Let $A$ be an associative unital algebra, $B_k$ its successive quotients of lower central series and $N_k$ the successive quotients of ideals generated by lower central series. The geometric and algebraic aspects of $B_k$ and $N_k$ have been of great interest since the pioneering work of \cite{feigin2007}. In this paper, we will concentrate on the case where $A$ is a noncommutative polynomial algebra over $\mathbb{Z}$ modulo a single homogeneous relation. Both the torsion part and the free part of $B_k$ and $N_k$ are explored. Many examples are demonstrated in detail, and several general theorems are proved. Finally we end up with an appendix about the torsion subgroups of $N_k(A_n(\mathbb{Z}))$ and some open problems.
In this paper we consider some classical varieties of linear algebras over the field which has characteristic 0. For every considered variety we take a category of the finite generated free algebras of this variety. And for every this category we calculate the quotient group of the group of the all automorphisms of this category over the subgroup of the all inner automorphisms. This quotient group measures difference between the geometric equivalence and automorphic equivalence of algebras from this variety.
We continue the study of the lower central series L_i(A) and its successive quotients B_i(A) of a noncommutative associative algebra A, defined by L_1(A)=A, L_{i+1}(A)=[A,L_i(A)], and B_i(A)=L_i(A)/L_{i+1}(A). We describe B_{2}(A) for A a quotient of the free algebra on two or three generators by the two-sided ideal generated by a generic homogeneous element. We prove that it is isomorphic to a certain quotient of Kaehler differentials on the non-smooth variety associated to the abelianization of A.
For the algebra $\mI_1= K<x, \frac{d}{dx}, \int>$ of polynomial integro-differential operators over a field $K$ of characteristic zero, a classification of simple modules is given. It is proved that $\mI_1$ is a left and right coherent algebra. The {\em Strong Compact-Fredholm Alternative} is proved for $\mI_1$. The endomorphism algebra of each simple $\mI_1$-module is a {\em finite dimensional} skew field. In contrast to the first Weyl algebra, the centralizer of a non-scalar integro-differential operator can be a noncommutative, non-Noetherian, non-finitely generated algebra which is not a d…
In 1988, S. V. Okhitin proved that for any field k of characteristic zero, the T-space CP(M_2(k)) is finitely based, and he raised the question as to whether CP(A) is finitely based for every (unitary) associative algebra A with nonzero T-ideal of identities that is properly contained CP(A). V. V. Shchigolev (2001) showed that for any field k of characteristic zero, every T-space of the infinite dimensional free associative k algebra is finitely based, and it follows from this that every T-space of the infinite dimensional free unitary k algebra is also finitely based. This more than answers O…
Let $K$ be a field, let $σ$ be an automorphism of $K$, and let $δ$ be a derivation of $K$. We show that if $D$ is one of $K(x;σ)$ or $K(x;δ)$, then $D$ either contains a free algebra over its center on two generators, or every finitely generated subalgebra of $D$ satisfies a polynomial identity. As a corollary, we are able to show that the quotient division ring of any iterated Ore extension of an affine domain satisfying a polynomial identity either again satisfies a polynomial identity or it contains a free algebra over its center on two variables.
Let $L_i(R)$ denote the $i^{\text{th}}$ term of the lower central series of an associative algebra $R$, and let $B_i(R)=L_i(R)/L_{i+1}(R)$. We show that $B_2(\mathbb{C}<x, y>/ P)\cong Ω^2((\mathbb{C}<x, y>/ P)_{ab})$, for all homogeneous or quasihomogeneous $P$ with square-free abelianization. Our approach generalizes that of Balagovic and Balasubramanian in 2010, which in turn developed from that of Dobrovolska, Kim, and Ma in 2007. We also use ideas of Feign and Shoikhet in 2006, who initiated the study of the groups $B_i(R)$.
In this paper it is shown how the Heisenberg group of order 27 can be used to construct quotients of degenerate Sklyanin algebras. These quotients have properties similar to the classical Sklyanin case in the sense that they have the same Hilbert series, the same character series and a central element of degree 3. Regarding the central element of a 3-dimensional Sklyanin algebra, a better way to view this using Heisenberg-invariants is shown.
In this paper we discuss finite presentability of the universal central extensions of Lie algebras ${\mathfrak{sl}_n(R)}$, where $n\geq 3$ and $R$ is a unital associative $k$-algebra. We show that a universal central extension is finitely presented if and only if the algebra $R$ is finitely presented.
Let $D$ be a division algebra such that $D\t D^o$ is a Noetherian algebra, then any division subalgebra of $D$ is a {\em finitely generated} division algebra. Let $\D $ be a finite set of commuting derivations or automorphisms of the division algebra $D$, then the group $\Ev (\D)$ of common eigenvalues (i.e. {\em weights}) is a {\em finitely generated abelian} group. Typical examples of $D$ are the quotient division algebra ${\rm Frac} (\CD (X))$ of the ring of differential operators $\CD (X)$ on a smooth irreducible affine variety $X$ over a field $K$ of characteristic zero, and the quotient …
For central simple finitely generated algebras of finite Gelfand-Kirillov dimension and for their division algebras upper bounds are obtained for the transcendence degree of their commutative subalgebras and subfields respectively. In the formulae for the upper bounds are used only two dimensions of the algebras: the Gelfand-Kirillov and the filter dimension. In the case of the ring of differential operators on a smooth irreducible affine algebraic variety the upper bounds are the exact upper bounds. A similar upper bound is given for the transcendence degree of isotropic subalgebras of strong…
In this paper we study adjunction relations between some natural functors on the representation category of the category of finite linearly ordered sets and order-preserving injections. We also prove that the Nakayama functor induces an equivalence from the Serre quotient of the category of finitely generated modules by the category of finitely generated torsion modules to the category of finite dimensional modules.
Feigin and Shoikhet conjectured in math/0610410 that successive quotients $B_m(A_n)$ of the lower central series filtration of a free associative algebra $A_n$ have polynomial growth. In this paper we give a proof of this conjecture, using the structure of $W_n$-representation on $B_m(A_n)$ which was defined in math/0610410 . We also prove that the number of squares in a Young diagram $D$ corresponding to an irreducible $W_n$-module in the Jordan-Holder series of $B_m(A_n)$ is bounded above by the integer $(m-1)^2+2[(n-2)/2](m-1)$. This bound combined with MAGMA computations by Rains in math/0…
We prove that the space of coinvariants of functions on an affine variety by a Lie algebra of vector fields whose flow generates finitely many leaves is finite-dimensional. Cases of the theorem include Poisson (or more generally Jacobi) varieties with finitely many symplectic leaves under Hamiltonian flow, complete intersections in Calabi-Yau varieties with isolated singularities under the flow of incompressible vector fields, quotients of Calabi-Yau varieties by finite volume-preserving groups under the incompressible vector fields, and arbitrary varieties with isolated singularities under th…
We continue the study of the lower central series and its associated graded components for a free associative algebra with n generators, as initiated by B. Feigin and B. Shoikhet. We establish a linear bound on the degree of tensor field modules appearing in the Jordan-Hoelder series of each graded component, which is conjecturally tight. We also bound the leading coefficient of the Hilbert polynomial of each graded component. As applications, we confirm conjectures of P. Etingof and B. Shoikhet concerning the structure of the third graded component.
In the category of finitely generated modules over an artinian ring, we classify all the abelian exact subcategories closed under predecessors or, equivalently, all the split torsion pairs with torsion-free class closed under quotients.
This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. Section 2 provides some preliminaries on quasi-projective modules over commutative rings. Section 3 investigates the correlation with well-known Prufer conditions; namely, we prove that this class of rings stands strictly between the two classes of arithmetical rings and Gaussian rings. Thereby, we generalize Osofsky's theorem on the weak global dimension of arithmetical rings and partially resolve Bazzoni-Glaz's related conjecture on Gaussian rings. We also e…
We study conditions under which subdirect products of various types of algebraic structures are finitely generated or finitely presented. In the case of two factors, we prove general results for arbitrary congruence permutable varieties, which generalise previously known results for groups, and which apply to modules, rings, $K$-algebras and loops. For instance, if $C$ is a fiber product of $A$ and $B$ over a common quotient $D$, and if $A$, $B$ and $D$ are finitely presented, then $C$ is finitely generated. For subdirect products of more than two factors we establish a general connection with…
We prove that, for any positive integer $c$, the quotient group $γ_{c}(M_{3})/γ_{c+1}(M_{3})$ of the lower central series of the McCool group $M_{3}$ is isomorphic to two copies of the quotient group $γ_{c}(F_{3})/γ_{c+1}(F_{3})$ of the lower central series of a free group $F_{3}$ of rank $3$ as $\mathbb{Z}$-modules. Furthermore, we give a necessary and sufficient condition whether the associated graded Lie algebra ${\rm gr}(M_{3})$ of $M_3$ is naturally embedded into the Johnson Lie algebra ${\cal L}({\rm IA}(F_{3}))$ of the IA-automorphisms of $F_{3}$.
Let $d\ge1$ be an integer, $W_d$ and $\mathcal{K}_d$ be the Witt algebra and the weyl algebra over the Laurent polynomial algebra $A_d=\mathbb{C} [x_1^{\pm1}, x_2^{\pm1}, ..., x_d^{\pm1}]$, respectively. For any $\mathfrak{gl}_d$-module $M$ and any admissible module $P$ over the extended Witt algebra $\widetilde W_d$, we define a $W_d$-module structure on the tensor product $P\otimes M$. We prove in this paper that any simple $W_d$-module that is finitely generated over the cartan subalgebra is a quotient module of the $W_d$-module $P \otimes M$ for a finite dimensional simple $\mathfrak{gl}_d…
