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 this paper we examine how the notion of algebra of quotients for Lie algebras ties up with the corresponding well-known concept in the associative case. Specifically, we completely characterize when a Lie algebra $Q$ is an algebra of quotients of a Lie algebra $L$ in terms of the associative algebras generated by the adjoint operators of $L$ and $Q$ respectively. In a converse direction, we also provide with new examples of algebras of quotients of Lie algebras and these come from associative algebras of quotients. In the course of our analysis, we make use of the notions of density and mul…
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 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.
We study the quotient Q_i(A) of a free algebra A by the ideal M_i(A) generated by relation that the i-th commutator of any elements is zero. In particular, we completely describe such quotient for i=4 (for i<=3 this was done previously by Feigin and Shoikhet). We also study properties of the ideals M_i(A), e.g. when M_i(A)M_j(A) is contained in M_{i+j-1}(A) (by a result of Gupta and Levin, it is always contained in M_{i+j-2}(A)).
We give the algebraic classification of alternative, left alternative, Jordan, bicommutative, left commutative, assosymmetric, Novikov and left symmetric central extensions of null-filiform associative algebras.
We address a Jordan version of Johnson theorem on (associative) algebras of quotients, namely whether a strongly nonsingular (the Jordan version of nonsingularity) has a von Neumann regular algebra of quotients. Although the answer is negative in general, we prove that the result holds in the important case of algebras satisfying a polynomial identity.
Let $\mathfrak{a},\mathfrak{b},\mathfrak{e}$ be algebras over a field $k$. Then $\mathfrak{e}$ is an extension of $\mathfrak{a}$ by $\mathfrak{b}$ if $\mathfrak{a}$ is an ideal of $\mathfrak{e}$ and $\mathfrak{b}$ is isomorphic to the quotient algebra $\mathfrak{e}/\mathfrak{a}$. In this paper, by using Gröbner-Shirshov bases theory for associative (resp. Lie) algebras, we give complete characterizations of associative (resp. Lie) algebra extensions of $\mathfrak{a}$ by $\mathfrak{b}$, where $\mathfrak{b}$ is presented by generators and relations.
The motivation for this paper has been to study the relation between the zero component of the maximal graded algebra of quotients and the maximal graded algebra of quotients of the zero component, both in the Lie case and when considering Martindale algebras of quotients in the associative setting. We apply our results to prove that the finitary complex Lie algebras are (graded) strongly nondegenerate and compute their maximal algebras of quotients.
For n even, we prove Pozhidaev's conjecture on the existence of associative enveloping algebras for simple n-Lie algebras. More generally, for n even and any (n+1)-dimensional n-Lie algebra L, we construct a universal associative enveloping algebra U(L) and show that the natural map from L to U(L) is injective. We use noncommutative Grobner bases to present U(L) as a quotient of the free associative algebra on a basis of L and to obtain a monomial basis of U(L). In the last section, we provide computational evidence that the construction of U(L) is much more difficult for n odd.
We continue the study of the lower central series of a free associative algebra, initiated by B. Feigin and B. Shoikhet (arXiv:math/0610410). We generalize via Schur functors the constructions of the lower central series to any symmetric tensor category; specifically we compute the modified first quotient \bar{B}_1, and second and third quotients B_2, and B_3 of the series for a free algebra T(V) in any symmetric tensor category, generalizing the main results of (arXiv:math/0610410) and (arXiv:0902.4899). In the case A_{m|n}:=T(\CC^{m|n}), we use these results to compute the explicit Hilbert s…
This paper concerns the associative lower central series ideals $M_i$ of the free algebra $A_n$ on $n$ generators. Namely, we study the successive quotients $N_i=M_i/M_{i+1}$, which admit an action of the Lie algebra $W_n$ of vector fields on $\Bbb C^n$. We bound the degree $|λ|$ of tensor field modules $F_λ$ appearing in the Jordan-Hölder series of each $N_i$, confirming a recent conjecture of Arbesfeld and Jordan. As an application, we compute these decompositions for small $n$ and $i$.
Consider the free algebra A_n generated over Q by n generators x_1, ..., x_n. Interesting objects attached to A = A_n are members of its lower central series, L_i = L_i(A), defined inductively by L_1 = A, L_{i+1} = [A,L_{i}], and their associated graded components B_i = B_i(A) defined as B_i=L_i/L_{i+1}. These quotients B_i, for i at least 2, as well as the reduced quotient \bar{B}_1=A/(L_2+A L_3), exhibit a rich geometric structure, as shown by Feigin and Shoikhet and later authors, (Dobrovolska-Kim-Ma,Dobrovolska-Etingof,Arbesfeld-Jordan,Bapat-Jordan). We study the same problem over the inte…
We give an accessible introduction into the theory of lower central series of associative algebras, exhibiting the interplay between algebra, geometry and representation theory that is characteristic for this subject, and to discuss some open questions. In particular, we provide shorter and clearer proofs of the main results of this theory. We also discuss some new theoretical and computational results and conjectures on the lower central series of the free algebra in two generators modulo a generic homogeneous relation.
Lie superautomorphisms of prime associative superalgebras are considered. A definitive result is obtained for central simple superalgebras: their Lie superautomorphisms are of standard forms, except when the dimension of the superalgebra in question is 2 or 4.
In this paper, we introduce the notion of algebras of quotients of Hom-Lie algebras and investigate some properties which can be lifted from a Hom-Lie algebra to its algebra of quotients. We also give some necessary and sufficient conditions for Hom-Lie algebras having algebras of quotients. We also examine the relationship between a Hom-Lie algebra and the associative algebra generated by inner derivations of the corresponding Hom-Lie algebra of quotients. Moreover, we introduce the notion of dense extensions and get a proposition about Hom-Lie algebras of quotients via dense extensions.
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…
In this paper, the definitions of algebras of quotients and Martandale-like qoutients of Leibniz algebras are introduced and the interactions between the two quotients are determined. Firstly, some important properties which not only hold for a Leibniz algebras but also can been lifted to its algebras of quotients are investigated. Secondly, for any semiprime Leibniz algebra, its maximal algebra of quotients is constucted and a Passman-like characterization of the maximal algebra is described. Thirdly, the relationship between a Leibniz algebra and the associative algebra which is generated by…
We describe families of nonassociative finite unital rings that occur as quotients of natural nonassociative orders in generalized nonassociative cyclic division algebras over number fields. These natural orders have already been used to systematically construct fully diverse fast-decodable space-time block codes. We show how the quotients of natural orders can be employed for coset coding. Previous results by Oggier and Sethuraman involving quotients of orders in associative cyclic division algebras are obtained as special cases.
We classify nilpotent associative algebras of dimensions up to 4 over any field. This is done by constructing the nilpotent associative algebras as central extensions of algebras of smaller dimension, analogous to methods known for nilpotent Lie algebras.
Let $F$ be a field and let $F \langle X \rangle$ be the free unital associative $F$-algebra on the free generating set $X = \{ x_1, x_2, \dots \}$. A subalgebra (a vector subspace) $V$ in $F \langle X \rangle$ is called a $T$-subalgebra (a $T$-subspace) if $ϕ(V) \subseteq V$ for all endomorphisms $ϕ$ of $F \langle X \rangle$. For an algebra $G$, its central polynomials form a $T$-subalgebra $C(G)$ in $F \langle X \rangle$. Over a field of characteristic $p > 2$ there are algebras $G$ whose algebras of all central polynomials $C (G)$ are not finitely generated as $T$-subspaces in $F \langle X \…
The lower central series invariants M_k of an associative algebra A are the two-sided ideals generated by k-fold iterated commutators; the M_k provide a filtration of A. We study the relationship between the geometry of X = Spec A_ab and the associated graded components N_k of this filtration. We show that the N_k form coherent sheaves on a certain nilpotent thickening of X, and that Zariski localization on X coincides with noncommutative localization of A. Under certain freeness assumptions on A, we give an alternative construction of N_k purely in terms of the geometry of X (and in particula…
Let $C(B,N)$ be the free associative conformal algebra generated by a set $B$ with a bounded locality $N$. Let $S$ be a subset of $C(B,N)$. A Composition-Diamond lemma for associative conformal algebras is firstly established by Bokut, Fong, and Ke in 2004 \cite{BFK04} which claims that if (i) $S$ is a Gröbner-Shirshov basis in $C(B,N)$, then (ii) the set of $S$-irreducible words is a linear basis of the quotient conformal algebra $C(B,N|S)$, but not conversely. In this paper, by introducing some new definitions of normal $S$-words, compositions and compositions to be trivial, we give a new Co…
A central extension is a regular epimorphism in a Barr exact category $\mathscr{C}$ satisfying suitable conditions involving a given Birkhoff subcategory of $\mathscr{C}$ (joint work with G. M. Kelly, 1994). In this paper we take $\mathscr{C}$ to be the category of (not-necessarily-unital) algebras over a (unital) commutative ring and consider central extensions with respect to the category of commutative algebras. We propose a new approach that avoids the intermediate notion of central extension due to A. Fröhlich in showing that $α:A\to B$ is a central extension if and only if $aa'=a'a$ for …
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 $B_i(R)$.
The Gelfand-Kirillov dimension measures the asymptotic growth rate of algebras. For every associative dialgebra $\mathcal{D}$, the quotient $\mathcal{A}_\mathcal{D}:=\mathcal{D}/\mathsf{Id}(S)$, where $\mathsf{Id}(S)$ is the ideal of $\mathcal{D}$ generated by the set $S:=\{x \vdash y-x\dashv y \mid x,y\in \mathcal{D}\}$, is called the associative algebra associated to $\mathcal{D}$. Here we show that the Gelfand--Kirillov dimension of $\mathcal{D}$ is bounded above by twice the Gelfand--Kirillov dimension of $\mathcal{A}_\mathcal{D}$. Moreover, we prove that no associative dialgebra has Gelfa…
We prove that the only proper right ideal of the universal enveloping algebra of a finite-dimensional central simple Lie triple system over a field of characteristic zero is its augmentation ideal.
The aim of this paper is to introduce and study BiHom-associative dialgebras. We give various constructions and study its connections with BiHom-Poisson dialgebras and BiHom-Leibniz algebras. Next we discuss the central extensions of BiHom-diassociative and we describe the classification of n-dimensional BiHom-diassociative algebras for n less or equal to 4. Finally, we discuss derivations of BiHom-associative dialgebras.
Interrelations between discrete deformations of the structure constants for associative algebras and discrete integrable systems are reviewed. A theory of deformations for associative algebras is presented. Closed left ideal generated by the elements representing the multiplication table plays a central role in this theory. Deformations of the structure constants are generated by the Deformation Driving Algebra and governed by the central system of equations. It is demonstrated that many discrete equations like discrete Boussinesq equation, discrete WDVV equation, discrete Schwarzian KP and BK…
Let $A$ be an $n$-dimensional algebra over a field $k$ and $a(A)$ its quantum symmetry semigroup. We prove that the automorphisms group ${\rm Aut}_{\rm Alg} (A)$ of $A$ is isomorphic to the group $U \bigl( G(a (A)^{\rm o} ) \bigl)$ of all invertible group-like elements of the finite dual $a (A)^{\rm o}$. For a group $G$, all $G$-gradings on $A$ are explicitly described and classified: the set of isomorphisms classes of all $G$-gradings on $A$ is in bijection with the quotient set $ {\rm Hom}_{\rm BiAlg} \, \bigl( a (A) , \, k[G] \bigl)/\approx$ of all bialgebra maps $a (A) \, \to k[G]$, via th…
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 theor…
Algebraic scheme for constructing deformations of structure constants for associative algebras generated by a deformation driving algebras (DDAs) is discussed. An ideal of left divisors of zero plays a central role in this construction. Deformations of associative three-dimensional algebras with the DDA being a three-dimensional Lie algebra and their connection with integrable systems are studied.
We study the universal enveloping associative conformal algebra for the central extension of a current Lie conformal algebra at the locality level $N=3$. A standard basis of defining relations for this algebra is explicitly calculated. As a corollary, we find a linear basis of the free commutative conformal algebra relative to the locality $N=3$ on the generators.
Let G be a connected reductive algebraic group and let G'=[G,G] be its derived subgroup. Let (G,V) be a multiplicity free representation with a one dimensional quotient (see definition below). We prove that the algebra D(V)^{G'} of G'-invariant differential operators with polynomial coefficients on V, is a quotient of a so-called Smith algebra over its center. Over C this class of algebras was introduced by S.P. Smith as a class of algebras similar to the enveloping algebra U(sl(2)) of sl(2). Our result generalizes the case of the Weil representation, where the associative algebra generated by…
We develop an elementary method for proving the PBW theorem for associative algebras with an ascending filtration. The idea is roughly the following. At first, we deduce a proof of the PBW property for the {\it ascending} filtration (with the filtered degree equal to the total degree in $x_i$'s) to a suitable PBW-like property for the {\it descending} filtration (with the filtered degree equal to the power of a polynomial parameter $\hbar$, introduced to the problem). This PBW property for the descending filtration guarantees the genuine PBW property for the ascending filtration, for almost al…
A non-associative algebra over a field $\mathbb{K}$ is a $\mathbb{K}$-vector space $A$ equipped with a bilinear operation \[ {A\times A\to A\colon\; (x,y)\mapsto x\cdot y=xy}. \] The collection of all non-associative algebras over $\mathbb{K}$, together with the product-preserving linear maps between them, forms a variety of algebras: the category $\mathsf{Alg}_\mathbb{K}$. The multiplication need not satisfy any additional properties, such as associativity or the existence of a unit. Familiar categories such as the varieties of associative algebras, Lie algebras, etc. may be found as subvarie…
Using actions of free monoids and free associative algebras, we establish some Schreier-type formulas involving the ranks of actions and the ranks of subactions in free actions or Grassmann-type relations for the ranks of intersections of subactions of free actions. The coset action of the free group is used to establish the generalization of the Schreier formula to the case of subgroups of infinite index. We also study and apply large modules over free associative algebras in the spirit of the paper Olshanskii, A. Yu.; Osin, D.V., Large groups and their periodic quotients, Proc. Amer. Math. S…
Let Z<X> be the free unitary associative ring freely generated by an infinite countable set X = {x_1, x_2,...}. Define a left-normed commutator [x_1, x_2, ..., x_n] by [a,b] = ab - ba, [a,b,c] = [[a,b],c]. For n \ge 2, let T^(n) be the ideal in Z<X> generated by all commutators [a_1,a_2,..., a_n] (a_i \in Z<X>). It can be easily seen that the additive group of the quotient ring Z<X> /T^(2) is a free abelian group. Recently Bhupatiraju, Etingof, Jordan, Kuszmaul and Li have noted that the additive group of Z<X> /T^(3) is free abelian as well. In the present note we show that this is not the cas…
Discrete and q-difference deformations of the structure constants for a class of associative noncommutative algebras are studied. It is shown that these deformations are governed by a central system of discrete or q-difference equations which in particular cases represent discrete and q-difference versions of the oriented associativity equation. It is demonstrated also that the celebrated Hirota-Miwa bilinear equation for the AKP and BKP hierarchies describes discrete deformations of certain finite-dimensional algebras.
Deformations of the structure constants for a class of associative noncommutative algebras generated by Deformation Driving Algebras (DDA's) are defined and studied. These deformations are governed by the Central System (CS). Such a CS is studied for the case of DDA being the algebra of shifts. Concrete examples of deformations for the three-dimensional algebra governed by discrete and mixed continuous-discrete Boussinesq (BSQ) and WDVV equations are presented. It is shown that the theory of the Darboux transformations, at least for the BSQ case, is completely incorporated into the proposed sc…
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 study fundamental model-theoretic questions for free associative algebras, namely, first-order classification, decidability of the first-order theory, and definability of the set of free bases. We show that two free associative algebras of finite rank over fields are elementarily equivalent if and only if their ranks are the same and the fields are equivalent in the weak second order logic. In particular, two free associative algebras of finite rank over the same field are elementarily equivalent if and only if they are isomorphic. We prove that if an arbitrary ring $B$ with a…
We use computer algebra to study the 512-dimensional associative algebra Q B_3, the rational monoid algebra of 3 x 3 Boolean matrices. We obtain a basis for the radical in bijection with the 42 non-regular elements of B_3. The center of the 470-dimensional semisimple quotient has dimension 14; we use a splitting algorithm to find a basis of orthogonal primitive idempotents. We show that the semisimple quotient is the direct sum of simple two-sided ideals isomorphic to d x d rational matrix algebras for d = 1, 1, 1, 2, 3, 3, 3, 3, 6, 6, 7, 9, 9, 12. We construct the irreducible representations …
Axial algebras are a class of non-associative algebra with a strong link to finite (especially simple) groups which have recently received much attention. Of primary interest are the axial algebras of Monster type $(α, β)$, of which the Griess algebra (with the Monster as its automorphism group) is an important motivating example. In this paper, we complete the classification of the symmetric $2$-generated primitive axial algebras of Monster type $(α, β)$. By previous work of Yabe, and Franchi and Mainardis, any such algebra is either explicitly known, or is a quotient of the infinite-dimensio…
The word problem is an old and central problem in (computational) group theory. It is well-known that the word problem is undecidable in general, but decidable for specific types of presentations. Consistent polycyclic presentations are an important class of group presentations with solvable word problem. These presentations play a fundamental role in the algorithmic theory of polycyclic groups. Problems analogous to the word problem arise when computing with other algebraic structures. Various aspects of this topic are considered in the literature. The aim of this paper is to provide a genera…
Quantum deformations of the structure constants for a class of associative noncommutative algebras are studied. It is shown that these deformations are governed by the quantum central systems which has a geometrical meaning of vanishing Riemann curvature tensor for Christoffel symbols identified with the structure constants. A subclass of isoassociative quantum deformations is described by the oriented associativity equation and, in particular, by the WDVV equation. It is demonstrated that a wider class of weakly (non)associative quantum deformations is connected with the integrable soliton eq…
We study a structure of the group of unitriangular automorphisms of a free associative algebra and a polynomial algebra and prove that this group is a semi direct product of abelian groups. Using this decomposition we describe a structure of the lower central series and the series of derived subgroups for the group of unitriangular automorphisms and prove that every element from the derived subgroup is a commutator. In addition we prove that the group of unitriangular automorphisms of a free associative algebra of rang more than 2 is not linear and describe some two-generated subgroups from th…
We show that every central simple algebra A over a field k is Brauer equivalent to a quotient of a finite dimensional Hopf algebra over the same field (that is- A is Hopf Schur). If the characteristic of the field is zero, or if the algebra has a Galois splitting field of degree prime to the characteristic of k, we can take this Hopf algebra to be semisimple. We also show that if F is any finite extension of k, then F is a quotient of a finite dimensional Hopf algebra over k. We use it in order to show why the algebric closeness assumption is necessary in a weak form of Kaplansky's tenth conje…
We study relatively free associative algebras $F^{(n)}_r$ of ranks $r=2,3$ with the identity $[x_1,\dots, x_n]=0$ of Lie nilpotency of step $n\geqslant 3$ over a field $K$ of characteristic $\neq 2,3$. First we prove a Theorem on the inclusion $ T^{(m)}T^{(n)}\subseteq T^{(m+n-1)}$ for an associative algebra $A$ of rank $3$, where $T^{(n)}=T^{(n)}(A)$ is a T-ideal of $A$ generated by the commutator $[x_1,\dots, x_n]$; the restriction on rank is essential. Further, we describe 3-variable identities of the algebra $F^{(n)}$. In particular, the obtained description implies that $Z(F^{(n)}_r)=(T^{…
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.
