zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Frobenius extensions and weak Hopf algebras. (English) Zbl 1018.16020
Let M/N be a Frobenius extension of k-algebras with Frobenius homomorphism E and dual bases {x i }, {y i }. Let U=C M (N). The extension is called symmetric if E commutes with every uU, and Markov if the extension is strongly separable (i.e. E(1)=1 and i x i y i =λ -1 1) and there is a (Markov) trace T:Nk such that T(1)=1 k and T 0 =TE:Mk is a trace. The basic construction theorem says that if NM is a symmetric Markov extension and M 1 =M N M=End(M N ) then M 1 /M is a symmetric Markov extension; the Frobenius endomorphism E M and the dual bases are described, and the Markov trace is T 0 . If in addition U is Kanzaki separable, T 0 | U is non-degenerate and i x i y i = i y i x i then V=C M 1 (M) is Kansaki separable and the restriction of T 1 =T 0 E M to V is non-degenerate. The construction can be iterated to obtain the Jones tower NMM 1 M 2 . Let A=C M 1 (N) and B=C M 2 (M). Assuming the existence of dual bases for E M resp. E M 1 in A resp. B (depth 2 condition) the authors prove some properties of algebra extensions involving A, B, U and V. Examples of extensions with depth 2 are provided in the final appendix. Then semisimple weak Hopf algebra structures are defined in A and B, and also A resp. B-module algebra structures on M resp. M 1 . Two isomorphisms M 1 M#A and M 2 M 1 #B are also provided. Finally, in the absence of a trace, the basic construction is iterated to the right, extending a result of M. Pimsner and S. Popa [Trans. Am. Math. Soc. 310, No. 1, 127-133 (1988; Zbl 0706.46047)].
MSC:
16W30Hopf algebras (assoc. rings and algebras) (MSC2000)
16S40Smash products of general Hopf actions
16S50Endomorphism rings: matrix rings
16H05Separable associative algebras
16L60Quasi-Frobenius rings