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Let X be a complex Banach space, and denote by \mathcal {B}(X) the algebra of all bounded linear operators on X. Let C,D\in \mathcal {B} \left (X\right) be fixed operators. In this paper, we characterize linear, continuous and bijective maps \varphi and \psi on \mathcal {B}\left (X\right) for which there exist invertible operators T_0, W_0 \in \mathcal { B}(X) such that \varphi (T_0), \psi (W_0) \in \mathcal {B}(X) are both invertible, having the property that \varphi \left (A\right) \psi \left (B\right) =D in \mathcal {B}(X) whenever AB=C in \mathcal {B}(X). As a corollary, we deduce the form of linear, bijective and continuous maps \varphi on \mathcal {B}(X) having the property that \varphi \left (A\right) \varphi \left (B\right) =D in \mathcal {B}(X) whenever AB=C. [ABSTRACT FROM AUTHOR]

In this short paper, we derive a new result on Umehara algebra. As a consequence, we prove that an indefinite complex hyperbolic space and an indefinite complex projective space do not share a common complex submanifold with induced metrics, answering a question raised in Cheng et al. [ABSTRACT FROM AUTHOR]

Important connections in representation theory arise from resolving a finite-dimensional algebra by an endomorphism algebra of a generator-cogenerator with finite global dimension; for instance, Auslander's correspondence, classical Schur–Weyl duality and Soergel's Struktursatz. Here, the module category of the resolution and the module category of the algebra being resolved are linked via an exact functor known as the Schur functor. In this paper, we investigate how to measure the quality of the connection between module categories of (projective) Noetherian algebras, B, and module categories of endomorphism algebras of generator-relative cogenerators over B which are split quasi-hereditary Noetherian algebras. In particular, we are interested in finding, if it exists, the highest degree n so that the endomorphism algebra of a generator-cogenerator provides an n-faithful cover, in the sense of Rouquier, of B. The degree n is known as the Hemmer–Nakano dimension of the standard modules. We prove that, in general, the Hemmer–Nakano dimension of standard modules with respect to a Schur functor from a split highest weight category over a field to the module category of a finite-dimensional algebra B is bounded above by the number of non-isomorphic simple modules of B. We establish methods for reducing computations of Hemmer–Nakano dimensions in the integral setup to computations of Hemmer–Nakano dimensions over finite-dimensional algebras, and vice-versa. In addition, we extend the framework to study Hemmer–Nakano dimensions of arbitrary resolving subcategories. In this setup, we find that the relative dominant dimension over (projective) Noetherian algebras is an important tool in the computation of these degrees, extending the previous work of Fang and Koenig. In particular, this theory allows us to derive results for Schur algebras and the BGG category \mathcal {O} in the integral setup from the finite-dimensional case. More precisely, we use the relative dominant dimension of Schur algebras to completely determine the Hemmer–Nakano dimension of standard modules with respect to Schur functors between module categories of Schur algebras over regular Noetherian rings and module categories of group algebras of symmetric groups over regular Noetherian rings. We exhibit several structural properties of deformations of the blocks of the Bernstein-Gelfand-Gelfand category \mathcal {O} establishing an integral version of Soergel's Struktursatz. We show that deformations of the combinatorial Soergel's functor have better homological properties than the classical one. [ABSTRACT FROM AUTHOR]

We describe some recent computer investigations with the 'Constraint Logic Programming over Finite Domains' library in the Prolog programming environment to search for new simple Lie algebras over the field \operatorname {GF}(2) of 2 elements. Motivated by a paper of Grishkov et al., we specifically look for those with a thin decomposition , and we settle one of their conjectures. We extrapolate from our results the existence of two new infinite families of simple Lie algebras, in addition to finding seven new sporadic examples in dimension 31. We also better contextualise some previously discovered simple algebras, putting them into families which do not seem to have ever appeared in the literature, and give an updated table of those currently known. [ABSTRACT FROM AUTHOR]

In this paper, we prove a result similar to results of Itoh [J. Algebra 150 (1992), pp. 101–117] and Hong-Ulrich [J. Lond. Math. Soc. (2) 90 (2014), pp. 861–878], proving that integral closure of an ideal is compatible with specialization by a general element of that ideal for ideals of height at least two in a large class of rings. Moreover, we show integral closure of sufficiently large powers of the ideal is compatible with specialization by a general element of the original ideal. In a polynomial ring over an infinite field, we give a class of squarefree monomial ideals for which the integral closure is compatible with specialization by a general linear form. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study a poset isomorphism between two support \tau-tilting posets. We take several information from combinatorial properties of support \tau-tilting posets and give an analogue of Happel-Unger's reconstruction theorem. [ABSTRACT FROM AUTHOR]

Building on work of Ruelle and Putnam in the Smale space case, Thomsen defined the homoclinic and heteroclinic C^\ast-algebras for an expansive dynamical system. In this paper we define a class of expansive dynamical systems, called synchronizing dynamical systems, that exhibit hyperbolic behavior almost everywhere. Synchronizing dynamical systems generalize Smale spaces (and even finitely presented systems). Yet they still have desirable dynamical properties such as having a dense set of periodic points. We study various C^\ast-algebras associated with a synchronizing dynamical system. Among other results, we show that the homoclinic algebra of a synchronizing system contains an ideal which behaves like the homoclinic algebra of a Smale space. [ABSTRACT FROM AUTHOR]

Let (A,\mathfrak {m}) be an excellent normal domain of dimension two containing a field k \cong A/\mathfrak {m}. An \mathfrak {m}-primary ideal I is a p_g-ideal if the Rees algebra A[It] is a Cohen-Macaulay normal domain. If k is algebraically closed then Okuma, Watanabe and Yoshida proved that A has p_g-ideals and furthermore product of two p_g-ideals is a p_g ideal. Previously we showed that if k has characteristic zero then A has p_g-ideals. In this paper we prove that if k is perfect field of positive characteristic then also A has p_g ideals. [ABSTRACT FROM AUTHOR]

Let S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I be the determinantal ideal of maximal minors or Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S-module structures of Ext^j_S(S/I^t, S) and Ext^j_S(S/Pf^t, S), from which we get the degrees of generators of these Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H^j_\mathfrak {m}(S/I^t). [ABSTRACT FROM AUTHOR]

Upper triangular, and more generally, block-triangular matrices, are rather important in Linear Algebra, and also in Ring theory, namely in the theory of PI algebras (algebras that satisfy polynomial identities). The group gradings on such algebras have been extensively studied during the last decades. In this paper we prove that for any group grading on a block-triangular matrix algebra, over an arbitrary field, the Jacobson radical is a graded (homogeneous) ideal. As noted by F. Yasumura [Arch. Math. (Basel) 110 (2018), pp. 327–332] this yields the classification of the group gradings on these algebras and confirms a conjecture made by A. Valenti and M. Zaicev [Arch. Math. (Basel) 89 (2007), pp. 33–40]. [ABSTRACT FROM AUTHOR]

We show that the celebrated operad of pre-Lie algebras is very rigid: it has no "non-obvious" degrees of freedom from either of the three points of view: deformations of maps to and from the "three graces of operad theory", homotopy automorphisms, and operadic twisting. Examining the latter, it is possible to answer two questions of Markl from 2005 [Czechoslovak Math. J. 57 (2007), pp. 253–268; J. Lie Theory 17 (2007), pp. 241–261], including a Lie-theoretic version of the Deligne conjecture. [ABSTRACT FROM AUTHOR]

We extend Nekrashevych's KK-duality for C^*-algebras of regular, recurrent, contracting self-similar group actions to regular, contracting self-similar groupoid actions on a graph, removing the recurrence condition entirely and generalising from a finite alphabet to a finite graph. More precisely, given a regular and contracting self-similar groupoid (G,E) acting faithfully on a finite directed graph E, we associate two C^*-algebras, \mathcal {O}(G,E) and \widehat {\mathcal {O}}(G,E), to it and prove that they are strongly Morita equivalent to the stable and unstable Ruelle C*-algebras of a Smale space arising from a Wieler solenoid of the self-similar limit space. That these algebras are Spanier-Whitehead dual in KK-theory follows from the general result for Ruelle algebras of irreducible Smale spaces proved by Kaminker, Putnam, and the last author. [ABSTRACT FROM AUTHOR]

Spin^h manifolds are the quaternionic analogue to \text {spin}^c manifolds. We compute the \text {spin}^h bordism groups at the prime 2 by proving a structure theorem for the cohomology of the \text {spin}^h bordism spectrum \mathrm {MSpin^h} as a module over the mod 2 Steenrod algebra. This provides a 2-local splitting of \mathrm {MSpin^h} as a wedge sum of familiar spectra. We also compute the decomposition of H^*(\mathrm {MSpin^h};\mathbb {Z}/2\mathbb {Z}) explicitly in degrees up through 30 via a counting process. [ABSTRACT FROM AUTHOR]

Consider ideals I of the form \[ I=(x_1^2,\dots, x_n^2)+\operatorname {RLex}(x_ix_j) \] where \operatorname {RLex}(x_ix_j) is the ideal generated by all the square-free monomials which are greater than or equal to x_ix_j in the reverse lexicographic order. We will determine some interesting properties regarding the shape of the Hilbert series of I. Using a theorem of Lindsey [Proc. Amer. Math. Soc. 139 (2011), no. 1, 79–92], this allows for a short proof that any algebra defined by I has the strong Lefschetz property when the underlying field is of characteristic zero. Building on recent work by Phuong and Tran [Colloq. Math. 173 (2023), no. 1, 1–8], this result is then extended to fields of sufficiently high positive characteristic. As a consequence, this shows that for any possible number of minimal generators for an artinian quadratic ideal there exists such an ideal minimally generated by that many monomials and defining an algebra with the strong Lefschetz property. [ABSTRACT FROM AUTHOR]

We work out the exact relationship between algebraic modular forms for a two-by-two general unitary group over a definite quaternion algebra, and those arising from genera of positive-definite quinary lattices, relating stabilisers of local lattices with specific open compact subgroups, paramodular at split places, and with Atkin-Lehner operators. Combining this with the recent work of Rösner and Weissauer, proving conjectures of Ibukiyama on Jacquet-Langlands type correspondences (mildly generalised here), provides an effective tool for computing Hecke eigenvalues for Siegel modular forms of degree two and paramodular level. It also enables us to prove examples of congruences of Hecke eigenvalues connecting Siegel modular forms of degrees two and one. These include some of a type conjectured by Harder at level one, supported by computations of Fretwell at higher levels, and a subtly different congruence discovered experimentally by Buzzard and Golyshev. [ABSTRACT FROM AUTHOR]

We prove that for any fixed unitary matrix U, any abelian self-adjoint algebra of matrices that is invariant under conjugation by U can be embedded into a maximal abelian self-adjoint algebra that is still invariant under conjugation by U. We use this result to analyse the structure of matrices A for which A^*A commutes with AA^*, and to characterize matrices that are unitarily equivalent to weighted permutations. [ABSTRACT FROM AUTHOR]

Let G be a reductive group over a non-archimedean local field F of residue characteristic p. We prove that the Hecke algebras of G(F), with coefficients in any noetherian \mathbb {Z}_{\ell }-algebra R with \ell \neq p, are finitely generated modules over their centers, and that these centers are finitely generated R-algebras. Following Bernstein's original strategy, we then deduce that "second adjointness" holds for smooth representations of G(F) with coefficients in any \mathbb {Z}[\frac {1}{p}]-algebra. These results had been conjectured for a long time. The crucial new tool that unlocks the problem is the Fargues-Scholze morphism between a certain "excursion algebra" defined on the Langlands parameters side and the Bernstein center of G(F). Using this bridge, our main results are representation theoretic counterparts of the finiteness of certain morphisms between coarse moduli spaces of local Langlands parameters that we also prove here, which may be of independent interest. [ABSTRACT FROM AUTHOR]

We establish a general framework for representability of a metric group on a (well-behaved) class of Banach spaces. More precisely, let \mathcal {G} be a topological group, and \mathcal {A} a unital symmetric C^*-subalgebra of \mathrm {UC}(\mathcal {G}), the algebra of bounded uniformly continuous functions on \mathcal {G}. Generalizing the notion of a stable metric, we study \mathcal {A}-metrics \delta, i.e., the function \delta (e, \cdot) belongs to \mathcal {A}; the case \mathcal {A}=W\hskip -0.7mm A\hskip -0.2mm P(\mathcal {G}), the algebra of weakly almost periodic functions on \mathcal {G}, recovers stability. If the topology of G is induced by a left invariant metric d, we prove that \mathcal {A} determines the topology of \mathcal {G} if and only if d is uniformly equivalent to a left invariant \mathcal {A}-metric. As an application, we show that the additive group of C[0,1] is not reflexively representable; this is a new proof of Megrelishvili [ Topological transformation groups: selected topics , Elsevier, 2007, Question 6.7] (the problem was already solved by Ferri and Galindo [Studia Math. 193 (2009), pp. 99–108] with different methods and later the results were generalized by Yaacov, Berenstein, and Ferri [Math. Z. 267 (2011), pp.129–138]). Let now \mathcal {G} be a metric group, and assume \mathcal {A}\subseteq \mathrm {LUC}(\mathcal {G}), the algebra of bounded left uniformly continuous functions on \mathcal {G}, is a unital C^*-algebra which is the uniform closure of coefficients of representations of \mathcal {G} on members of \mathscr {F}, where \mathscr {F} is a class of Banach spaces closed under \ell _2-direct sums. We prove that \mathcal {A} determines the topology of \mathcal {G} if and only if \mathcal {G} embeds into the isometry group of a member of \mathscr {F}, equipped with the weak operator topology. As applications, we obtain characterizations of unitary and reflexive representability. [ABSTRACT FROM AUTHOR]

We study the maximal abelian von Neumann algebra corresponding to L^\infty (\mathbb {R}) via the Bargmann transform. It is naturally an algebra of operators on the Fock space F^2, but it can also be realized as a function algebra contained in F^2. This provides an interesting example of a C^* algebra whose elements are analytic functions. [ABSTRACT FROM AUTHOR]

We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the fundamental theorem of algebra for quaternions to polynomials with two monomials in the leading form, while showing that it fails for three. [ABSTRACT FROM AUTHOR]

We determine the defining equations of the Rees algebra and of the special fiber ring of the ideal of maximal minors of a 2\times n sparse matrix. We prove that their initial algebras are ladder determinantal rings. This allows us to show that the Rees algebra and the special fiber ring are Cohen-Macaulay domains, they are Koszul, they have rational singularities in characteristic zero and are F-rational in positive characteristic. [ABSTRACT FROM AUTHOR]

For each odd integer n \geq 3, we construct a rank-3 graph \Lambda _n with involution \gamma _n whose real C^*-algebra C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n) is stably isomorphic to the exotic Cuntz algebra \mathcal E_n. This construction is optimal, as we prove that a rank-2 graph with involution (\Lambda,\gamma) can never satisfy C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda, \gamma)\sim _{ME} \mathcal E_n, and Boersema reached the same conclusion for rank-1 graphs (directed graphs) in [Münster J. Math. 10 (2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank-1 graph with involution (\Lambda, \gamma) whose real C^*-algebra C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda, \gamma) is stably isomorphic to the suspension S \mathbb {R}. In the Appendix, we show that the i-fold suspension S^i \mathbb {R} is stably isomorphic to a graph algebra iff -2 \leq i \leq 1. [ABSTRACT FROM AUTHOR]

The \imathHall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of 1-periodic complexes of coherent sheaves on the weighted projective line over a finite field. We show that this Hall algebra provides a realization of the \imathquantum loop algebra, which is a generalization of the \imathquantum group arising from the quantum symmetric pair of split affine type ADE in its Drinfeld type presentation. The \imathHall algebra of the \imathquiver algebra of split affine type A was known earlier to realize the same algebra in its Serre presentation. We then establish a derived equivalence which induces an isomorphism of these two \imathHall algebras, explaining the isomorphism of the \imathquantum group of split affine type A under the two presentations. [ABSTRACT FROM AUTHOR]

Over the last several decades, improvements in the fields of analytic combinatorics and computer algebra have made determining the asymptotic behaviour of sequences satisfying linear recurrence relations with polynomial coefficients largely a matter of routine, under assumptions that hold often in practice. The algorithms involved typically take a sequence, encoded by a recurrence relation and initial terms, and return the leading terms in an asymptotic expansion up to a big-O error term. Less studied, however, are effective techniques giving an explicit bound on asymptotic error terms. Among other things, such explicit bounds typically allow the user to automatically prove sequence positivity (an active area of enumerative and algebraic combinatorics) by exhibiting an index when positive leading asymptotic behaviour dominates any error terms. In this article, we present a practical algorithm for computing such asymptotic approximations with rigorous error bounds, under the assumption that the generating series of the sequence is a solution of a differential equation with regular (Fuchsian) dominant singularities. Our algorithm approximately follows the singularity analysis method of Flajolet and Odlyzko [SIAM J. Discrete Math. 3 (1990), pp. 216–240], except that all big-O terms involved in the derivation of the asymptotic expansion are replaced by explicit error terms. The computation of the error terms combines analytic bounds from the literature with effective techniques from rigorous numerics and computer algebra. We implement our algorithm in the SageMath computer algebra system and exhibit its use on a variety of applications (including our original motivating example, solution uniqueness in the Canham model for the shape of genus one biomembranes). [ABSTRACT FROM AUTHOR]

Let R be a finitely generated \mathbb N-graded algebra domain over a Noetherian ring and let I be a homogeneous ideal of R. Given P\in Ass(R/I) one defines the v-invariant v_P(I) of I at P as the least c\in \mathbb N such that P=I:f for some f\in R_c. A classical result of Brodmann [Proc. Amer. Math. Soc. 74 (1979), pp. 16–18] asserts that Ass(R/I^n) is constant for large n. So it makes sense to consider a prime ideal P\in Ass(R/I^n) for all the large n and investigate how v_P(I^n) depends on n. We prove that v_P(I^n) is eventually a linear function of n. When R is the polynomial ring over a field this statement has been proved independently also by Ficarra and Sgroi in their recent preprint [ Asymptotic behaviour of the \text {v}-number of homogeneous ideals , https://arxiv.org/abs/2306.14243, 2023]. [ABSTRACT FROM AUTHOR]

Lei and Lin [Comm. Pure Appl. Math. 64 (2011), pp. 1297–1304] have recently given a proof of a global mild solution of the three-dimensional Navier-Stokes equations in function spaces based on the Wiener algebra. An alternative proof of existence of these solutions was then developed by Bae [Proc. Amer. Math. Soc. 143 (2015), pp. 2887–2892], and this new proof allowed for an estimate of the radius of analyticity of the solutions at positive times. We adapt the Bae proof to prove existence of the Lei-Lin solution in the spatially periodic setting, finding an improved bound for the radius of analyticity in this case. [ABSTRACT FROM AUTHOR]

The notion of an extriangulated category gives a unification of existing theories in exact or abelian categories and in triangulated categories. In this article, we develop Auslander–Reiten theory for extriangulated categories. This unifies Auslander–Reiten theories developed in exact categories and triangulated categories independently. We give two different sets of sufficient conditions on the extriangulated category so that existence of almost split extensions becomes equivalent to that of an Auslander–Reiten–Serre duality. We also show that existence of almost split extensions is preserved under taking relative extriangulated categories, ideal quotients, and extension-closed subcategories. Moreover, we prove that the stable category \underline {\mathscr {C}} of an extriangulated category \mathscr {C} is a \tau-category (see O. Iyama [Algebr. Represent. Theory 8 (2005), pp. 297–321]) if \mathscr {C} has enough projectives, almost split extensions and source morphisms. This gives various consequences on \underline {\mathscr {C}}, including Igusa–Todorov's Radical Layers Theorem (see K. Igusa and G. Todorov [J. Algebra 89 (1984), pp. 105–147]), Auslander–Reiten Combinatorics on dimensions of Hom-spaces, and Reconstruction Theorem of the associated completely graded category of \underline {\mathscr {C}} via the complete mesh category of the Auslander–Reiten species of \underline {\mathscr {C}}. Finally we prove that any locally finite symmetrizable \tau-quiver (=valued translation quiver) is an Auslander–Reiten quiver of some extriangulated category with sink morphisms and source morphisms. [ABSTRACT FROM AUTHOR]

We show that R is regular if Tor_{i}^{R}(R^{+},k) = 0 for some i\geq 1 assuming further that R is a \mathbb {N}-graded ring of dimension 2 finitely generated over an algebraically closed equicharacteristic zero field k. This answers a question of Bhatt, Iyengar, and Ma [Comm. Algebra 47 (2019), pp. 2367–2383]. We use almost mathematics over R^{+} to deduce properties of the noetherian ring R and rational surface singularities. Moreover we observe that R^{+} in equicharacteristic zero has a rich module-theoretic structure; it is m-adically ideal(wise) separated, (weakly) intersection flat, and Ohm-Rush. As an application we show that the hypothesis can be astonishingly vacuous for i \ll dim(R). We show that a positive answer to an old question of Aberbach and Hochster [J. Pure Appl. Algebra 122 (1997), pp. 171–184] also answers this question and we use our techniques to study a question of André and Fiorot [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23 (2022), pp. 81–114] regarding 'fpqc analogues' of splinters. [ABSTRACT FROM AUTHOR]

We prove that every slightly degenerate braided fusion category admits a minimal nondegenerate extension, and hence that every pseudo-unitary super modular tensor category admits a minimal modular extension. This completes the program of characterizing minimal nondegenerate extensions of braided fusion categories. Our proof relies on the new subject of fusion 2-categories. We study in detail the Drinfel'd centre \mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}}) of the fusion 2-category {_{}\mathrm {Mod}\text {-}\mathcal {B}} of module categories of a braided fusion 1-category \mathcal {B}. We show that minimal nondegenerate extensions of \mathcal {B} correspond to certain trivializations of \mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}}). In the slightly degenerate case, such trivializations are obstructed by a class in H^5(K(\mathbb {Z}_2, 2); \mathbb {k}^\times) and we use a numerical invariant—defined by evaluating a certain two-dimensional topological field theory on a Klein bottle—to prove that this obstruction always vanishes. Along the way, we develop techniques to explicitly compute in braided fusion 2-categories which we expect will be of independent interest. In addition to the model of \mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}}) in terms of braided \mathcal {B}-module categories, we develop a computationally useful model in terms of certain algebra objects in \mathcal {B}. We construct an S-matrix pairing for any braided fusion 2-category, and show that it is nondegenerate for \mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}}). As a corollary, we identify components of \mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}}) with blocks in the annular category of \mathcal {B} and with the homomorphisms from the Grothendieck ring of the Müger centre of \mathcal {B} to the ground field. [ABSTRACT FROM AUTHOR]