Narrow Results

We classify, up to isomorphism, all gradings by an arbitrary abelian group on simple finitary Lie algebras of linear transformations (special linear, orthogonal and symplectic) on infinite-dimensional vector spaces over an algebraically closed field of characteristic different from 2.

Known classification results allow us to find the number of (equivalence classes of) fine gradings on matrix algebras and on classical simple Lie algebras over an algebraically closed field 𝔽 (assuming char 𝔽 ≠ 2 in the Lie case). The computation is easy for matrix algebras and especially for simple Lie algebras of type B_r (the answer is just r + 1), but involves counting orbits of certain finite groups in the case of Series A, C and D. For X ∈ {A, C, D}, we determine the exact number of fine gradings, N_X(r), on the simple Lie algebras of type X_r with r ≤ 100 as well as the asymptotic behavior of the average, , for large r. In particular, we prove that there exist positive constants b and c such that . The analogous average for matrix algebras M_n(𝔽) is proved to be a ln n + O(1) where a is an explicit constant depending on char 𝔽.

Let $A$ be a finite dimensional simple real algebra with a division grading by a finite abelian group $G$. In this paper, we provide a finite basis for the $T_G$-ideal of graded polynomial identities for $A$ and a finite basis for the $T_G$-space of graded central polynomials for $A$.

Let $F$ be an infinite field of characteristic different from 2, and let $E$ be the Grassmann algebra of a countable of dimensional $F$-vector space $L$. In this paper, we study the graded central polynomials of gradings on $E$ by the groups ${\mathbb{Z}}_2$ and $\mathbb{Z}$, where the basis of the vector space $L$ is homogeneous. More specifically, we provide a basis for the $T_G$-space of graded central polynomials for $E$, where the group $G$ is ${\mathbb{Z}}_2$ and $\mathbb{Z}$.

We show that the right ideal of a Novikov algebra generated by the square of a right nilpotent subalgebra is nilpotent. We also prove that a $G$-graded Novikov algebra $N$ over a field $K$ with solvable $0$-component $N_0$ is solvable, where $G$ is a finite additive abelean group and the characteristic of $K$ does not divide the order of the group $G$. We also show that any Novikov algebra $N$ with a finite solvable group of automorphisms $G$ is solvable if the algebra of invariants $N^G$ is solvable.

The aim of this paper is to present some connections of infinite Cogalois Theory with Clifford extensions and Hopf algebras.

A magma is just a set $A$ with a single binary operation

We provide an explicit description of homogeneous locally nilpotent derivations of the algebra of regular functions on affine trinomial hypersurfaces. As an application, we describe the set of roots of trinomial algebras.

A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a classification of finite-dimensional graded-central graded-division algebras over an arbitrary field $𝔽$ can be reduced to the following three classifications, for each finite Galois extension $𝕃$ of $𝔽$: (1) finite-dimensional central division algebras over $𝕃$, up to isomorphism; (2) twisted group algebras of finite groups over $𝕃$, up to graded-isomorphism; (3) $𝔽$-forms of certain graded matrix algebras with coefficients in $\mathrm{\Delta }{\otimes }_{𝕃}{𝒞}$ where $\mathrm{\Delta }$ is as in (1) and ${𝒞}$ is as in (2). As an application, we classify, up to graded-isomorphism, the finite-dimensional graded-division algebras over the field of real numbers (or any real closed field) with an abelian grading group. We also discuss group gradings on fields.

It is important for applications of Homological Algebra in Representation Theory to have control over the behavior of (minimal) projective resolutions under various functors. In this paper, we describe three broad families of functors that preserve such resolutions. We will use these results in our work on Representation Theory of Schur algebras.

For ungraded quotients of an arbitrary ℤ-graded ring, we define the general PBW property, that covers the classical PBW property and the N-type PBW property studied via the N-Koszulity by several authors (see [2–4]). In view of the noncommutative Gröbner basis theory, we conclude that every ungraded quotient of a path algebra (or a free algebra) has the general PBW property. We remark that an earlier result of Golod [5] concerning Gröbner bases can be used to give a homological characterization of the general PBW property in terms of Shafarevich complex. Examples of application are given.

Let k be a field of arbitrary characteristic and let k[x₁,…,x_n] be the polynomial k-algebra. We describe, in some special cases, polynomial graded subalgebras of k[x₁,…,x_n] containing , where p is a prime number.

Let K〈X〉=K〈X₁,…,X_n〉 be the free K-algebra on X={X₁,…,X_n} over a field K, which is equipped with a weight ℕ-gradation (i.e., each X_i is assigned a positive degree), and let be a finite homogeneous Gröbner basis for the ideal of K〈X〉 with respect to some monomial ordering ≺ on K〈X〉. It is shown that if the monomial algebra is semiprime, where is the set of leading monomials of with respect to ≺, then the ℕ-graded algebra A=K〈X〉 /I is semiprimitive in the sense of Jacobson. In the case that is a finite nonhomogeneous Gröbner basis with respect to a graded monomial ordering ≺_gr, and the ℕ-filtration FA of the algebra A=K〈X〉 /I induced by the ℕ-grading filtration FK〈X〉 of K〈X〉 is considered, if the monomial algebra is semiprime, then it is shown that the associated ℕ-graded algebra G(A) and the Rees algebra à of A determined by FA are all semiprimitive.

We consider the symmetric algebra of a class of monomial ideals generated by s-sequences. For these ideals with linear syzygies, we determine their Jacobian dual modules and study their duality properties.

Fix a prime N, and consider the action of the Hecke operator T_N on the space of modular forms of full level and varying weight κ. The coefficients of the matrix of T_N with respect to the basis {E₄ⁱ E₆^j | 4i + 6j = κ} for can be combined for varying κ into a generating function F_N. We show that this generating function is a rational function for all N, and present a systematic method for computing F_N. We carry out the computations for N = 2, 3, 5, and indicate and discuss generalizations to spaces of modular forms of arbitrary level.