Cartan Subalgebra Research Articles

On Bi-exactness of Discrete Quantum Groups

We define Ozawa's notion of bi-exactness to discrete quantum groups, and then prove some structural properties of associated von Neumann algebras. In particular, we prove that any non amenable subfactor of free quantum group von Neumann algebras, which is an image of a faithful normal conditional expectation, has no Cartan subalgebras.

Gauge Fixing and Classical Dynamical r-Matrices in ISO(2, 1)-Chern–Simons Theory

We apply Dirac's gauge fixing procedure to Chern-Simons theory with gauge group ISO(2,1) on manifolds RxS, where S is a punctured oriented surface of general genus. For all gauge fixing conditions that satisfy certain structural requirements, this yields an explicit description of the Poisson structure on the moduli space of flat ISO(2,1)-connections on S via the resulting Dirac bracket. The Dirac bracket is determined by classical dynamical r-matrices for ISO(2,1). We show that the Poisson structures and classical dynamical r-matrices arising from different gauge fixing conditions are related by dynamical ISO(2,1)-valued transformations that generalise the usual gauge transformations of classical dynamical r-matrices. By means of these transformations, it is possible to classify all Poisson structures and classical dynamical r-matrices obtained from such gauge fixings. Generically these Poisson structures combine classical dynamical r-matrices for non-conjugate Cartan subalgebras of iso(2,1).

English

We obtain an estimate of Voiculescu's (modified) free entropy dimension for generators of a ${II}_1$-factor $\mc{M}$ with a subfactor $\mc{N}$ containing an abelian subalgebra $\mc{A}$ of finite multiplicity. It implies in particular that the interpolated free group subfactors of finite Jones index do not have abelian subalgebras of finite multiplicity or Cartan subalgebras.

Torus Invariants of the Homogeneous Coordinate Ring of G/B – Connection with Coxeter Elements

In this article, we prove that for any indecomposable dominant character χ of a maximal torus T of a simple adjoint group G over ℂ such that there is a Coxeter element w in the Weyl group W for which , the graded algebra is a polynomial ring if and only if dim(H 0(G/B, ℒχ) T ) ≤rank of G. We also prove that the coordinate ring ℂ[𝔥] of the cartan subalgebra 𝔥 of the Lie algebra 𝔤 of G and are isomorphic if and only if is nonempty for some coxeter element w in W, where α0 denotes the highest long root.

Partially harmonic forms and models of H-series

Let G be a real reductive Lie group, let H=TA be the identity component of a Cartan subgroup, and let h be the corresponding Cartan subalgebra. This leads to a parabolic subgroup of G whose identity component is MAN. The unitary G-representations induced by MAN are known as the H-series. We study symplectic geometry of G×h and apply geometric quantization to construct unitary G-representations by partially harmonic forms. They are direct integrals of the H-series, indexed by the image of the moment map. We also perform symplectic reduction and symplectic induction, and consider their analogues in representation theory via geometric quantization.

Unique Cartan decomposition for II1 factors arising from arbitrary actions of free groups

We prove that for any free ergodic probability measure-preserving action \({\mathbb{F}_n \curvearrowright (X, \mu)}\) of a free group on n generators \({\mathbb{F}_n, 2\leq n \leq \infty}\), the associated group measure space II1 factor \({L^\infty (X)\rtimes \mathbb{F}_n}\) has L∞(X) as its unique Cartan subalgebra, up to unitary conjugacy. We deduce that group measure space II1 factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II1 factors arising from arbitrary actions of a much larger family of groups, including all free products of amenable groups and their direct products.

Amalgamated free product type III factors with at most one Cartan subalgebra

AbstractWe investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras ${\mathop{M{}_{1} \ast }\nolimits}_{B} {M}_{2} $ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $({M}_{1} , {\varphi }_{1} )\ast ({M}_{2} , {\varphi }_{2} )$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established by A. Ioana [Cartan subalgebras of amalgamated free product ${\mathrm{II} }_{1} $factors, arXiv:1207.0054]. Next, we prove that any countable nonsingular ergodic equivalence relation $ \mathcal{R} $ defined on a standard measure space and which splits as the free product $ \mathcal{R} = { \mathcal{R} }_{1} \ast { \mathcal{R} }_{2} $ of recurrent subequivalence relations gives rise to a nonamenable factor $\mathrm{L} ( \mathcal{R} )$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors ${\mathrm{L} }^{\infty } (X)\rtimes \Gamma $ arising from nonsingular free ergodic actions $\Gamma \curvearrowright (X, \mu )$ on standard measure spaces of amalgamated groups $\Gamma = {\mathop{\Gamma {}_{1} \ast }\nolimits}_{\Sigma } {\Gamma }_{2} $ over a finite subgroup $\Sigma $.

Some radicals, Frattini and Cartan subalgebras of Leibniz n-algebras

In the present work, we introduce notions such as -solvability, - and -nilpotency and the corresponding radicals. We prove that these radicals are invariant under derivations of Leibniz -algebras. The Frattini and Cartan subalgebras of Leibniz -algebras are studied. In particular, we construct examples that show a classical result on conjugacy of Cartan subalgebras of Lie algebras, which also holds in Leibniz algebras and Lie -algebras, is not true for Leibniz -algebras.

A cohomological proof of Peterson–Kacʼs theorem on conjugacy of Cartan subalgebras for affine Kac–Moody Lie algebras

This paper deals with the problem of conjugacy of Cartan subalgebras for affine Kac–Moody Lie algebras. Unlike the methods used by Peterson and Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of J. Tits on buildings.

A class of groups for which every action is W$^*$-superrigid

We prove the uniqueness of the group measure space Cartan subalgebra in crossed products A \rtimes \Gamma covering certain cases where \Gamma is an amalgamated free product over a non-amenable subgroup. In combination with Kida's work we deduce that if \Sigma <\mathrm{SL}(3,\mathbb{Z}) denotes the subgroup of matrices g with g_{31} = g_{32}=0 , then any free ergodic probability measure preserving action of \Gamma = \mathrm{SL}(3,\mathbb{Z})*_\Sigma \mathrm{SL}(3,\mathbb{Z}) is stably W*-superrigid. In the second part we settle a technical issue about the unitary conjugacy of group measure space Cartan subalgebras.

Some unique group-measure space decomposition results

Using an approach emerging from the theory of closable derivations on von Neumann algebras, we exhibit a class of groups CR satisfying the following property: given any groups G_1, G_2 in CR, then any free, ergodic, measure preserving action on a probability space G_1 x G_2 on X gives rise to a von Neumann algebra with unique group measure space Cartan subalgebra. Pairing this result with Popa's Orbit Equivalence Superrigidity Theorem we obtain new examples of W*-superrigid actions.

Low-dimensional cohomology of Lie superalgebra [formula omitted] with coefficients in Witt or Special superalgebras

Over a field of characteristic p>2, the Witt superalgebras are viewed as modules of the special linear Lie superalgebra slm|n by means of the adjoint representation. The low-dimensional cohomology groups of slm|n with coefficients in Witt superalgebras are computed by virtue of the direct sum decomposition of submodules and the weight space decompositions of these submodules relative to the standard Cartan subalgebra of slm|n. Then we are able to obtain the low-dimensional cohomology groups of slm|n with coefficients in Special superalgebras, which are slm|n-submodules of Witt superalgebras.

-ALGEBRAS ASSOCIATED WITH LAMBDA-SYNCHRONIZING SUBSHIFTS AND FLOW EQUIVALENCE

AbstractThe class of $\lambda $-synchronizing subshifts generalizes the class of irreducible sofic shifts. A $\lambda $-synchronizing subshift can be presented by a certain $\lambda $-graph system, called the $\lambda $-synchronizing $\lambda $-graph system. The $\lambda $-synchronizing $\lambda $-graph system of a $\lambda $-synchronizing subshift can be regarded as an analogue of the Fischer cover of an irreducible sofic shift. We will study algebraic structure of the ${C}^{\ast } $-algebra associated with a $\lambda $-synchronizing $\lambda $-graph system and prove that the stable isomorphism class of the ${C}^{\ast } $-algebra with its Cartan subalgebra is invariant under flow equivalence of $\lambda $-synchronizing subshifts.

Affine Mirković-Vilonen polytopes

Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra \(\mathfrak{g}\) has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given \(\mathfrak{g}\), there is a limit crystal, usually denoted by B(−∞), which contains all the other crystals. When \(\mathfrak{g}\) is finite dimensional, a convex polytope, called the Mirkovic-Vilonen polytope, can be associated to each element in B(−∞). This polytope sits in the dual space of a Cartan subalgebra of \(\mathfrak{g}\), and its edges are parallel to the roots of \(\mathfrak{g}\). In this paper, we generalize this construction to the case where \(\mathfrak{g}\) is a symmetric affine Kac-Moody algebra. The datum of the polytope must however be complemented by partitions attached to the edges parallel to the imaginary root δ. We prove that these decorated polytopes are characterized by conditions on their normal fans and on their 2-faces. In addition, we discuss how our polytopes provide an analog of the notion of Lusztig datum for affine Kac-Moody algebras. Our main tool is an algebro-geometric model for B(−∞) constructed by Lusztig and by Kashiwara and Saito, based on representations of the completed preprojective algebra Λ of the same type as \(\mathfrak{g}\). The underlying polytopes in our construction are described with the help of Buan, Iyama, Reiten and Scott’s tilting theory for the category \(\Lambda \text {\upshape -}\mathrm {mod}\). The partitions we need come from studying the category of semistable Λ-modules of dimension-vector a multiple of δ.

Discovering real lie subalgebras of $\mathfrak {e}_6$e6 using Cartan decompositions

The process of complexification is used to classify Lie algebras and identify their Cartan subalgebras. However, this method does not distinguish between real forms of a complex Lie algebra, which can differ in signature. In this paper, we show how Cartan decompositions of a complexified Lie algebra can be combined with information from the Killing form to identify real forms of a given Lie algebra. We apply this technique to \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(3,\mathbb {O})$\end{document}sl(3,O), a real form of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}_6$\end{document}e6 with signature (52, 26), thereby identifying chains of real subalgebras and their corresponding Cartan subalgebras within \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}_6$\end{document}e6. Motivated by an explicit construction of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(3,\mathbb {O})$\end{document}sl(3,O), we then construct an Abelian group of order 8 which acts on the real forms of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}_6$\end{document}e6, leading to the identification of 8 particular copies of the 5 real forms of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}_6$\end{document}e6, which can be distinguished by their relationship to the original copy of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(3,\mathbb {O})$\end{document}sl(3,O).

On the structural theory of [formula omitted] factors of negatively curved groups, II: Actions by product groups

This paper contains a series of structural results for von Neumann algebras arising from measure preserving actions by product groups on probability spaces. Expanding upon the methods used earlier by the first two authors, we obtain new examples of strongly solid factors as well as von Neumann algebras with unique or no Cartan subalgebra. For instance, we show that every II1 factor associated with a weakly amenable group in the class S of Ozawa is strongly solid. There is also the following product version of this result: any maximal abelian ⋆-subalgebra of any II1 factor associated with a finite product of weakly amenable groups in the class S of Ozawa has an amenable normalizing algebra. Finally, pairing some of these results with a cocycle superrigidity result of Ioana, it follows that compact actions by finite products of lattices in Sp(n,1), n≥2, are virtually W∗-superrigid.

Computing with real Lie algebras: Real forms, Cartan decompositions, and Cartan subalgebras

We describe algorithms for performing various tasks related to real simple Lie algebras. These algorithms form the basis of our software package CoReLG, written in the language of the computer algebra system GAP4. First, we describe how to efficiently construct real simple Lie algebras up to isomorphism. Second, we consider a real semisimple Lie algebra g. We provide an algorithm for constructing a maximally (non-)compact Cartan subalgebra of g; this is based on the theory of Cayley transforms. We also describe the construction of a Cartan decomposition g=k⊕p. Using these results, we provide an algorithm to construct all Cartan subalgebras of g up to conjugacy; this is a constructive version of a classification theorem due to Sugiura.

VOGAN DIAGRAMS OF AFFINE TWISTED LIE SUPERALGEBRAS

A Vogan diagram is a Dynkin diagram with a Cartan involution of twisted affine superlagebras based on maximally compact Cartan subalgebras. This article construct the Vogan diagrams of twisted affine superalgebras. AMS Subject Classification: 17B20, 17B22, 17B40

II1 FACTORS AND EQUIVALENCE RELATIONS WITH DISTINCT FUNDAMENTAL GROUPS

We construct a group measure space II1 factor that has two nonconjugate Cartan subalgebras. We show that the fundamental group of the II1 factor is trivial, while the fundamental group of the equivalence relation associated with the second Cartan subalgebra is nontrivial. This is not absurd as the second Cartan inclusion is twisted by a 2-cocycle.

Unique Cartan decomposition for II 1 factors arising from arbitrary actions of hyperbolic groups

Abstract We prove that for any free ergodic probability measure preserving action Γ → ( X , μ ) $\Gamma \rightarrow (X,\mu )$ of a non-elementary hyperbolic group, or a lattice in a rank one simple Lie group, the associated group measure space II 1 factor L ∞ ( X ) ⋊ Γ ${\mathord {\text{\rm L}}^\infty (X) \rtimes \Gamma }$ has L ∞ ( X ) ${\mathord {\text{\rm L}}^\infty (X)}$ as its unique Cartan subalgebra, up to unitary conjugacy.