Communications in Algebra

Hablicsek, Márton; Juhász, Máté L.

doi: 10.1080/00927872.2017.1344691pmid: N/A

Nikolai Durov introduced the theory of generalized rings and schemes to study Arakelov geometry in an alternative algebraic framework, and introduced the residue field at the infinite place, 𝔽∞. We show an elementary algebraic approach to modules and algebras over this object, define prime congruences, show that the polynomial ring of n variables is of Krull dimension n, and derive a prime decomposition theorem for these primes.

Birkenmeier, G. F.; Ghirati, M.; Ghorbani, A.; Naghdi, A.; Taherifar, A.

doi: 10.1080/00927872.2017.1360335pmid: N/A

In this corrigendum, we correct the statement and proof of Lemma 3.2, revise the proofs of Lemma 3.3 and Theorem 3.5, retract Proposition 3.10 and correct several typographical errors.

Hoshino, Mitsuo; Kameyama, Noritsugu; Koga, Hirotaka

doi: 10.1080/00927872.2017.1372458pmid: N/A

Let R be a ring and n≥2 an integer. We provide a systematic way to define new multiplications on Mn(R), the ring of n×n full matrices with entries in R. The obtained new rings Λ are Auslander–Gorenstein if and only if so is R.

Calderón, A. J.; Cheikh, D. M.

doi: 10.1080/00927872.2017.1376214pmid: N/A

A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g0, where g0 is some element of the grading group G such that g0 = 0 or 4g0≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras.

Chandrasekhar, Olivia; Penn, Michael; Shao, Hanbo

doi: 10.1080/00927872.2017.1388810pmid: N/A

By using quantum vertex operators we study the invariance of the rank n free-fermion vertex algebra under the action of the group ℤ∕2ℤ and obtain its minimal generating set. When n = 1, it is well known that this subalgebra is isomorphic to the Virasoro vertex algebra with central charge 1∕2. In the n = 2 case we show that invariant subalgebra is isomorphic to a simple quotient of a certain W-algebra, which we explicitly construct. For n≥3, our approach leads to a rediscovery of the spinor representation of the affine vertex algebra associated to the Lie algebra 𝔰𝔬(n) of I. Frenkel.

Clark, Pete L.

doi: 10.1080/00927872.2017.1392540pmid: N/A

The rank rk(R) of a ring R is the supremum of minimal cardinalities of generating sets of I as I ranges over ideals of R. Matsuda and Matson showed that every n∈ℤ+ (the positive integers) occurs as the rank of some ring R. Motivated by the result of Cohen and Gilmer that a ring of finite rank has Krull dimension 0 or 1, we give four different constructions of rings of rank n (for all n∈ℤ+). Two constructions use one-dimensional domains. Our third construction uses Artinian rings (dimension zero), and our last construction uses polynomial rings over local Artinian rings (dimension one, irreducible, not a domain).

Rogers, Sean

doi: 10.1080/00927872.2017.1407418pmid: N/A

Let G be a simply connected semisimple algebraic groups over ℂ and let ρ:G→GL(Vλ) be an irreducible representation of G of highest weight λ. Suppose that ρ has finite kernel. Springer defined an adjoint-invariant regular map with Zariski dense image from the group to the Lie algebra, 𝜃λ:G→𝔤, which depends on λ. This map, 𝜃λ, takes the maximal torus T of G to its Lie algebra 𝔱. Thus, for a given simple group G and an irreducible representation Vλ, one may write , where we take the simple coroots as a basis for 𝔱. We give a complete determination for these coefficients ci(t) for any simple group G as a sum over the weights of the torus action on Vλ.

Zhang, Juanjuan

doi: 10.1080/00927872.2018.1424885pmid: N/A

For the two Cartan type S subalgebras of the Witt algebra 𝒲n, called Lie algebras of divergence-zero vector fields, we determine all module structures on the universal enveloping algebra of their Cartan subalgebra 𝔥n. We also give all submodules of these modules.

Nagamine, Takanori

doi: 10.1080/00927872.2018.1439041pmid: N/A

In this paper, we study some properties of coordinates in the polynomial ring by introducing some concepts which are weaker than coordinates. In particular, in the polynomial ring in two variables over an algebraically closed field of characteristic zero, we show some relations between polynomials and their fibers on 𝔸2.

Can, Mahir Bilen

doi: 10.1080/00927872.2018.1439042pmid: N/A

We consider the conjugation action of symmetric group on the semigroup of all partial functions and develop a machinery to investigate character formulas and multiplicities. By interpreting these objects in terms of labeled rooted forests, we give a characterization of the labeled rooted trees whose Sn orbit afford the sign representation. Applications to rook theory are offered.