Topic
Centralizer and normalizer
About: Centralizer and normalizer is a research topic. Over the lifetime, 2752 publications have been published within this topic receiving 27760 citations. The topic is also known as: centralizer subgroup.
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Papers
TL;DR: In this article, it was shown that the index of a subfactor has to be either greater or equal than 4 or equal to 4cosZ(x) for some l~N, I>3 and that there exist subfactors for all these index values.
Abstract: In his paper [J-l] V. Jones introduced an index, which 'measures' the size of a subfactor in a II1 factor. The main result of that paper is that the index of a subfactor has to be either greater or equal than 4 or it has to be equal to 4cosZ(x//) for some l~N, I>3 and that there exist subfactors for all these index values. Similarly as for subgroups, the index alone does not characterize the subfactor up to conjugacy by automorphisms. The fact that there are only countably many possible index values < 4 seems to be related to another invariant. Subfactors with index less than 4 always have trivial centralizers, (or relative commutants), i.e. the only elements of the factor which commute with every element of the subfactor are multiples of the identity. On the other hand, the examples given in I-J-l] for subfactors with index greater than 4 all have nontrivial centralizers. Furthermore, all known examples of subfactors with trivial relative commutants have as index an algebraic integer. At the current state of knowledge, it is still unknown whether there are only countably many values possible for the index of subfactors with trivial centralizers. Note however, that the set of all possible index values of a subfactor with trivial centralizer in an arbitrary II~ factor has to be a closed subset of R (see [HW]). Our original motivation for this paper was to study how subfactors of the hyperfinite II1 factor can be constructed via AF algebras. We provide a method of computing the index and we give an upper bound for the size of the centralizer of the constructed subfactor. Our general results will then be applied to the series of complex Hecke algebras H,(q), n~N of type A,_I. Their standard generators gx, g2, -.., gn1 satisfy the same relations as a set of simple reflections of the symmetric group S, except that the reflection property g~ = 1 is replaced by g ~ = ( q 1 ) g i + q . It is well-known that H,(q) is isomorphic to C S , if q is not a root of unity. If the parameter is a root of unity, Hn(q) may no longer bc sernisimple and its structure is not known in general. This is, however, the most interesting case as far as subfactors are concerned. We define representations p of Ha(q) such that p(H,(q)) is semisimple for all n~N. Together with
478 citations
TL;DR: In this paper, the authors established sufficient conditions for a finite group to have a nontrivial center or a normal subgroup of odd order in order to be core-free in finite groups.
470 citations
TL;DR: In this paper, it was shown that the normalizer of any diffuse amenable subalgebra of a free group factor L(F-r) generates an amenable von Neumann sub-algebra.
Abstract: We prove that the normalizer of any diffuse amenable subalgebra of a free group factor L(F-r) generates an amenable von Neumann subalgebra. Moreover, any II1 factor of the form Q (circle times) over barL(F-r), with Q an arbitrary subfactor of a tensor product of free group factors, has no Cartan subalgebras. We also prove that if a free ergodic measure-preserving action of a free group F-r, 2 <= r <= infinity, on a probability space (X, mu) is profinite then the group measure space factor L-infinity (X) F-r has unique Cartan subalgebra, up to unitary conjugacy.
329 citations
TL;DR: In this paper, the authors used the term involution for a group element of order 2 and showed that there exist only a finite number of simple groups in which the normalizer of an involution is isomorphic to a given group.
Abstract: odd order are soluble. We shall use the term involution for a group element of order 2. If m is the total number of involutions of 65 and if we set n = g/m, the same method shows that 65 contains a normal subgroup V distinct from 5 such that the index of V is either 2 or is less than [n(n + 2)/2]! (where [x] denotes the largest integer not exceeding the real number x). If J is an involution in 65 and if n(J) is the order of its normalizer 91(J) in 5, then n < n(J). It then follows that there exist only a finite number of simple groups in which the normalizer of an involution is isomorphic to a given group.
244 citations
1 Jan 1985
TL;DR: In this paper, the authors give Harish-Chandra's result that there are no other spherical functions, besides those described in Chapter IV, on SL 2(R) where the proofs are short and easy.
Abstract: So far we have avoided to a large extent the more refined behavior of functions with respect to Lie derivatives. For the theory of spherical functions, we dealt with eigenvectors of convolution operators. The time has come to relate some invariants we have found in the representation theory with some of the invariant differential operators on G. Bargmann [Ba] saw how coefficient functions are eigenfunctions of such operators, Harish-Chandra got a complete insight into the situation by determining the center of the algebra of invariant differential operators, the centralizer of K in this algebra. Gelfand characterized spherical functions as eigenfunctions of this centralizer. In this chapter, we give Harish-Chandra’s result that there are no other spherical functions, besides those described in Chapter IV, on SL 2(R) where the proofs are short and easy.
231 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 35 |
| 2024 | 46 |
| 2023 | 108 |
| 2022 | 254 |
| 2021 | 118 |
| 2020 | 132 |