Journal of Algebra最新文献

We give a systematic study of the model theory of generic nilpotent groups and Lie algebras. We show that the Fraïssé limit of 2-nilpotent groups of exponent p studied by Baudisch is 2-dependent and NSOP₁. We prove that the class of c-nilpotent Lie algebras over an arbitrary field, in a language with predicates for a Lazard series, is closed under free amalgamation. We show that for $2<c$, the generic c-nilpotent Lie algebra over $F_p$ is strictly NSOP₄ and c-dependent. Via the Lazard correspondence, we obtain the same result for c-nilpotent groups of exponent p, for an odd prime $p>c$.

In this work, we obtain contraction results for a class of diagrams of ring morphisms which strictly includes the ones obtained by Lipman. Further, we present some applications on quotient and in the changing of the base ring in the saturation.

We prove that the stable image of an endomorphism of a virtually free group is computable. For an endomorphism φ, an element $x\in G$ and a subset $K\subseteq G$, we say that the φ-order of g relative to K, $\phi {\text{-ord}}_K\left(g\right)$, is the smallest nonnegative integer k such that $g{\phi }^k\in K$. We prove that the set of orders, which we call φ-spectrum, is computable in two extreme cases: when K is a finite subset and when K is a recognizable subset. The finite case is proved for virtually free groups and the recognizable case for finitely presented groups. The case of finitely generated virtually abelian groups and some variations of the problem are also discussed.

Let $g$ be a symmetrizable Kac-Moody Lie algebra, and let $V_{\hat{g},ħ}^{\ell }$, $L_{\hat{g},ħ}^{\ell }$ be the quantum affine vertex algebras constructed in [11]. For any complex numbers ℓ and ${\ell }^{\prime }$, we present an ħ-adic quantum vertex algebra homomorphism Δ from $V_{\hat{g},ħ}^{\ell +{\ell }^{\prime }}$ to the twisted tensor product ħ-adic quantum vertex algebra $V_{\hat{g},ħ}^{\ell }\hat{\otimes }{V}_{\hat{g},ħ}^{{\ell }^{\prime }}$. In addition, if both ℓ and ${\ell }^{\prime }$ are positive integers, we show that Δ induces an ħ-adic quantum vertex algebra homomorphism from $L_{\hat{g},ħ}^{\ell +{\ell }^{\prime }}$ to the twisted tensor product ħ-adic quantum vertex algebra $L_{\hat{g},ħ}^{\ell }\hat{\otimes }{L}_{\hat{g},ħ}^{{\ell }^{\prime }}$. Moreover, we prove the coassociativity of Δ.

We show that a unital ring is generated by its commutators as an ideal if and only if there exists a natural number N such that every element is a sum of N products of pairs of commutators. We show that one can take $N\le 2$ for matrix rings, and that one may choose $N\le 3$ for rings that contain a direct sum of matrix rings – this in particular applies to C*-algebras that are properly infinite or have real rank zero. For Jiang-Su-stable C*-algebras, we show that $N\le 6$ can be arranged.

For arbitrary rings, we show that every element in the commutator ideal admits a power that is a sum of products of commutators. Using that a C*-algebra cannot be a radical extension over a proper ideal, we deduce that a C*-algebra is generated by its commutators as a not necessarily closed ideal if and only if every element is a finite sum of products of pairs of commutators.

In this paper we give positive answers for two open questions on extension bundles over weighted projective lines, raised by Kussin, Lenzing and Meltzer in the paper “Triangle singularities, ADE-chains and weighted projective lines”.

In this note we study line and conic arrangements associated with sextactic and type 9 points on the Fermat cubic F and we provide explicit coordinates for each of the 72 type 9 points on F.

Given a sequence of related modules $M_n$ defined over a sequence of related polynomial rings, one may ask how to simultaneously compute a finite Gröbner basis for each $M_n$. Furthermore, one may ask how to simultaneously compute the module of syzygies of each $M_n$. In this paper we address both questions. Working in the setting of OI-modules over a Noetherian polynomial OI-algebra, we provide OI-analogues of Buchberger's Criterion, Buchberger's Algorithm for computing Gröbner bases, and Schreyer's Theorem for computing syzygies. We also establish a stabilization result for Gröbner bases.

In this work we compute the triangulated Grothendieck groups for each of the family of discrete cluster categories of Dynkin type $A_{\infty }$ as introduced by Holm-Jørgensen. Subsequently, we also compute the Grothendieck group of a completion of these discrete cluster categories in the sense of Paquette-Yildirim.

We introduce and study flipped non-associative polynomial rings. In particular, we show that all Cayley–Dickson algebras naturally appear as quotients of a certain type of such rings; this extends the classical construction of the complex numbers (and quaternions) as a quotient of a (skew) polynomial ring to the octonions, and beyond. We also extend some classical results on algebraic properties of Cayley–Dickson algebras by McCrimmon to a class of flipped non-associative polynomial rings.