Israel Journal of Mathematics最新文献

This paper classifies all the Dirac series (that is, irreducible unitary representations having non-zero Dirac cohomology) of E₇₍₇₎. Enhancing the Helgason–Johnson bound in 1969 for the group E₇₍₇₎ is one key ingredient. Our calculation partially supports Vogan’s fundamental parallelepiped (FPP) conjecture. As applications, when passing to Dirac index, we continue to find cancellation between the even part and the odd part of Dirac cohomology. Moreover, for the first time, we find Dirac series whose spin lowest K-types have multiplicities.

This paper deals with countable products of countable Borel equivalence relations and equivalence relations “just above” those in the Borel reducibility hierarchy. We show that if E is strongly ergodic with respect to μ then E^ℕ is strongly ergodic with respect to μ^ℕ. We answer questions of Clemens and Coskey regarding their recently defined Γ-jump operations, in particular showing that the ℤ^k+1-jump of E_∞ is strictly above the ℤ^k-jump of E_∞. We study a notion of equivalence relations which can be classified by infinite sequences of “definably countable sets”. In particular, we define an interesting example of such an equivalence relation which is strictly above E^ℕ_∞, strictly below =⁺, and is incomparable with the Γ-jumps of countable equivalence relations.

We establish a characterization of strong ergodicity between Borel equivalence relations in terms of symmetric models, using results from [Sha21]. The proofs then rely on a fine analysis of the very weak choice principles “every sequence of E-classes admits a choice sequence”, for various countable Borel equivalence relations E.

Let A be an associative algebra endowed with a superautomorphism φ. In this paper we completely classify the finite-dimensional simple algebras with superautomorphism of order ≤ 2. Moreover, after generalizing the Wedderburn–Malcev Theorem in this setting, we prove that the sequence of φ-codimensions of A is polynomially bounded if and only if the variety generated by A does not contain the group algebra of ℤ₂ and the algebra of 2 × 2 upper triangular matrices with suitable superautomorphisms.

We investigate a problem of when commutative local domains have a finite number of trace ideals. The problem is left for the case of dimension one. In this paper, with a necessary assumption, we give a complete answer by using integrally closed ideals. We also explore properties of such domains related to birational extensions, reflexive ideals, and reflexive Ulrich modules. Special attention is given in the case of numerical semigroup rings of non-gap four. We then obtain a criterion for a ring to have a finite number of reflexive ideals up to isomorphism. Non-domains arising from fiber products are also explored.

Let M be a smooth, orientable, closed, connected 4-manifold and suppose that H₁(M; ℤ) is finitely generated and has no 2-torsion. We give a homotopy decomposition of the suspension of M in terms of spheres, Moore spaces and ΣℂP². This is used to calculate any reduced generalized cohomology theory of M as a group and to determine the homotopy types of certain current groups and gauge groups.

For a very general complex projective K3 surface S and a smooth projective surface A with trivial canonical class, we prove that there is no dominant rational map A → S, which is not an isomorphism.

We prove a quantitative equidistribution statement for certain adelic homogeneous subsets in positive characteristic. As an application, we describe a proof of property (τ) for arithmetic groups in this context.

We introduce a class of finite-dimensional superalgebras over an algebraically closed field of characteristic zero, whose Grassmann envelopes generate all ∗-minimal varieties. Moreover, we prove that any affine minimal variety of superalgebras with superinvolution is generated by a suitable element in this selected class.

On the unit tangent bundle of a nonflat compact nonpositively curved surface, we prove that there is a unique probability Borel measure invariant by a horocyclic flow which gives full measure to the set of rank 1 vectors recurrent by the geodesic flow. If we assume in addition that the surface has no flat strips, we show that the horocyclic flow is uniquely ergodic. These results are valid for any parametrization of the horocyclic flow.

We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type A. The statement is not true in other types, and we propose a weaker correspondence. Namely, we introduce distinguished subsets of the Weyl group on which the join operation conjecturally agrees with the intersections of Verma modules. We also relate our conjecture with a statement about the socles of the cokernels of inclusions between Verma modules. The latter determines the first Ext space between a simple module and a Verma module. We give a conjectural complete description of such socles which we verify in a number of cases. Along the way, we determine the poset structure of the join-irreducible elements in Weyl groups and obtain closed formulae for certain families of Kazhdan–Lusztig polynomials.