Field Of Zero Research Articles

Functional equations in formal power series

Let \(k\) be an algebraically closed field of characteristic zero, and \(k[[z]]\) the ring of formal power series over \(k\). In this paper, we study equations in the semigroup \(z^2k[[z]]\) with the semigroup operation being composition. We prove a number of general results about such equations and provide some applications. In particular, we answer a question of Horwitz and Rubel about decompositions of "even" formal power series. We also show that every right amenable subsemigroup of \(z^2k[[z]]\) is conjugate to a subsemigroup of the semigroup of monomials.

On the dualizability of fusion 2-categories

Over an arbitrary field, we prove that the relative 2-Deligne tensor product of two separable module 2-categories over a compact semisimple tensor 2-category exists. This allows us to consider the Morita 4-category of compact semisimple tensor 2-categories, separable bimodule 2-categories, and their morphisms. Categorifying a result of Douglas, Schommer-Pries, Snyder [Mem. Amer. Math. Soc. 268 (2021)], we prove that separable compact semisimple tensor 2-categories are fully dualizable objects therein. In particular, it then follows from the main theorem of Décoppet [Comp. Math.] that, over an algebraically closed field of characteristic zero, every fusion 2-category is a fully dualizable object of the above Morita 4-category. We explain how this can be extended to any field of characteristic zero. Finally, we discuss the field theoretic interpretation of our results.

On the ascent of atomicity to monoid algebras

A commutative cancellative monoid is atomic if every non-invertible element factors into irreducibles (also called atoms), while an integral domain is atomic if its multiplicative monoid is atomic. Back in the eighties, Gilmer posed the question of whether the fact that a torsion-free monoid M and an integral domain R are both atomic implies that the monoid algebra R[M] of M over R is also atomic. In general this is not true, and the first negative answer to this question was given by Roitman in 1993: he constructed an atomic integral domain whose polynomial extension is not atomic. More recently, Coykendall and the first author constructed finite-rank torsion-free atomic monoids whose monoid algebras over certain finite fields are not atomic. Still, the ascent of atomicity from finite-rank torsion-free monoids to their corresponding monoid algebras over fields of characteristic zero is an open problem. Coykendall and the first author also constructed an infinite-rank torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. As the primary result of this paper, we construct a rank-one torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. To do so, we introduce and study a methodological construction inside the class of rank-one torsion-free monoids that we call lifting, which consists in embedding a given monoid into another monoid that is often more tractable from the arithmetic viewpoint. For instance, we prove here that the embedding in the lifting construction preserves the ascending chain condition on principal ideals and the existence of maximal common divisors.

Counting gradings on Lie algebras of block-triangular matrices

We study the number of isomorphism classes of gradings on Lie algebras of block-triangular matrices. Let G be a finite abelian group, for m∈Ns we determine the number E(−)(G,m) of isomorphism classes of elementary G-gradings on the Lie algebra UT(m)(−) of block-triangular matrices over an algebraically closed field of characteristic zero. We study the asymptotic growth of E(−)(G,m) and as a consequence prove that the E(−)(G,⋅) determines G up to isomorphism. We also study the asymptotic growth of the number N(−)(G,m) of isomorphism classes of G-gradings on UT(m)(−) and prove that N(−)(G,m))∼|G|E(−)(G,m).

The finite presentation problem for Noetherian algebras

We prove that there exists an affine Noetherian algebra over a field of characteristic zero which is not finitely presented. Specifically, we prove that the Medvedev–Passman–Resco–Small algebra, recently shown to form a counterexample to the stability problem for Noetherian algebras, is not finitely presented. This answers a question due to Bergman and Irving in characteristic zero, a case explicitly left open by Resco and Small in 1993, who gave a counterexample to this question in positive characteristic.

Categorical resolutions of filtered schemes

We give an alternative proof of the theorem by Kuznetsov and Lunts, stating that any separated scheme of finite type over a field of characteristic zero admits a categorical resolution of singularities. Their construction makes use of the fact that every variety (over a field of characteristic zero) can be resolved by a finite sequence of blow-ups along smooth centres. We merely require the existence of (projective) resolutions. To accomplish this we put the \mathcal{A} -spaces of Kuznetsov and Lunts in a different light, viewing them instead as schemes endowed with finite filtrations. The categorical resolution is then constructed by gluing together differential graded categories obtained from a hypercube of finite length filtered schemes.

Algebraic characterizations of homeomorphisms between algebraic varieties

We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be a homeomorphism for the Zariski topology and for a strong topology that we introduce. Our answers involve a study of seminormalization and saturation for morphisms between algebraic varieties, together with an interpretation in terms of continuous rational functions on the closed points of an algebraic variety. The continuity refers to the strong topology which is the usual Euclidean topology in the complex case and which comes from the theory of real closed fields otherwise.

Representations Of Lie Algebras of Vector Fields on Pairing Between Gauge Modules and Rudakov Modules

For an irreducible affine variety X over an algebraically closed field of characteristic zero, we define two new classes of modules over the Lie algebra of vector fields on X: gauge modules and Rudakov modules. These modules admit a compatible action of the algebra of functions on X. We prove general simplicity theorems for these two types of modules, demonstrating their irreducibility under specific conditions. Additionally, we establish a pairing between gauge modules and Rudakov modules, highlighting the connections and interactions between these two classes of modules. We have established that gauge modules and Rudakov modules, corresponding to simple glN -modules, remain irreducible as modules over the Lie algebra of vector fields unless they appear in the de Rham complex. Additionally, we have studied the irreducibility of tensor products of Rudakov modules, providing a comprehensive understanding of these module structures and their applications.

On minimal varieties of superalgebras with superautomorphism

Let A be a finite-dimensional superalgebra with superautomorphism φ over a field of characteristic zero. In [11] the authors gave a positive answer to the Amitsur's conjecture in this setting showing that the φ-exponent of A exists and it is an integer. In the present paper we extend the notion of minimal variety to the context of superalgebras with superautomorphism and prove that a variety is minimal of fixed φ-exponent if, and only if, it is generated by a subalgebra of an upper block triangular matrix algebra equipped with a suitable elementary Z2-grading and superautomorphism. Along the way, we give a contribution on the isomorphism question within the theory of polynomial identities.

Minimal affine varieties of superalgebras with superinvolution: A characterization

We exhibit a class C of finite dimensional algebras with superinvolution over an algebraically closed field of characteristic zero, with the remarkable property that each member of C generates a minimal variety of algebras with superinvolution. This sums up to the fact that any affine minimal variety of algebras with superinvolution is generated by a suitable member of C, thus providing a complete characterization of the affine minimal varieties of algebras with superinvolution.

On algebras embeddable into bicommutative algebras

We consider algebras embeddable into free bicommutative algebras with respect to commutator and anti-commutator products over a field of characteristic zero. We show that every metabelian Lie algebra can be embedded into a bicommutative algebra with respect to the commutator product. Furthermore, we prove that the class of commutative algebras embeddable into bicommutative algebras with respect to the anti-commutator product forms a variety. As a consequence, we obtain that every metabelian Lie algebra can be embedded into a Novikov algebra with respect to the commutator product.

Nef vector bundles on a hyperquadric with first Chern class two

Abstract We classify nef vector bundles on a smooth hyperquadric of dimension ≥ 4 {\geq 4} with first Chern class two over an algebraically closed field of characteristic zero.

Automorphisms of quantum polynomial rings and Drinfeld Hecke algebras

We consider quantum (skew) polynomial rings and observe that their graded automorphisms coincide with those of quantum exterior algebras. This allows us to define a quantum determinant that gives a homomorphism of groups acting on quantum polynomial rings. We use quantum subdeterminants to classify the resulting Drinfeld Hecke algebras for the symmetric group, other infinite families of Coxeter and complex reflection groups, and mystic reflection groups (which satisfy a version of the Shephard–Todd–Chevalley theorem). This direct combinatorial approach replaces the technology of Hochschild cohomology used by Naidu and Witherspoon over fields of characteristic zero and allows us to extend some of their results to fields of arbitrary characteristic and also locate new deformations of skew group algebras.

Two criteria for quasihomogeneity

Let ( R , m R , k ) (R,\mathfrak {m}_R,\mathbb {k}) be a one-dimensional complete local reduced k \mathbb {k} -algebra over a field of characteristic zero. The ring R R is said to be quasihomogeneous if there exists a surjection Ω R ↠ m \Omega _R\twoheadrightarrow \mathfrak {m} where Ω R \Omega _R denotes the module of differentials. We present two characterizations of quasihomogeniety of R R in the case when R R is a domain. The first one on the valuation semigroup of R R and the other on the trace ideal of the module Ω R \Omega _R .

Classifying simple superalgebras with automorphism and pseudoautomorphism

Let F be an algebraically closed field of characteristic zero. We give a complete classification of finite dimensional simple superalgebras over F endowed with a graded automorphism or a pseudoautomorphism.

Derivations in disjointly complete commutative regular algebras

We show that any nonexpansive derivation on a subalgebra of a dis-jointly complete commutative regular algebra extends up to a derivation on . For an algebra of functions , continuous on a dense open subset of Stone compact X, we establish that the lack of nontrivial derivation is equivalent to σ-distributivity of the Boolean algebra of clopen subsets of X. The field is an arbitrary normed field of charachteristic zero containing a complete non-discrete subfield. Our work is motivated by two seemingly unrelated problems due to Ayupov [2] and Wickstead [32].

On simple modules and automorphisms of the quantum Galilei group

This paper is devoted to ring theoretic properties and representations of the quantum Galilei group F q ( G ) and its Hopf dual U q ( g ) . We classify the primitive ideals and simple modules of F q ( G ) and U q ( g ) over an algebraically closed field of arbitrary characteristic. We determine their groups of automorphisms over a field of characteristic zero.

On unitary algebras with graded involution of quadratic growth

Let F be a field of characteristic zero. By a ⁎-superalgebra we mean an algebra A with graded involution over F. Recently, algebras with graded involution have been extensively studied in PI-theory and the sequence of ⁎-graded codimensions {cngri(A)}n≥1 has been investigated by several authors. In this paper, we classify varieties generated by unitary ⁎-superalgebras having quadratic growth of ⁎-graded codimensions. As a result we obtain that a unitary ⁎-superalgebra with quadratic growth is T2⁎-equivalent to a finite direct sum of minimal unitary ⁎-superalgebras with at most quadratic growth, where at least one ⁎-superalgebra of this sum has quadratic growth. Furthermore, we provide a method to determine explicitly the factors of those direct sums.

Matrices for finite group representations that respect Galois automorphisms

AbstractWe are given a finite group H, an automorphism $$\tau $$ τ of H of order r, a Galois extension L/K of fields of characteristic zero with cyclic Galois group $$\langle \sigma \rangle $$ ⟨ σ ⟩ of order r, and an absolutely irreducible representation $$\rho :H\rightarrow \textsf {GL} (n,L)$$ ρ : H → GL ( n , L ) such that the action of $$\tau $$ τ on the character of $$\rho $$ ρ is the same as the action of $$\sigma $$ σ . Then the following are equivalent. $$\bullet $$ ∙ $$\rho $$ ρ is equivalent to a representation $$\rho ':H\rightarrow \textsf {GL} (n,L)$$ ρ ′ : H → GL ( n , L ) such that the action of $$\sigma $$ σ on the entries of the matrices corresponds to the action of $$\tau $$ τ on H, and $$\bullet $$ ∙ the induced representation $$\textsf {ind} _{H,H\rtimes \langle \tau \rangle }(\rho )$$ ind H , H ⋊ ⟨ τ ⟩ ( ρ ) has Schur index one; that is, it is similar to a representation over K. As examples, we discuss a three dimensional irreducible representation of $$A_5$$ A 5 over $$\mathbb {Q}[\sqrt{5}]$$ Q [ 5 ] and a four dimensional irreducible representation of the double cover of $$A_7$$ A 7 over $$\mathbb {Q}[\sqrt{-7}]$$ Q [ - 7 ] .

Annihilator Ideals of Indecomposable Modules of Finite-Dimensional Pointed Hopf Algebras of Rank One

Let [Formula: see text] be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero. We describe all annihilator ideals of indecomposable [Formula: see text]-modules by generators. In particular, we give the classification of all ideals of the finite-dimensional pointed rank one Hopf algebra of nilpotent type over the Klein 4-group.