Nonlinear lie-type derivations on full matrix algebras

Let ${\mathcal{R}}$ be a 2-torsion free commutative ring with identity and ${{\rm M}_n(\mathcal{R}) (n\geq 2)}$ be the full matrix algebra over ${\mathcal{R}}$ . In this note, we prove that every nonlinear Lie triple derivation on ${{\rm M}_n(\mathcal{R})}$ is of the standard form, i.e. it can be expressed as a sum of an inner derivation, an additive induced derivation and a functional annihilating all second commutators of ${{\rm M}_n(\mathcal{R})}$ . A open conjecture about Lie n-derivations is posed at the end of this note.     

On uniformly bounded spherical functions in Hilbert space

Let G be a commutative group, written additively, with a neutral element 0, and let K be a finite group. Suppose that K acts on G via group automorphisms ${G \ni a \mapsto ka \in G}$ , ${k \in K}$ . Let ${{\mathfrak{H}}}$ be a complex Hilbert space and let ${{\mathcal L}({\mathfrak{H}})}$ be the algebra of all bounded linear operators on ${{\mathfrak{H}}}$ . A mapping ${u \colon G \to {\mathcal L}({\mathfrak{H}})}$ is termed a K-spherical function if it satisfies (1) ${|K|^{-1} \sum_{k\in K} u (a+kb)=u (a) u (b)}$ for any ${a,b\in G}$ , where |K| denotes the cardinality of K, and (2) ${u (0) = {\rm id}_{\mathfrak {H}},}$ where ${{\rm id}_{\mathfrak {H}}}$ designates the identity operator on ${{\mathfrak{H}}}$ . The main result of the paper is that for each K-spherical function ${u \colon G \to {\mathcal {L}}({\mathfrak {H}})}$ such that ${\| u \|_{\infty} = \sup_{a\in G} \| u (a)\|_{{\mathcal L}({\mathfrak{H}})} < \infty,}$ there is an invertible operator S in ${{\mathcal L}({\mathfrak{H}})}$ with ${\| S \| \, \| S^{-1}\| \leq |K| \, \| u \|_{\infty}^2}$ such that the K-spherical function ${{\tilde{u}} \colon G \to {\mathcal L}({\mathfrak{H}})}$ defined by ${{\tilde{u}}(a) = S u (a) S^{-1},\,a \in G,}$ satisfies ${{\tilde{u}}(-a) = {\tilde{u}}(a)^*}$ for each ${a \in G}$ . It is shown that this last condition is equivalent to insisting that ${{\tilde{u}}(a)}$ be normal for each ${a \in G}$ .     

Howe correspondence and Springer correspondence for real reductive dual pairs

We consider a real reductive dual pair (G′, G) of type I, with rank ${({\rm G}^{\prime}) \leq {\rm rank(G)}}$ . Given a nilpotent coadjoint orbit ${\mathcal{O}^{\prime} \subseteq \mathfrak{g}^{{\prime}{*}}}$ , let ${\mathcal{O}^{\prime}_\mathbb{C} \subseteq \mathfrak{g}^{{\prime}{*}}_\mathbb{C}}$ denote the complex orbit containing ${\mathcal{O}^{\prime}}$ . Under some condition on the partition λ′ parametrizing ${\mathcal{O}^{\prime}}$ , we prove that, if λ is the partition obtained from λ by adding a column on the very left, and ${\mathcal{O}}$ is the nilpotent coadjoint orbit parametrized by λ, then ${\mathcal{O}_\mathbb{C}= \tau (\tau^{\prime -1}(\mathcal{O}_\mathbb{C}^{\prime}))}$ , where ${\tau, \tau^{\prime}}$ are the moment maps. Moreover, if ${chc(\hat\mu_{\mathcal{O}^{\prime}}) \neq 0}$ , where chc is the infinitesimal version of the Cauchy-Harish-Chandra integral, then the Weyl group representation attached by Wallach to ${\mu_{\mathcal{O}^{\prime}}}$ with corresponds to ${\mathcal{O}_\mathbb{C}}$ via the Springer correspondence.     

Modular Decomposition of the Orlik-Terao Algebra

Let ${\mathcal{A}}$ be a collection of n linear hyperplanes in ${\mathbb{k}^\ell}$ , where ${\mathbb{k}}$ is an algebraically closed field. The Orlik-Terao algebra of ${\mathcal{A}}$ is the subalgebra ${{\rm R}(\mathcal{A})}$ of the rational functions generated by reciprocals of linear forms vanishing on hyperplanes of ${\mathcal{A}}$ . It determines an irreducible subvariety ${Y (\mathcal{A})}$ of ${\mathbb{P}^{n-1}}$ . We show that a flat X of ${\mathcal{A}}$ is modular if and only if ${{\rm R}(\mathcal{A})}$ is a split extension of the Orlik-Terao algebra of the subarrangement ${\mathcal{A}_X}$ . This provides another refinement of Stanley’s modular factorization theorem [34] and a new characterization of modularity, similar in spirit to the fibration theorem of [27]. We deduce that if ${\mathcal{A}}$ is supersolvable, then its Orlik-Terao algebra is Koszul. In certain cases, the algebra is also a complete intersection, and we characterize when this happens.     

On the pullback of an arithmetic theta function

In this paper, we describe a relationship between the simplest examples of arithmetic theta series. The first of these are the weight 1 theta series ${\widehat{\phi}_{\mathcal C}(\tau)}$ defined using arithmetic 0-cycles on the moduli space ${\mathcal C}$ of elliptic curves with CM by the ring of integers ${O_{\kappa}}$ of an imaginary quadratic field. The second such series ${\widehat{\phi}_{\mathcal M}(\tau)}$ has weight 3/2 and takes values in the arithmetic Chow group ${\widehat{{\rm CH}}^1(\mathcal{M})}$ of the arithmetic surface associated to an indefinite quaternion algebra ${B/\mathbb{Q}}$ . For an embedding ${O_\kappa \rightarrow O_B}$ , a maximal order in B, and a two sided O _B -ideal Λ, there is a morphism ${j_\Lambda:{\mathcal C} \rightarrow {\mathcal M}}$ and a pullback ${j_\Lambda^*: \widehat{{\rm CH}}^1(\mathcal{M}) \rightarrow \widehat{{\rm CH}}^1(\mathcal C)}$ . Our main result is an expression for the pullback ${j^*_\Lambda \widehat{\phi}_{\mathcal M}(\tau)}$ as a linear combination of products of ${\widehat{\phi}_{\mathcal C}(\tau)}$ ’s and classical weight ${\frac{1}{2}}$ theta series.     

On the Topology of Kac–Moody groups

We study the topology of spaces related to Kac–Moody groups. Given a Kac–Moody group over $\mathbb C $ , let $\text {K}$ denote the unitary form with maximal torus ${{\mathrm{T}}}$ having normalizer ${{\mathrm{N}}}({{\mathrm{T}}})$ . In this article we study the cohomology of the flag manifold $\text {K}/{{{\mathrm{T}}}}$ as a module over the Nil-Hecke algebra, as well as the (co)homology of $\text {K}$ as a Hopf algebra. In particular, if $\mathbb F $ has positive characteristic, we show that $\text {H}_*(\text {K},\mathbb F )$ is a finitely generated algebra, and that $\text {H}^*(\text {K},\mathbb F )$ is finitely generated only if $\text {K}$ is a compact Lie group . We also study the stable homotopy type of the classifying space $\text {BK}$ and show that it is a retract of the classifying space $\text {BN(T)}$ of ${{\mathrm{N}}}({{\mathrm{T}}})$ . We illustrate our results with the example of rank two Kac–Moody groups.     

Unbounded Induced Representations of ∗-Algebras

Induced representations of *-algebras by unbounded operators in Hilbert space are investigated. Conditional expectations of a *-algebra ${{\mathcal{A}}}$ onto a unital *-subalgebra ${{\mathcal{B}}}$ are introduced and used to define inner products on the corresponding induced modules. The main part of the paper is concerned with group graded *-algebras ${{\mathcal{A}}}=\oplus_{g\in G}{{\mathcal{A}}}_g$ for which the *-subalgebra ${{\mathcal{B}}}:={{\mathcal{A}}}_e$ is commutative. Then the canonical projection $p:{{\mathcal{A}}}\to{{\mathcal{B}}}$ is a conditional expectation and there is a partial action of the group G on the set ${{\mathcal{B}}}p$ of all characters of ${{\mathcal{B}}}$ which are nonnegative on the cone $\sum{{\mathcal{A}}}^2{{\mathcal{A}}}p{{\mathcal{B}}}.$ The complete Mackey theory is developed for *-representations of ${{\mathcal{A}}}$ which are induced from characters of ${{\widehat{{{\mathcal{B}}}}^+}}.$ Systems of imprimitivity are defined and two versions of the Imprimitivity Theorem are proved in this context. A concept of well-behaved *-representations of such *-algebras ${{\mathcal{A}}}$ is introduced and studied. It is shown that well-behaved representations are direct sums of cyclic well-behaved representations and that induced representations of well-behaved representations are again well-behaved. The theory applies to a large variety of examples. For important examples such as the Weyl algebra, enveloping algebras of the Lie algebras su(2), su(1,1), and of the Virasoro algebra, and *-algebras generated by dynamical systems our theory is carried out in great detail.     

Characterization of the automorphism group of quaternary linear Hadamard codes

A quaternary linear Hadamard code ${\mathcal{C}}$ is a code over ${\mathbb{Z}_4}$ such that, under the Gray map, gives a binary Hadamard code. The permutation automorphism group of a quaternary linear code ${\mathcal{C}}$ of length n is defined as ${{\rm PAut}(\mathcal{C}) = \{\sigma \in S_{n} : \sigma(\mathcal{C}) = \mathcal{C}\}}$ . In this paper, the order of the permutation automorphism group of a family of quaternary linear Hadamard codes is established. Moreover, these groups are completely characterized by computing the orbits of the action of ${{\rm PAut}(\mathcal{C})}$ on ${\mathcal{C}}$ and by giving the generators of the group. Since the dual of a Hadamard code is an extended 1-perfect code in the quaternary sense, the permutation automorphism group of these codes is also computed.     

Complemented modular lattices with involution and orthogonal geometry

With each orthogeometry (P, ⊥) we associate ${{\mathbb {L}}(P, \bot)}$ , a complemented modular lattice with involution (CMIL), consisting of all subspaces X and X ^⊥ such that dim X < ?₀, and we study its rôle in decompositions of (P, ⊥) as directed (resp., disjoint) union. We also establish a 1–1 correspondence between ?-varieties ${\mathcal {V}}$ of CMILs with ${\mathcal {V}}$ generated by its finite dimensional members and ‘quasivarieties’ ${\mathcal {G}}$ of orthogeometries: ${\mathcal {V}}$ consists of the CMILs representable within some geometry from ${\mathcal {G}}$ and ${\mathcal {G}}$ of the (P, ⊥) with ${{\mathbb {L}}(P, \bot) \in {\mathcal {V}}}$ . Here, ${\mathcal {V}}$ is recursively axiomatizable if and only if so is ${\mathcal {G}}$ . It follows that the equational theory of ${\mathcal {V}}$ is decidable provided that the equational theories of the ${\{{\mathbb {L}}(P, \bot)\, |\, (P, \bot) \in \mathcal {G}, {\rm{dim}} P = n\}}$ are uniformly decidable.     

Covering properties of ideals

Elekes proved that any infinite-fold cover of a σ-finite measure space by a sequence of measurable sets has a subsequence with the same property such that the set of indices of this subsequence has density zero. Applying this theorem he gave a new proof for the random-indestructibility of the density zero ideal. He asked about other variants of this theorem concerning I-almost everywhere infinite-fold covers of Polish spaces where I is a σ-ideal on the space and the set of indices of the required subsequence should be in a fixed ideal ${{\mathcal{J}}}$ on ω. We introduce the notion of the ${{\mathcal{J}}}$ -covering property of a pair ${({\mathcal{A}}, I)}$ where ${{\mathcal{A}}}$ is a σ-algebra on a set X and ${{I \subseteq \mathcal{P}(X)}}$ is an ideal. We present some counterexamples, discuss the category case and the Fubini product of the null ideal ${\mathcal{N}}$ and the meager ideal ${\mathcal{M}}$ . We investigate connections between this property and forcing-indestructibility of ideals. We show that the family of all Borel ideals ${{\mathcal{J}}}$ on ω such that ${\mathcal{M}}$ has the ${{\mathcal{J}}}$ -covering property consists exactly of non weak Q-ideals. We also study the existence of smallest elements, with respect to Katětov–Blass order, in the family of those ideals ${\mathcal{J}}$ on ω such that ${\mathcal{N}}$ or ${\mathcal{M}}$ has the ${\mathcal{J}}$ -covering property. Furthermore, we prove a general result about the cases when the covering property “strongly” fails.     

Conditional Reducibility of Certain Unbounded Nonnegative Hamiltonian Operator Functions

Let J and ${{\mathfrak{J}}}$ be operators on a Hilbert space ${{\mathcal{H}}}$ which are both self-adjoint and unitary and satisfy ${J{\mathfrak{J}}=-{\mathfrak{J}}J}$ . We consider an operator function ${{\mathfrak{A}}}$ on [0, 1] of the form ${{\mathfrak{A}}(t)={\mathfrak{S}}+{\mathfrak{B}}(t)}$ , ${t \in [0, 1]}$ , where ${\mathfrak{S}}$ is a closed densely defined Hamiltonian ( ${={\mathfrak{J}}}$ -skew-self-adjoint) operator on ${{\mathcal{H}}}$ with ${i {\mathbb{R}} \subset \rho ({\mathfrak{S}})}$ and ${{\mathfrak{B}}}$ is a function on [0, 1] whose values are bounded operators on ${{\mathcal{H}}}$ and which is continuous in the uniform operator topology. We assume that for each ${t \in [0,1] \,{\mathfrak{A}}(t)}$ is a closed densely defined nonnegative (=J-accretive) Hamiltonian operator with ${i {\mathbb{R}} \subset \rho({\mathfrak{A}}(t))}$ . In this paper we give sufficient conditions on ${{\mathfrak{S}}}$ under which ${{\mathfrak{A}}}$ is conditionally reducible, which means that, with respect to a natural decomposition of ${{\mathcal{H}}}$ , ${{\mathfrak{A}}}$ is diagonalizable in a 2×2 block operator matrix function such that the spectra of the two operator functions on the diagonal are contained in the right and left open half planes of the complex plane. The sufficient conditions involve bounds on the resolvent of ${{\mathfrak{S}}}$ and interpolation of Hilbert spaces.     

Irregular and Singular Loci of Commuting Varieties

Let $ {\user1{\mathcal{C}}} $ be the commuting variety of the Lie algebra $ \mathfrak{g} $ of a connected noncommutative reductive algebraic group G over an algebraically closed field of characteristic zero. Let $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ be the singular locus of $ {\user1{\mathcal{C}}} $ and let $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ be the locus of points whose G-stabilizers have dimension > rk G. We prove that: (a) $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ is a nonempty subset of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ ; (b) $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{irr}}}} = 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ where the maximum is taken over all simple ideals $ \mathfrak{a} $ of $ \mathfrak{g} $ and $ l{\left( \mathfrak{a} \right)} $ is the “lacety” of $ \mathfrak{a} $ ; and (c) if $ \mathfrak{t} $ is a Cartan subalgebra of $ \mathfrak{g} $ and $ \alpha \in \mathfrak{t}^{*} $ root of $ \mathfrak{g} $ with respect to $ \mathfrak{t} $ , then $ \overline{{G{\left( {{\text{Ker}}\,\alpha \times {\text{Ker }}\alpha } \right)}}} $ is an irreducible component of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ of codimension 4 in $ {\user1{\mathcal{C}}} $ . This yields the bound $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ and, in particular, $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 2 $ . The latter may be regarded as an evidence in favor of the known longstanding conjecture that $ {\user1{\mathcal{C}}} $ is always normal. We also prove that the algebraic variety $ {\user1{\mathcal{C}}} $ is rational.     

On the annihilators and attached primes of top local cohomology modules

Let ${\mathfrak{a}}$ be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. It is shown that ${{\rm Ann}_R(H_{\mathfrak{a}}^{{\rm dim} M}(M))= {\rm Ann}_R(M/T_R(\mathfrak{a}, M))}$ , where ${T_R(\mathfrak{a}, M)}$ is the largest submodule of M such that ${{\rm cd}(\mathfrak{a}, T_R(\mathfrak{a}, M)) < {\rm cd}(\mathfrak{a}, M)}$ . Several applications of this result are given. Among other things, it is shown that there exists an ideal ${\mathfrak{b}}$ of R such that ${{\rm Ann}_R(H_{\mathfrak{a}}^{{\rm dim} M}(M))={\rm Ann}_R(M/H_{\mathfrak{b}}^{0}(M))}$ . Using this, we show that if ${ H_{\mathfrak{a}}^{{\rm dim} R}(R)=0}$ , then ${{{\rm Att}_R} H^{{\rm dim} R-1}_{\mathfrak a}(R)= \{\mathfrak{p} \in {\rm Spec} R | \,{\rm cd}(\mathfrak{a}, R/\mathfrak{p}) = {\rm dim} R-1\}.}$ These generalize the main results of Bahmanpour et al. (see [2, Theorem 2.6]), Hellus (see [7, Theorem 2.3]), and Lynch (see [10, Theorem 2.4]).     

Characterization of (generalized) Lie derivations on {\mathcal{J}}-subspace lattice algebras by local action

Let ${\mathcal{L}}$ be a ${\mathcal{J}}$ -subspace lattice on a Banach space X over the real or complex field ${\mathbb{F}}$ with dim X ≥ 2 and Alg ${\mathcal{L}}$ be the associated ${\mathcal{J}}$ -subspace lattice algebra. For any scalar ${\xi \in \mathbb{F}}$ , there is a characterization of any linear map L : Alg ${\mathcal{L} \rightarrow {\rm Alg} {\mathcal{L}}}$ satisfying ${L([A,B]_\xi) = [L(A),B]_\xi + [A,L(B)]_\xi}$ for any ${A, B \in{\rm Alg} {\mathcal{L}}}$ with AB = 0 (rep. ${[A,B]_ \xi = AB - \xi BA = 0}$ ) given. Based on these results, a complete characterization of (generalized) ξ-Lie derivations for all possible ξ on Alg ${\mathcal{L}}$ is obtained.     

Continuity of derivations of algebras of locally measurable operators

We prove that any derivation of the *-algebra ${LS({\mathcal{M}})}$ of all locally measurable operators affiliated with a properly infinite von Neumann algebra ${{\mathcal{M}}}$ is continuous with respect to the local measure topology ${t({\mathcal{M}})}$ . Building an extension of a derivation ${\delta:{\mathcal{M}}\rightarrow LS({\mathcal{M}})}$ up to a derivation from ${LS({\mathcal{M}})}$ into ${LS({\mathcal{M}})}$ , it is further established that any derivation from ${{\mathcal{M}}}$ into ${LS({\mathcal{M}})}$ is ${t({\mathcal{M}})}$ -continuous.     

Symmetric Graphs from Polytopes of High Rank

Given a polytope ${{\mathcal{P}}}$ of rank 2n, the faces of middle ranks n ? 1 and n constitute the vertices of a bipartite graph, the medial layer graph ${{M(\mathcal{P})}}$ of ${{\mathcal{P}}}$ . The group ${{D(\mathcal{P})}}$ of automorphisms and dualities of ${{\mathcal{P}}}$ has a natural action on this graph. We prove algebraic and combinatorial conditions on ${{\mathcal{P}}}$ that ensure this action is transitive on k-arcs in ${{M(\mathcal{P})}}$ for some small k (in particular focussing on k?=?3), and provide examples of families of polytopes that satisfy these conditions. We also examine how ${{D(\mathcal{P})}}$ acts on the k-stars based at vertices of ${{M(\mathcal{P})},}$ and describe self-dual regular polytopes (in particular those of rank 6) for which this action is transitive on the k-stars for small k.     

Categories of partial algebras for critical points between varieties of algebras

We denote by Con_c A the ${(\vee, 0)}$ -semilattice of all finitely generated congruences of an algebra A. A lifting of a ${(\vee, 0)}$ -semilattice S is an algebra A such that ${S \cong {\rm Con}_{\rm c} A}$ . The assignment Conc can be extended to a functor. The notion of lifting is generalized to diagrams of ${(\vee, 0)}$ -semilattices. A gamp is a partial algebra endowed with a partial subalgebra together with a semilattice-valued distance; gamps form a category that lends itself to a universal algebraic-type study. The raison d’être of gamps is that any algebra can be approximated by its finite subgamps, even in case it is not locally finite. Let ${\mathcal{V}}$ and ${\mathcal{W}}$ be varieties of algebras (on finite, possibly distinct, similarity types). Let P be a finite lattice. We assume the existence of a combinatorial object, called an ${\aleph_0}$ -lifter of P, of infinite cardinality ${\lambda}$ . Let ${\vec{A}}$ be a P-indexed diagram of finite algebras in ${\mathcal{V}}$ . If ${{\rm Con}_{\rm c} \circ \vec{A}}$ has no partial lifting in the category of gamps of ${\mathcal{W}}$ , then there is an algebra ${A \in \mathcal{V}}$ of cardinality ${\lambda}$ such that Con_c A is not isomorphic to Con_c B for any ${B \in \mathcal{W}}$ . This makes it possible to generalize several known results. In particular, we prove the following theorem, without assuming that ${\mathcal{W}}$ is locally finite. Let ${\mathcal{V}}$ be locally finite and let ${\mathcal{W}}$ be congruence-proper (i.e., congruence lattices of infinite members of ${\mathcal{W}}$ are infinite). The following equivalence holds. Every countable ${(\vee, 0)}$ -semilattice with a lifting in ${\mathcal{V}}$ has a lifting in ${\mathcal{W}}$ if and only if every ${\omega}$ -indexed diagram of finite ${(\vee, 0)}$ -semilattices with a lifting in ${\mathcal{V}}$ has a lifting in ${\mathcal{W}}$ . Gamps are also applied to the study of congruence-preserving extensions. Let ${\mathcal{V}}$ be a non-semidistributive variety of lattices and let n ≥ 2 be an integer. There is a bounded lattice ${A \in \mathcal{V}}$ of cardinality ${\aleph_1}$ with no congruence n-permutable, congruence-preserving extension. The lattice A is constructed as a condensate of a square-indexed diagram of lattices.