acad. Sci. U.S.A.

137

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Examples of finite-dimensional Hopf algebras over a field, whose antipodes have arbitrary even orders >4 as mappings, are furnished. The + 1 0 1 0 Y. We define an algebra homomorphism e from R to K by specifying e(X,) = 1 for 1 < i < n, and e(Y) = 0. One easily checks that (e 0) IR)A = IR = (IX 0 e)A (identifying K 0) R and R 0 K with R), as these are algebra homomorphisms agreeing on the Xi and Y, so that e is a counit. Hence, R is a bialgebra. Let I be the ideal of R generated by the union of the following four sets of elements of R:(1) {X1 -11 < i < n} { Ye}We assert that I is a bi-ideal of R, i.e., AI C I 0 R + R 0 1 and e(I) = 0. As A and e are algebra homomorphisms, and since J = I 0 R + R 0 I is an ideal of R 0 R, it suffices to check these conditions on the generators of I. Since e(X,) = 1, we have e(Xi" -1) = e(XiXj -XjX1) = 0, and since e(Y) = 0, we have e(YX1 -wXiY) = e(Yq) = 0. Working modulo J, A(Xiq-1) = Xiq Xiq-1 1-1-01 = 0. For (2),

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Section: Further Remarksmentioning

confidence: 99%

The Order of the Antipode of Finite-dimensional Hopf Algebra

Taft

1971

Proc. Natl. Acad. Sci. U.S.A.

137

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“…A short proof of Radford's formula Radford's formula for S 4 in the case of Hopf algebra was first proved in full generality in [17] and with predecessors in [19] and [11]. A more recent proof, that is in the spirit of the one we present below, appears in [18].…”

Section: Observation 1 It Is Worth Noticing That One Can Construct Bmentioning

confidence: 99%

Radford's Formula for Bifrobenius Algebras and Applications

Santos

Haim

2008

Communications in Algebra

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Abstract. In a biFrobenius algebra H, in particular in the case that H is a finite dimensional Hopf algebra, the antipode S : H → H can be decomposed as S =tc•c φ where c φ : H → H * and tc : H * → H are the Frobenius and coFrobenius isomorphisms. We use this decomposition to present an easy proof of Radford's formula for S 4 . Then, in the case that the map S satisfies the additional condition that S ⋆ id = id ⋆ S = uε, we prove the trace formula: tr(S 2 ) = ε(t)φ(1) . We finish by applying the above results to study the semisimplicity and cosemisimplicity of H.

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“…Same notion has appeared in Heyneman et al [11]. Johnson introduced differential module structures on certain modules of Kähler differentials [4].…”

Section: Introductionmentioning

confidence: 95%

Tensor,symmetric and exterior algebras K\ {a}hler modules

Şahin¹,

Uluçay²,

Olgun³

2016

ntmsci

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Let k be an algebraically closed field of characteristic zero, R an affine k-algebra and let Ω (q) (R/k) denote its universal finite Kähler module of differentials over k. In this paper, we consider the tensor, exterior and symmetric algebras of Kähler modules introduced by H. Osborn [9]. We explore some interesting properties of the algebras of Kähler modules, which have not been considered before.

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Integrals for Hopf Algebras

Cited by 214 publications

References 5 publications

The Order of the Antipode of Finite-dimensional Hopf Algebra

The Order of the Antipode of Finite-dimensional Hopf Algebra

Radford's Formula for Bifrobenius Algebras and Applications

Tensor,symmetric and exterior algebras K\ {a}hler modules

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