{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T16:10:09Z","timestamp":1773245409867,"version":"3.50.1"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"3","license":[{"start":{"date-parts":[[2010,10,22]],"date-time":"2010-10-22T00:00:00Z","timestamp":1287705600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proc. Amer. Math. Soc."],"abstract":"<p>\n                    We investigate when a weak Hopf algebra\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H\">\n                        <mml:semantics>\n                          <mml:mi>H<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">H<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is Frobenius. We show this is not always true, but it is true if the semisimple base algebra\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A\">\n                        <mml:semantics>\n                          <mml:mi>A<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">A<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    has all its matrix blocks of the same dimension. However, if\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A\">\n                        <mml:semantics>\n                          <mml:mi>A<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">A<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a semisimple algebra not having this property, there is a weak Hopf algebra\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H\">\n                        <mml:semantics>\n                          <mml:mi>H<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">H<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with base\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A\">\n                        <mml:semantics>\n                          <mml:mi>A<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">A<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    which is not Frobenius (and consequently, it is not Frobenius \u201cover\u201d\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A\">\n                        <mml:semantics>\n                          <mml:mi>A<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">A<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    either). Moreover, we give a categorical counterpart of the result that a Hopf algebra is a Frobenius algebra for a noncoassociative generalization of a weak Hopf algebra.\n                  <\/p>","DOI":"10.1090\/s0002-9939-09-10121-1","type":"journal-article","created":{"date-parts":[[2009,11,24]],"date-time":"2009-11-24T11:12:46Z","timestamp":1259061166000},"page":"837-845","source":"Crossref","is-referenced-by-count":10,"special_numbering":"609","title":["When weak Hopf algebras are Frobenius"],"prefix":"10.1090","volume":"138","author":[{"given":"Miodrag","family":"Iovanov","sequence":"first","affiliation":[]},{"given":"Lars","family":"Kadison","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2009,10,22]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"552","DOI":"10.1016\/S0021-8693(03)00175-3","article-title":"Integrals for (dual) quasi-Hopf algebras. 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