13,896 research outputs found
Centralizers of Finite Subgroups of the Mapping Class Group
In this paper, we study the action of finite subgroups of the mapping class
group of a surface on the curve complex. We prove that if the diameter of the
almost fixed point set of a finite subgroup H is big enough, then the
centralizer of H is infinite.Comment: 16 page
A note on the distance spectra of co-centralizer graphs
Let be a finite non abelian group. The centralizer graph of is a
simple undirected graph , whose vertex set consists of proper
centralizers of and two vertices are adjacent if and only if their
cardinalities are identical [6]. We call the complement of the centralizer
graph as the co-centralizer graph. In this paper, we investigate the distance,
distance (signless) Laplacian spectra of co-centralizer graphs of some classes
of finite non-abelian groups, and obtain some conditions on a group so that the
co-centralizer graph is distance, distance (signless) Laplacian integral.Comment: arXiv admin note: substantial text overlap with arXiv:2208.0061
Centralizers of p-Subgroups in Simple Locally Finite Groups
In Ersoy et al. [J. Algebra481 (2017), 1–11], we have proved that if G is a locally finite group with an elementary abelian p-subgroup A of order strictly greater than p2 such that CG(A) is Chernikov and for every non-identity α ∈ A the centralizer CG(α) does not involve an infinite simple group, then G is almost locally soluble. This result is a consequence of another result proved in Ersoy et al. [J. Algebra481 (2017), 1–11], namely: if G is a simple locally finite group with an elementary abelian group A of automorphisms acting on it such that the order of A is greater than p2, the centralizer CG(A) is Chernikov and for every non-identity α ∈ A the set of fixed points CG(α) does not involve an infinite simple groups then G is finite. In this paper, we improve this result about simple locally finite groups: Indeed, suppose that G is a simple locally finite group, consider a finite non-abelian subgroup P of automorphisms of exponent p such that the centralizer CG(P) is Chernikov and for every non-identity α ∈ P the set of fixed points CG(α) does not involve an infinite simple group. We prove that if Aut(G) has such a subgroup, then G ≅PSLp(k) where char k ≠p and P has a subgroup Q of order p2 such that CG(P) = Q
Classification of finite dimensional irreducible modules over W-algebras
Finite W-algebras are certain associative algebras arising in Lie theory.
Each W-algebra is constructed from a pair of a semisimple Lie algebra g (our
base field is algebraically closed and of characteristic 0) and its nilpotent
element e. In this paper we classify finite dimensional irreducible modules
with integral central character over W-algebras. In more detail, in a previous
paper the first author proved that the component group A(e) of the centralizer
of the nilpotent element under consideration acts on the set of finite
dimensional irreducible modules over the W-algebra and the quotient set is
naturally identified with the set of primitive ideals in U(g) whose associated
variety is the closure of the adjoint orbit of e. In this paper for a given
primitive ideal with integral central character we compute the corresponding
A(e)-orbit. The answer is that the stabilizer of that orbit is basically a
subgroup of A(e) introduced by G. Lusztig. In the proof we use a variety of
different ingredients: the structure theory of primitive ideals and
Harish-Chandra bimodules of semisimple Lie algebras, the representation theory
of W-algebras, the structure theory of cells and Springer representations, and
multi-fusion monoidal categories.Comment: 52 pages, preliminarly version, comments welcome; v2 53 pages, small
correction
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