Bulletin of the London Mathematical Society最新文献

Let $�F$ be a field of characteristic zero and $�G$ a finite abelian group. In this paper, we prove that an affine variety of $�G$-graded PI-algebras is minimal if and only if it is generated by a graded algebra $��U�T�(�A_1�,�\cdots �,�A_m�;�\gamma �)�$ of upper block triangular matrices where $��A_1�,�\cdots �,�A_m�$ are finite-dimensional $�G$-simple algebras.

In this paper, we prove the following geometric inequalities in the Euclidean space $��R^n���(�n�⩾�3�)��$, which are weighted Alexandrov–Fenchel type inequalities,

We generalize the definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type due to Koziarz and Maubon to the context of singular klt varieties, where the natural fundamental groups to consider are those of smooth loci. Assuming that the rank of the target Lie group is not greater than two, we show that the Toledo invariant satisfies a Milnor–Wood-type inequality and we characterize the corresponding maximal representations.

We prove that the product of a subset and a normal subset inside any finite simple non-abelian group $�G$ grows rapidly. More precisely, if $�A$ and $�B$ are two subsets with $�B$ normal and neither of them is too large inside $�G$, then $���\left|�A�B�\right|��⩾�{�\left|�A�\right|�\left|�B�\right|�}^{�1�-�\epsilon �}�$ where $��\epsilon �>�0�$ can be taken arbitrarily small. This is a somewhat surprising strengthening of a theorem of Liebeck, Schul, and Shalev.

Important correspondences in representation theory can be regarded as restrictions of the Morita–Tachikawa correspondence. Moreover, this correspondence motivates the study of many classes of algebras like Morita algebras and gendo-symmetric algebras. Explicitly, the Morita–Tachikawa correspondence describes that endomorphism algebras of generators–cogenerators over finite-dimensional algebras are exactly the finite-dimensional algebras with dominant dimension at least two. In this paper, we introduce the concepts of quasi-generators and quasi-cogenerators that generalise generators and cogenerators, respectively. Using these new concepts, we present higher versions of the Morita–Tachikawa correspondence that take into account relative dominant dimension with respect to a self-orthogonal module with arbitrary projective and injective dimensions. These new versions also hold over Noetherian algebras that are finitely generated and projective over a commutative Noetherian ring.

In Kechris et al. [J. Symb. Log. 83 (2018), no. 3, 1190–1203], it was asked if equality on the reals is sharp as a lower bound for the complexity of topological isomorphism between oligomorphic groups. We prove that under the assumption of weak elimination of imaginaries, this is indeed the case. Our methods are model theoretic and they also have applications on the classical problem of reconstruction of isomorphisms of permutation groups from (topological) isomorphisms of automorphisms groups. As a concrete application, we give an explicit description of $��\mathrm{Aut}�\left(�\mathrm{GL}�\right(�V�\left)�\right)�$ for any vector space $�V$ of dimension $�{\aleph }_0$ over a finite field, in affinity with the classical description for finite-dimensional spaces due to Schreier and van der Waerden.

In this short note, we prove the existence of infinitely many pairwise nonisomorphic, non-hyperelliptic Riemann surfaces with automorphism group acting transitively on the Weierstrass points. We also find all compact Riemann surfaces with automorphism group acting transitively on the Weierstrass points, under the assumption that they are simple.

We give applications of equivariant Gromov–Hausdorff convergence in various contexts. Namely, using equivariant Gromov–Hausdorff convergence, we prove a stability result in the setting of compact finite-dimensional Alexandrov spaces. Moreover, we introduce the notion of an almost commutative diagram and show that its use simplifies both exposition and argument.