21,857 research outputs found
Continuous spectral decompositions of Abelian group actions on C*-algebras
Let G be a locally compact Abelian group. Following Ruy Exel, we view Fell
bundles over the Pontrjagin dual of G as continuous spectral decompositions of
G-actions on C*-algebras. We classify such spectral decompositions using
certain dense subspaces related to Marc Rieffel's theory of
square-integrability. There is a unique continuous spectral decomposition if
the group acts properly on the primitive ideal space of the C*-algebra. But
there are also examples of group actions without or with several inequivalent
spectral decompositions.Comment: 34 page
Krull-Schmidt decompositions for thick subcategories
Following Krause \cite{Kr}, we prove Krull-Schmidt type decomposition
theorems for thick subcategories of various triangulated categories including
the derived categories of rings, Noetherian stable homotopy categories, stable
module categories over Hopf algebras, and the stable homotopy category of
spectra. In all these categories, it is shown that the thick ideals of small
objects decompose uniquely into indecomposable thick ideals. We also discuss
some consequences of these decomposition results. In particular, it is shown
that all these decompositions respect K-theory.Comment: Added more references, fixed some typos, to appear in Journal of Pure
and Applied Algebra, 22 pages, 1 figur
PT symmetry, Cartan decompositions, Lie triple systems and Krein space related Clifford algebras
Gauged PT quantum mechanics (PTQM) and corresponding Krein space setups are
studied. For models with constant non-Abelian gauge potentials and extended
parity inversions compact and noncompact Lie group components are analyzed via
Cartan decompositions. A Lie triple structure is found and an interpretation as
PT-symmetrically generalized Jaynes-Cummings model is possible with close
relation to recently studied cavity QED setups with transmon states in
multilevel artificial atoms. For models with Abelian gauge potentials a hidden
Clifford algebra structure is found and used to obtain the fundamental symmetry
of Krein space related J-selfadjoint extensions for PTQM setups with
ultra-localized potentials.Comment: 11 page
Extensions and block decompositions for finite-dimensional representations of equivariant map algebras
Suppose a finite group acts on a scheme and a finite-dimensional Lie
algebra . The associated equivariant map algebra is the Lie
algebra of equivariant regular maps from to . The irreducible
finite-dimensional representations of these algebras were classified in
previous work with P. Senesi, where it was shown that they are all tensor
products of evaluation representations and one-dimensional representations. In
the current paper, we describe the extensions between irreducible
finite-dimensional representations of an equivariant map algebra in the case
that is an affine scheme of finite type and is reductive.
This allows us to also describe explicitly the blocks of the category of
finite-dimensional representations in terms of spectral characters, whose
definition we extend to this general setting. Applying our results to the case
of generalized current algebras (the case where the group acting is trivial),
we recover known results but with very different proofs. For (twisted) loop
algebras, we recover known results on block decompositions (again with very
different proofs) and new explicit formulas for extensions. Finally,
specializing our results to the case of (twisted) multiloop algebras and
generalized Onsager algebras yields previously unknown results on both
extensions and block decompositions.Comment: 41 pages; v2: minor corrections, formatting changed to match
published versio
- …