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Albert algebras, a specific kind of Jordan algebra, are naturally distinguished objects among commutative nonassociative algebras and also arise naturally in the context of simple affine group schemes of type
$\mathsf {F}_4$
,
$\mathsf {E}_6$
, or
$\mathsf {E}_7$
. We study these objects over an arbitrary base ring R, with particular attention to the case
$R = \mathbb {Z}$
. We prove in this generality results previously in the literature in the special case where R is a field of characteristic different from 2 and 3.
Let G be a connected reductive group, T a maximal torus of G, N the normalizer of T and
$W=N/T$
the Weyl group of G. Let
${\mathfrak {g}}$
and
${\mathfrak {t}}$
be the Lie algebras of G and T. The affine variety
$\mathfrak {car}={\mathfrak {t}}/\!/W$
of semisimple G-orbits of
${\mathfrak {g}}$
has a natural stratification
indexed by the set of G-conjugacy classes of Levi subgroups: the open stratum is the set of regular semisimple orbits and the closed stratum is the set of central orbits.
In [17], Rider considered the triangulated subcategory
$D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\mathrm {nil}}/G])^{\mathrm {Spr}}$
of
$D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\mathrm {nil}}/G])$
generated by the direct summand of the Springer sheaf. She proved that it is equivalent to the derived category of finitely generated dg modules over the smash product algebra
${\overline {\mathbb {Q}}_{\ell }}[W]\# H^{\bullet }_G(G/B)$
where
$H^{\bullet }_G(G/B)$
is the G-equivariant cohomology of the flag variety. Notice that the later derived category is
$D_{\mathrm {c}}^{\mathrm {b}}(\mathrm {B}(N))$
where
$\mathrm {B}(N)=[\mathrm {Spec}(k)/N]$
is the classifying stack of N-torsors.
The aim of this paper is to understand geometrically and generalise Rider’s equivalence of categories: For each
$\lambda $
we construct a cohomological correspondence inducing an equivalence of categories between
$D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {t}}_{\lambda }/N])$
and
$D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\lambda }/G])^{\mathrm {Spr}}$
.
Let k be an algebraically closed field of prime characteristic p. Let
$kGe$
be a block of a group algebra of a finite group G, with normal defect group P and abelian
$p'$
inertial quotient L. Then we show that
$kGe$
is a matrix algebra over a quantised version of the group algebra of a semidirect product of P with a certain subgroup of L. To do this, we first examine the associated graded algebra, using a Jennings–Quillen style theorem.
As an example, we calculate the associated graded of the basic algebra of the nonprincipal block in the case of a semidirect product of an extraspecial p-group P of exponent p and order
$p^3$
with a quaternion group of order eight with the centre acting trivially. In the case of
$p=3$
, we give explicit generators and relations for the basic algebra as a quantised version of
$kP$
. As a second example, we give explicit generators and relations in the case of a group of shape
$2^{1+4}:3^{1+2}$
in characteristic two.
We generalize the works of Pappas–Rapoport–Zhu on twisted affine Grassmannians to the wildly ramified case under mild assumptions. This rests on a construction of certain smooth affine
$\mathbb {Z}[t]$
-groups with connected fibers of parahoric type, motivated by previous work of Tits. The resulting
$\mathbb {F}_p(t)$
-groups are pseudo-reductive and sometimes non-standard in the sense of Conrad–Gabber–Prasad, and their
$\mathbb {F}_p [\hspace {-0,5mm}[ {t} ]\hspace {-0,5mm}] $
-models are parahoric in a generalized sense. We study their affine Grassmannians, proving normality of Schubert varieties and Zhu’s coherence theorem.
Given a profinite group G of finite p-cohomological dimension and a pro-p quotient H of G by a closed normal subgroup N, we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H. We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H, we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H. We apply our study to prove lower bounds on the p-ranks of class groups of certain nonabelian extensions of
$\mathbb {Q}$
and to give a new proof of the vanishing of Massey triple products in Galois cohomology.
We establish a Hirzebruch–Riemann–Roch-type theorem and a Grothendieck–Riemann–Roch-type theorem for matrix factorizations on quotient Deligne–Mumford stacks. For this, we first construct a Hochschild–Kostant–Rosenberg-type isomorphism explicit enough to yield a categorical Chern character formula. Then, we find an expression of the canonical pairing of Shklyarov under the isomorphism.
We study non-abelian versions of the Mellin transformations, originally introduced by Gabber-Loeser on complex affine tori. Our main result is a generalisation to the non-abelian context and with arbitrary coefficients of the t-exactness of Gabber-Loeser’s Mellin transformation. As an intermediate step, we obtain vanishing results for the Sabbah specialisation functors. Our main application is to construct new examples of duality spaces in the sense of Bieri-Eckmann, generalising results of Denham-Suciu.
We prove that the derived categories of abelian categories have unique enhancements—all of them, the unbounded, bounded, bounded above and bounded below derived categories. The unseparated and left completed derived categories of a Grothendieck abelian category are also shown to have unique enhancements. Finally, we show that the derived category of complexes with quasi-coherent cohomology and the category of perfect complexes have unique enhancements for quasi-compact and quasi-separated schemes.
Let K be a number field, let A be a finite-dimensional K-algebra, let
$\operatorname {\mathrm {J}}(A)$
denote the Jacobson radical of A and let
$\Lambda $
be an
$\mathcal {O}_{K}$
-order in A. Suppose that each simple component of the semisimple K-algebra
$A/{\operatorname {\mathrm {J}}(A)}$
is isomorphic to a matrix ring over a field. Under this hypothesis on A, we give an algorithm that, given two
$\Lambda $
-lattices X and Y, determines whether X and Y are isomorphic and, if so, computes an explicit isomorphism
$X \rightarrow Y$
. This algorithm reduces the problem to standard problems in computational algebra and algorithmic algebraic number theory in polynomial time. As an application, we give an algorithm for the following long-standing problem: Given a number field K, a positive integer n and two matrices
$A,B \in \mathrm {Mat}_{n}(\mathcal {O}_{K})$
, determine whether A and B are similar over
$\mathcal {O}_{K}$
, and if so, return a matrix
$C \in \mathrm {GL}_{n}(\mathcal {O}_{K})$
such that
$B= CAC^{-1}$
. We give explicit examples that show that the implementation of the latter algorithm for
$\mathcal {O}_{K}=\mathbb {Z}$
vastly outperforms implementations of all previous algorithms, as predicted by our complexity analysis.
It was established by Boalch that Euler continuants arise as Lie group valued moment maps for a class of wild character varieties described as moduli spaces of points on
$\mathbb {P}^1$
by Sibuya. Furthermore, Boalch noticed that these varieties are multiplicative analogues of certain Nakajima quiver varieties originally introduced by Calabi, which are attached to the quiver
$\Gamma _n$
on two vertices and n equioriented arrows. In this article, we go a step further by unveiling that the Sibuya varieties can be understood using noncommutative quasi-Poisson geometry modelled on the quiver
$\Gamma _n$
. We prove that the Poisson structure carried by these varieties is induced, via the Kontsevich–Rosenberg principle, by an explicit Hamiltonian double quasi-Poisson algebra defined at the level of the quiver
$\Gamma _n$
such that its noncommutative multiplicative moment map is given in terms of Euler continuants. This result generalises the Hamiltonian double quasi-Poisson algebra associated with the quiver
$\Gamma _1$
by Van den Bergh. Moreover, using the method of fusion, we prove that the Hamiltonian double quasi-Poisson algebra attached to
$\Gamma _n$
admits a factorisation in terms of n copies of the algebra attached to
$\Gamma _1$
.
This paper gives a description of the full space of Bridgeland stability conditions on the bounded derived category of a contraction algebra associated to a
$3$
-fold flop. The main result is that the stability manifold is the universal cover of a naturally associated hyperplane arrangement, which is known to be simplicial and in special cases is an ADE root system. There are four main corollaries: (1) a short proof of the faithfulness of pure braid group actions in both algebraic and geometric settings, the first that avoid normal forms; (2) a classification of tilting complexes in the derived category of a contraction algebra; (3) contractibility of the stability space associated to the flop; and (4) a new proof of the
$K(\unicode{x3c0} \,,1)$
-theorem in various finite settings, which includes ADE braid groups.
We present a categorical point of view on dynamical quantum groups in terms of categories of Harish-Chandra bimodules. We prove Tannaka duality theorems for forgetful functors into the monoidal category of Harish-Chandra bimodules in terms of a slight modification of the notion of a bialgebroid. Moreover, we show that the standard dynamical quantum groups
$F(G)$
and
$F_q(G)$
are related to parabolic restriction functors for classical and quantum Harish-Chandra bimodules. Finally, we exhibit a natural Weyl symmetry of the parabolic restriction functor using Zhelobenko operators and show that it gives rise to the action of the dynamical Weyl group.
We state a sufficient condition for a fusion system to be saturated. This is then used to investigate localities with kernels: that is, localities that are (in a particular way) extensions of groups by localities. As an application of these results, we define and study certain products in fusion systems and localities, thus giving a new method to construct saturated subsystems of fusion systems.
In this paper, we prove a stronger form of the Bogomolov–Gieseker (BG) inequality for stable sheaves on two classes of Calabi–Yau threefolds, namely, weighted hypersurfaces inside the weighted projective spaces
$\mathbb {P}(1, 1, 1, 1, 2)$
and
$\mathbb {P}(1, 1, 1, 1, 4)$
. Using the stronger BG inequality as a main technical tool, we construct open subsets in the spaces of Bridgeland stability conditions on these Calabi–Yau threefolds.
Let q be a nontrivial odd prime power, and let
$n \ge 2$
be a natural number with
$(n,q) \ne (2,3)$
. We characterize the groups
$PSL_n(q)$
and
$PSU_n(q)$
by their
$2$
-fusion systems. This contributes to a programme of Aschbacher aiming at a simplified proof of the classification of finite simple groups.
We show that the dimer algebra of a connected Postnikov diagram in the disc is bimodule internally
$3$
-Calabi–Yau in the sense of the author’s earlier work [43]. As a consequence, we obtain an additive categorification of the cluster algebra associated to the diagram, which (after inverting frozen variables) is isomorphic to the homogeneous coordinate ring of a positroid variety in the Grassmannian by a recent result of Galashin and Lam [18]. We show that our categorification can be realised as a full extension closed subcategory of Jensen–King–Su’s Grassmannian cluster category [28], in a way compatible with their bijection between rank
$1$
modules and Plücker coordinates.
For every $\infty $-category $\mathscr {C}$, there is a homotopy n-category $\mathrm {h}_n \mathscr {C}$ and a canonical functor $\gamma _n \colon \mathscr {C} \to \mathrm {h}_n \mathscr {C}$. We study these higher homotopy categories, especially in connection with the existence and preservation of (co)limits, by introducing a higher categorical notion of weak colimit. Using homotopy n-categories, we introduce the notion of an n-derivator and study the main examples arising from $\infty $-categories. Following the work of Maltsiniotis and Garkusha, we define K-theory for $\infty $-derivators and prove that the canonical comparison map from the Waldhausen K-theory of $\mathscr {C}$ to the K-theory of the associated n-derivator $\mathbb {D}_{\mathscr {C}}^{(n)}$ is $(n+1)$-connected. We also prove that this comparison map identifies derivator K-theory of $\infty $-derivators in terms of a universal property. Moreover, using the canonical structure of higher weak pushouts in the homotopy n-category, we also define a K-theory space $K(\mathrm {h}_n \mathscr {C}, \mathrm {can})$ associated to $\mathrm {h}_n \mathscr {C}$. We prove that the canonical comparison map from the Waldhausen K-theory of $\mathscr {C}$ to $K(\mathrm {h}_n \mathscr {C}, \mathrm {can})$ is n-connected.
The Alperin–McKay conjecture is a longstanding open conjecture in the representation theory of finite groups. Späth showed that the Alperin–McKay conjecture holds if the so-called inductive Alperin–McKay (iAM) condition holds for all finite simple groups. In a previous paper, the author has proved that it is enough to verify the inductive condition for quasi-isolated blocks of groups of Lie type. In this paper, we show that the verification of the iAM-condition can be further reduced in many cases to isolated blocks. As a consequence of this, we obtain a proof of the Alperin–McKay conjecture for
$2$
-blocks of finite groups with abelian defect.
This is the first of a series of papers concerning what might be thought of as ‘locally grouped spaces’, in a loose analogy with the locally ringed spaces of algebraic geometry. The spaces that we have in mind are simplicial sets that generalise the simplicial sets that underlie and determine the classifying spaces of finite (or compact) groups. If the analogy is pursued, then the role of ‘structure sheaf’ is provided by the ‘fusion systems’ associated with these spaces. Our approach here will be purely algebraic and combinatorial, so we will not be concerned with topological realisations. All of the groups to be considered will be finite; but a parallel series of papers representing some joint work with Alex Gonzalez will considerably broaden the scope.