Isomorphism Classes Research Articles

Differential Brauer monoids

The differential Brauer monoid of a differential commutative ring R R is defined. Its elements are the isomorphism classes of differential Azumaya R R algebras with operation from tensor product subject to the relation that two such algebras are equivalent if matrix algebras over them, with entry-wise differentiation, are differentially isomorphic. The fine Bauer monoid, which is a group, is the same thing without the differential requirement. The differential Brauer monoid is then determined from the fine Brauer monoids of R R and R D R^D and the submonoid of the Brauer monoid whose underlying Azumaya algebras are matrix rings.

Characterization of Gromov-type geodesics

It is well known that, when endowed with the Gromov-Hausdorff distance dGH, the collection M of all isometry classes of compact metric spaces is a complete and separable space. It is also known that (M,dGH) is a geodesic metric space, but there is no known structural characterization of geodesics in M.In this paper we provide two characterizations of geodesics in M. We call a Gromov-Hausdorff geodesic γ:[0,1]→MHausdorff-realizable if there exists a compact metric space Z containing isometric copies of γ(t) for each t∈[0,1] such that the Hausdorff distance satisfies dHZ(γ(s),γ(t))=dGH(γ(s),γ(t)) for all s,t∈[0,1]. In this way, γ is actually a geodesic in the Hausdorff hyperspace of Z, and we call it a Hausdorff geodesic. We prove that in fact every Gromov-Hausdorff geodesic is Hausdorff-realizable. Inspired by this characterization, we further elucidate a structural connection between Hausdorff geodesics and Wasserstein geodesics: we show that every Hausdorff geodesic is equivalent to a so-called Hausdorff displacement interpolation. This equivalence allows us to establish that every Gromov-Hausdorff geodesic is dynamic, a notion which we develop in analogy with dynamic optimal couplings in the theory of optimal transport.Besides geodesics in M, we also study geodesics on the collection Mw of isomorphism classes of compact metric measure spaces. Sturm constructed a family of Gromov-type distances on Mw, which we denote dGW,pS (for p∈[1,∞)), as an analogue of dGH, and proved that (Mw,dGW,pS) is also a geodesic space. We define a notion of Wasserstein-realizabledGW,pS geodesics in a sense similar to Hausdorff-realizable geodesics and show that the set of all Wasserstein-realizabledGW,pS geodesics is dense in the set of all dGW,pS geodesics. We further identify a rich class of dGW,pS geodesics which are Wasserstein-realizable.

Free pre-Lie algebras of finite posets

We first recall the construction of a twisted pre-Lie algebra structure on the species of finite connected topological spaces. Then, we construct a corresponding non-coassociative permutative (NAP) coproduct on the subspecies of finite connected [Formula: see text] topological spaces, i.e. finite connected posets, and we prove that the vector space generated by isomorphism classes of finite posets is a free pre-Lie algebra and also a cofree NAP coalgebra. Furthermore, we give an explicit duality between the non-associative permutative product and the proposed NAP coproduct. Finally, we prove that the results in this paper remain true for finite connected topological spaces.

Holomorphic differentials of Klein four covers

Let k be an algebraically closed field of characteristic two, and let G be isomorphic to Z/2×Z/2. Suppose X is a smooth projective irreducible curve over k with a faithful G-action, and assume that the cover X→X/G is totally ramified, in the sense that it is ramified and every branch point is totally ramified. We study to what extent the lower ramification groups of the closed points of X determine the isomorphism types of the indecomposable kG-modules and the multiplicities with which they occur as direct summands of the space H0(X,ΩX/k) of holomorphic differentials of X over k. In the case when X/G=Pk1, we completely determine the decomposition of H0(X,ΩX/k) into a direct sum of indecomposable kG-modules. Moreover, we show that the isomorphism classes of indecomposable kG-modules that actually occur as direct summands belong to an infinite list of non-isomorphic indecomposable kG-modules that contain modules of arbitrarily large k-dimension. In particular, our results show that [14, Theorem 6.4] is incorrect.

Upper triangular matrices of graded division algebras and their identities

In the present paper we study UT(D1,…,Dn), a G-graded algebra of block triangular matrices where G is a group and the diagonal blocks D1,…,Dn are graded division algebras. We prove that any two such algebras are G-isomorphic if and only if they satisfy the same graded polynomial identities. We also discuss the number of different isomorphism classes obtained by varying the grading and we exhibit its connection with the factorability of the T-ideal of graded identities. Moreover we give some results about the generators of the graded polynomial identities for these algebras. In particular we generalize the results about the graded identities of UTn to the case in which the diagonal blocks D1,…,Dn are all isomorphic.

Extensions and torsors for finite group schemes

We give an explicit description of the category of central extensions of a group scheme by a sheaf of Abelian groups. Based on this, we describe a framework for computing with central extensions of finite locally free commutative group schemes, torsors under such group schemes and groups of isomorphism classes of these objects.

On the functional graph of the power map over finite groups

In this paper we study the description of the functional graphs associated with the power maps over finite groups. We present a structural result which describes the isomorphism class of these graphs for abelian groups and also for flower groups, which is a special class of non abelian groups introduced in this paper. Unlike the abelian case where all the trees associated with periodic points are isomorphic, in the case of flower groups we prove that several different classes of trees can occur. The class of central trees (i.e. associated with periodic points that are in the center of the group) are in general non-elementary and a recursive description is given in this work. Flower groups include many non abelian groups such as dihedral and generalized quaternion groups, and the projective general linear group of order two over a finite field. In particular, we provide improvements on past works regarding the description of the dynamics of the power map over these groups.

Extended Affine Root Supersystems of Types $C(I, J)$ and $BC(1, 1)$

In this paper, we complete the characterization of tame irreducible extended affine root supersystems. We give a complete description of tame irreducible extended affine root supersystems of type X = C(1, 1),C(1, 2), C(2, 2) and BC(1, 1) and determine isomorphic classes.

Tri-linear birational maps in dimension three

A tri-linear rational map in dimension three is a rational map ϕ : ( P C 1 ) 3 ⇢ P C 3 \phi : (\mathbb {P}_\mathbb {C}^1)^3 \dashrightarrow \mathbb {P}_\mathbb {C}^3 defined by four tri-linear polynomials without a common factor. If ϕ \phi admits an inverse rational map ϕ − 1 \phi ^{-1} , it is a tri-linear birational map. In this paper, we address computational and geometric aspects about these transformations. We give a characterization of birationality based on the first syzygies of the entries. More generally, we describe all the possible minimal graded free resolutions of the ideal generated by these entries. With respect to geometry, we show that the set B i r ( 1 , 1 , 1 ) \mathfrak {Bir}_{(1,1,1)} of tri-linear birational maps, up to composition with an automorphism of P C 3 \mathbb {P}_\mathbb {C}^3 , is a locally closed algebraic subset of the Grassmannian of 4 4 -dimensional subspaces in the vector space of tri-linear polynomials, and has eight irreducible components. Additionally, the group action on B i r ( 1 , 1 , 1 ) \mathfrak {Bir}_{(1,1,1)} given by composition with automorphisms of ( P C 1 ) 3 (\mathbb {P}_\mathbb {C}^1)^3 defines 19 orbits, and each of these orbits determines an isomorphism class of the base loci of these transformations.

Inducing braces and Hopf Galois structures

Let p be a prime number and let n be an integer not divisible by p and such that every group of order np has a normal subgroup of order p. (This holds in particular for p>n.) Under these hypotheses, we obtain a one-to-one correspondence between the isomorphism classes of braces of size np and the set of pairs (Bn,[τ]), where Bn runs over the isomorphism classes of braces of size n and [τ] runs over the classes of group morphisms from the multiplicative group of Bn to Zp⁎ under a certain equivalence relation. This correspondence gives the classification of braces of size np from the one of braces of size n. From this result we derive a formula giving the number of Hopf Galois structures of abelian type Zp×E on a Galois extension of degree np in terms of the number of Hopf Galois structures of abelian type E on a Galois extension of degree n. For a prime number p≥7, we apply the obtained results to describe all left braces of size 12p and determine the number of Hopf Galois structures of abelian type on a Galois extension of degree 12p.

Truncated Affine Rozansky–Witten Models as Extended TQFTs

We construct extended TQFTs associated to Rozansky–Witten models with target manifolds T^*mathbb {C}^n. The starting point of the construction is the 3-category whose objects are such Rozansky–Witten models, and whose morphisms are defects of all codimensions. By truncation, we obtain a (non-semisimple) 2-category mathcal C of bulk theories, surface defects, and isomorphism classes of line defects. Through a systematic application of the cobordism hypothesis we construct a unique extended oriented 2-dimensional TQFT valued in mathcal C for every affine Rozansky–Witten model. By evaluating this TQFT on closed surfaces we obtain the infinite-dimensional state spaces (graded by flavour and R-charges) of the initial 3-dimensional theory. Furthermore, we explicitly compute the commutative Frobenius algebras that classify the restrictions of the extended theories to circles and bordisms between them.

Non-weight representations of Lie superalgebras of Block type, I

In this paper, we study non-weight representations of two classes of Lie algebras and two classes of Lie superalgebras of Block type. More precisely, we completely classify the free U(h)-modules of rank one or two over these Lie superalgebras. Then we determine the simplicity and isomorphism classes of these modules. As a byproduct, we also obtain the non-weight representation results for a class of non-semisimple finitely graded extensions of the Virasoro algebra.

Unusual fluid of one-dimensional disordered bosons at finite temperature

In previous important examinations by Aleiner et al. [Nat. Phys. 6, 900 (2010)] and Michal et al. [Proc. Natl. Acad. Sci. (U.S.A.) 113, E4455 (2016)], the fluid insulator transition has been explored in one-dimensional (1D) disordered bosons by analytical methods. However, the nature of this ``fluid'' is still not quite understood. Here we shed more light on this ``fluid'' by studying its properties under changes in the average particle number $\ensuremath{\langle}N\ensuremath{\rangle}$ and temperature $T$ in the absence of superfluidity. For this purpose, the worm algorithm path-integral Monte Carlo method is utilized in the grand-canonical ensemble. A measure for the width of path fluctuations is introduced that we call ${\ensuremath{\sigma}}_{J}$. It is found that by increasing $\ensuremath{\langle}N\ensuremath{\rangle}$, ${\ensuremath{\sigma}}_{J}$ rises as the bosons become less repulsive and gain more mobility. The compressibility $\ensuremath{\chi}(\ensuremath{\mu})$, $\ensuremath{\mu}$ being the chemical potential, reveals fluid behavior at the lower $\ensuremath{\gamma}$ and a proposed Bose-glass phase at the larger $\ensuremath{\gamma}$ (the Lieb-Liniger interaction parameter). The kinetic energy ${E}_{K}$ versus $\ensuremath{\gamma}$ at various $T$ signals a transition by its convergence towards a common value. ${E}_{K}$ versus $\ensuremath{\mu}$ reveals a peculiar inversion as one goes from small to larger values of the speckle strength $\ensuremath{\langle}V\ensuremath{\rangle}$ demonstrating a complex influence of the disorder. A tipover of ${E}_{K}$ versus $\ensuremath{\gamma}$ from an increasing to a decreasing trend at the lower $\ensuremath{\gamma}$ also signals a transition. Further, Luttinger's liquid theory cannot be applied to describe 1D bosons in strong disorder at finite $T$ because of the absence of superfluidity. Moreover, $\ensuremath{\langle}V\ensuremath{\rangle}$ largely controls the behavior of $\ensuremath{\chi}(\ensuremath{\mu})$, the pair-correlation function at the origin ${g}_{2}(0)$, and ${E}_{K}$. Surprisingly, the diffusion constant ${D}_{0}$ does not obey the classical isomorphism as it rises with $T$. We also emphasize the importance of this work via its applicability to quantum wires.

A study on anabelian geometry of higher local fields

Anabelian geometry has been developed over a much wider class of fields than Grothendieck, who is the originator of anabelian geometry, conjectured. So, it is natural to ask the following question: What kinds of fields are suitable for the base fields of anabelian geometry? In this paper, we consider this problem for higher local fields. First, to consider “anabelianness” of higher local fields themselves, we give mono-anabelian reconstruction algorithms of various invariants of higher local fields from their absolute Galois groups. As a result, the isomorphism classes of certain types of higher local fields are completely determined by their absolute Galois groups. Next, we prove that mixed-characteristic higher local fields are Kummer-faithful. This result affirms the above question for these higher local fields to a certain extent.

Some Properties of n-semidualizing Modules

Let R be a commutative noetherian ring. The n-semidualizing modules of R are generalizations of its semidualizing modules. We will prove some basic properties of n-semidualizing modules. Our main result and example shows that the divisor class group of a Gorenstein determinantal ring over a field is the set of isomorphism classes of its 1-semidualizing modules. Finally, we pose some questions about n-semidualizing modules.

The tensor Harish-Chandra–Itzykson–Zuber integral I: Weingarten calculus and a generalization of monotone Hurwitz numbers

We study a generalization of the Harish-Chandra–Itzykson–Zuber integral to tensors and its expansion in terms of trace invariants of the two external tensors. This gives rise to natural generalizations of monotone double Hurwitz numbers, which count certain families of constellations. We find an expression of these numbers in terms of monotone simple Hurwitz numbers, thereby also providing expressions for monotone double Hurwitz numbers of arbitrary genus in terms of the single ones. We give an interpretation of the different combinatorial quantities at play in terms of enumeration of nodal surfaces. In particular, our generalization of Hurwitz numbers is shown to count certain isomorphism classes of branched coverings of a bouquet of D 2-spheres that touch at one common non-branch node.

A note on Quebbemann’s extremal lattices of rank 64

By constructing explicit examples, we show that the method of Quebbemann yields many isomorphism classes of extremal lattices of rank 64. Many of these examples have no non-trivial automorphisms.

On the enumeration of finite 𝐿-algebras

We use Constraint Satisfaction Methods to construct and enumerate finite L L -algebras up to isomorphism. These objects were recently introduced by Rump and appear in Garside theory, algebraic logic, and the study of the combinatorial Yang–Baxter equation. There are 377,322,225 isomorphism classes of L L -algebras of size eight. The database constructed suggests the existence of bijections between certain classes of L L -algebras and well-known combinatorial objects. We prove that Bell numbers enumerate isomorphism classes of finite linear L L -algebras. We also prove that finite regular L L -algebras are in bijective correspondence with infinite-dimensional Young diagrams.

Two-dimensional perfect evolution algebras over domains

We will study evolution algebras A that are free modules of dimension two over domains. We start by making some general considerations about algebras over domains: They are sandwiched between a certain essential D-submodule and its scalar extension over the field of fractions of the domain. We introduce the notion of quasiperfect algebras and we characterize the perfect and quasiperfect evolution algebras in terms of the determinant of its structure matrix. We classify the two-dimensional perfect evolution algebras over domains parametrizing the isomorphism classes by a convenient moduli set.

The size of a stratifying system can be arbitrarily large

In this short note we construct two families of examples of large stratifying systems in module categories of algebras. The first examples consist on stratifying systems of infinite size in the module category of an algebra A. In the second family of examples we show that the size of a finite stratifying system in the module category of a finite dimensional algebra A can be arbitrarily large in comparison to the number of isomorphism classes of simple A-modules. We note that both families of examples are built using well-established results in higher homological algebra.