arXiv subjects

Lars Kadison

Publications and source records attributed to Lars Kadison.

41 records · Page 3Linked to original sources

arXiv2001

Are biseparable extensions Frobenius?

In Secion~1 we describe what is known of the extent to which a separable extension of unital associative rings is a Frobenius extension. A problem of this kind is suggested by asking if three algebraic axioms for finite Jones index subfactors are dependent. In Section~2 the problem in the title is formulated in terms of separable bimodules. In Section~3 we specialize the problem to ring extensions, noting that a biseparable extension is a two-sided finitely generated projective, split, separable extension. Some reductions of the problem are discussed and solutions in special cases are provided. In Section~4 various examples are provided of projective separable extensions that are neither finitely generated nor Frobenius and which give obstructions to weakening the hypotheses of the question in the title. We show in Section~5 that existing characterizations of the separable extensions among the Frobenius extensions in are special cases of a result for adjoint functors.

math.RA↗

arXiv2001

Hopf algebra actions on strongly separable extensions of depth two

We bring together ideas in analysis of Hopf *-algebra actions on II_1 subfactors of finite Jones index and algebraic characterizations of Frobenius, Galois and cleft Hopf extensions to prove a non-commutative algebraic analogue of the classical theorem: a finite field extension is Galois iff it is separable and normal. Suppose N < M is a separable Frobenius extension of k-algebras split as N-bimodules with a trivial centralizer C_M(N). Let M_1 := End(M)_N and M_2 := End(M_1)_M be the endomorphism algebras in the Jones tower N < M < M_1 < M_2. We show that under depth 2 conditions on the second centralizers A := C_{M_1}(N) and B : = C_{M_2}(M) the algebras A and B are semisimple Hopf algebras dual to one another and such that M_1 is a smash product of M and A, and that M is a B-Galois extension of N.

math.RA↗

arXiv2001

An Approach to Hopf Algebras via Frobenius Coordinates I

In Section 1 we introduce Frobenius coordinates in the general setting that includes Hopf subalgebras. In Sections 2 and 3 we review briefly the theories of Frobenius algebras and augmented Frobenius algebras with some new material in Section 3. In Section 4 we study the Frobenius structure of an FH-algebra H \cite{Par72} and extend two recent theorems in \cite{EG}. We obtain two Radford formulas for the antipode in H and generalize in Section 7 the results on its order in \cite{FMS}. We study the Frobenius structure on an FH-subalgebra pair in Sections 5 and 6. In Section 8 we show that the quantum double of H is symmetric and unimodular.

math.RA↗

arXiv2001

An approach to Hopf algebras via Frobenius coordinates II

We study a Hopf algebra $H$, which is finitely generated and projective over a commutative ring $k$, as a $P$-Frobenius algebra. We define modular functions in this setting, and provide a complete proof of Radford's formula for the fourth power of the antipode, using Frobenius algebraic techniques. As further applications, we extend Etingof and Gelaki's result that a separable and coseparable Hopf algebra has antipode of order two, the result of Schneider that Hopf subalgebras are twisted Frobenius extensions, and show that the quantum double is always a Frobenius algebra.

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arXiv2001

Frobenius extensions and weak Hopf algebras

We study a symmetric Markov extension of k-algebras N \into M, a certain kind of Frobenius extension with conditional expectation that is tracial on the centralizer and dual bases with a separability property. We place a depth two condition on this extension, which is essentially the requirement that the Jones tower N \into M \into M_1 \into M_2 can be obtained by taking relative tensor products with centralizers A = C_{M_1}(N) and B = C_{M_2}(M). Under this condition, we prove that N \into M is the invariant subalgebra pair of a weak Hopf algebra action by A, i.e., that N = M^A. The endomorphism algebra M_1 = \End_N M is shown to be isomorphic to the smash product algebra M # A. We also extend results of Szymanski, Vainerman and the second author, and the authors.

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