Journal of Algebra最新文献

Groups in which the non-moduar subgroups fall into finitely many isomorphism classes are considered, and it is proved that a (generalized) soluble group with this property either has modular subgroup lattice or is a minimax group. The corresponding result for (generalized) soluble groups with finitely many isomorphism classes of non-permutable subgroups is also obtained.

A modular tensor category is a non-degenerate ribbon finite tensor category and a ribbon factorizable Hopf algebra is a Hopf algebra whose finite-dimensional representations form a modular tensor category. In this paper, we provide a method of constructing ribbon factorizable Hopf algebras using central extensions. We then apply this method to n-rank Taft algebras, which are considered finite-dimensional quantum groups associated with abelian Lie algebras (see Section 2 for the definition), and obtain a family of non-semisimple ribbon factorizable Hopf algebras $E_q$, thus producing non-semisimple modular tensor categories using their representation categories. And we provide a prime decomposition of $\mathrm{Rep}\left(E_q\right)$ (the representation category of $E_q$). By further studying the simplicity of $E_q$ (whether it is a simple Hopf algebra or not), we conclude that

• (1)

  there exists a twist J of $u_q\left(s{l}_2^{\oplus 3}\right)$ such that $u_q{\left(s{l}_2^{\oplus 3}\right)}^J$ is a simple Hopf algebra,

• (2)

  there is no relation between the simplicity of a Hopf algebra H and the primality of $\mathrm{Rep}\left(H\right)$,

• (3)

  there are many ribbon factorizable Hopf algebras that are distinct from some known ones, i.e., not isomorphic to any tensor products of trivial Hopf algebras (group algebras or their dual), Drinfeld doubles, and small quantum groups.

Utilizing previously established results concerning costratification in relative tensor-triangular geometry, we classify the colocalizing subcategories of the singularity category of a locally hypersurface ring and then we generalize this classification to singularity categories of schemes with hypersurface singularities.

The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's Positivestellensatz on a compact basic semi-algebraic set S and provide a new proof and new improved bounds on the degree of the representation of positive polynomials. These new bounds involve a parameter ε measuring the non-vanishing of the positive function, the constant $c$ and exponent L of a Łojasiewicz inequality for the semi-algebraic distance function associated to the inequalities $g=(g_1,\dots ,g_r)$ defining S. They are polynomial in $c$ and ${\epsilon }^{-1}$ with an exponent depending only on L. We analyse in details the Łojasiewicz inequality when the defining inequalities g satisfy the Constraint Qualification Condition. We show that, in this case, the Łojasiewicz exponent L is 1 and we relate the Łojasiewicz constant $c$ with the distance of g to the set of singular systems.

We continue our study of cyclic orbifolds of lattice vertex operator algebras and their full automorphism groups. We consider some special isometry $g\in O\left(L\right)$ such that $g^i$ is fixed point free on L for any $1\le i\le \left|g\right|-1$. We show that when $L\left(2\right)=\varnothing$ and $g^i$ is fixed point free on L for any $1\le i\le \left|g\right|-1$, $V_L^{\hat{g}}$ has extra automorphisms implies either (1) the order of g is a prime or (2) L is isometric to the Leech lattice or some coinvariant sublattices of the Leech lattice.

We analyse the structure of infinite weakly soluble left braces that satisfy the minimal condition for subbraces. We observe that they can be characterised as the left braces with Chernikov additive group. We also present an example of left braces satisfying the minimal condition for ideals, but that do not satisfy the minimal condition for subbraces.

Connected bipartite graphs whose binomial edge ideals are Cohen–Macaulay have been classified by Bolognini et al. In this paper, we compute the depth, Castelnuovo–Mumford regularity, and dimension of the generalized binomial edge ideals of these graphs.

Boij-Söderberg theory gives a combinatorial description of the set of Betti tables belonging to finite length modules over the polynomial ring $S=k[x_1,\dots ,x_n]$. We posit that a similar combinatorial description can be given for analogous numerical invariants of graded differential S-modules, which are natural generalizations of chain complexes. We prove several results that lend evidence in support of this conjecture, including a categorical pairing between the derived categories of graded differential S-modules and coherent sheaves on $P^{n-1}$ and a proof of the conjecture in the case where $S=k\left[t\right]$.

We study the rational Cherednik algebra $H_{t,c}(S_3,h)$ of type $A_2$ in positive characteristic p, and its irreducible category $O$ representations $L_{t,c}\left(\tau \right)$. For every possible value of $p,t,c$, and τ we calculate the Hilbert polynomial and the character of $L_{t,c}\left(\tau \right)$, and give explicit generators of the maximal proper graded submodule of the Verma module.

Let $T_X$ be the full transformation monoid over a finite set X, and fix some $a\in T_X$ of rank r. The variant $T_X^a$ has underlying set $T_X$, and operation $f\star g=fag$. We study the congruences of the subsemigroup $P=\mathrm{Reg}\left(T_X^a\right)$ consisting of all regular elements of $T_X^a$, and the lattice $\mathrm{Cong}\left(P\right)$ of all such congruences. Our main structure theorem ultimately decomposes $\mathrm{Cong}\left(P\right)$ as a specific subdirect product of $\mathrm{Cong}\left(T_r\right)$, and the full equivalence relation lattices of certain combinatorial systems of subsets and partitions. We use this to give an explicit classification of the congruences themselves, and we also give a formula for the height of the lattice.