Free Associative Algebra Research Articles

Hamiltonians for the quantised Volterra hierarchy

This paper builds upon our recent work, published in Carpentier et al (2022 Lett. Math. Phys. 112 94), where we established that the integrable Volterra lattice on a free associative algebra and the whole hierarchy of its symmetries admit a quantisation dependent on a parameter ω. We also uncovered an intriguing aspect: all odd-degree symmetries of the hierarchy admit an alternative, non-deformation quantisation, resulting in a non-commutative algebra for any choice of the quantisation parameter ω. In this study, we demonstrate that each equation within the quantum Volterra hierarchy can be expressed in the Heisenberg form. We provide explicit expressions for all quantum Hamiltonians and establish their commutativity. In the classical limit, these quantum Hamiltonians yield explicit expressions for the classical ones of the commutative Volterra hierarchy. Furthermore, we present Heisenberg equations and their Hamiltonians in the case of non-deformation quantisation. Finally, we discuss commuting first integrals, central elements of the quantum algebra, and the integrability problem for periodic reductions of the Volterra lattice in the context of both quantisations.

Homogeneous approximations of nonlinear control systems with output and weak algebraic equivalence

In the paper, we consider nonlinear control systems that are linear with respect to controls with output; vector fields defining the system and the output are supposed to be real analytic. Following the algebraic approach, we consider series $S$ of iterated integrals corresponding to such systems. Iterated integrals form a free associative algebra, and all our constructions use its properties. First, we consider the set of all (formal) functions of such series $f(S)$ and define the set $N_S$ of terms of minimal order for all such functions. We introduce the definition of the maximal graded Lie generated left ideal ${\mathcal J}_S^{\rm max}$ which is orthogonal to the set $N_S$. We describe the relation between this maximal left ideal and the left ideal ${\mathcal J}_S$ generated by the core Lie subalgebra of the system which realizes the series. Namely, we show that ${\mathcal J}_S\subset {\mathcal J}_S^{\rm max}$. In particular, this implies that the graded Lie subalgebra that generates the left ideal ${\mathcal J}_S^{\rm max}$ has a finite codimension. Also, we give the algorithm which reduces the series $S$ to the triangular form and propose the definition of the homogeneous approximation for the series $S$. Namely, homogeneous approximation is a homogeneous series with components that are terms of minimal order in each component of this triangular form. We prove that the set $N_S$ coincides with the set of all shuffle polynomials of components of a homogeneous approximation. Unlike the case when the output is identical, the homogeneous approximation is not completely defined by the ideal ${\mathcal J}_S^{\rm max}$. In order to describe this property, we introduce two different concepts of equivalence of series: algebraic equivalence (when two series have the same homogeneous approximation) and weak algebraic equivalence (when two series have the same maximal left ideal and therefore have the same minimal realizing system). We prove that if two series are algebraically equivalent, then they are weakly algebraically equivalent. The examples show that in general the converse is not true.

The Relation between constants in generic and degenerate subspaces of free unital associative complex algebra

From the study of the constants in the generic and the degenerate weight subspaces of the free unitary associative complex algebra B, it follows that the constants in the degenerate weight subspaces of the algebra B can be constructed from the corresponding constants in the generic case by a certain specialization procedure. Here we consider that each constant in each generic weight subspace of the algebra B can be expressed by certain iterated q-commutators.

Symmetric Polynomials in Free Associative Algebras—II

Let K⟨Xd⟩ be the free associative algebra of rank d≥2 over a field, K. In 1936, Wolf proved that the algebra of symmetric polynomials K⟨Xd⟩Sym(d) is infinitely generated. In 1984 Koryukin equipped the homogeneous component of degree n of K⟨Xd⟩ with the additional action of Sym(n) by permuting the positions of the variables. He proved finite generation with respect to this additional action for the algebra of invariants K⟨Xd⟩G of every reductive group, G. In the first part of the present paper, we established that, over a field of characteristic 0 or of characteristic p>d, the algebra K⟨Xd⟩Sym(d) with the action of Koryukin is generated by (noncommutative version of) the elementary symmetric polynomials. Now we prove that if the field, K, is of positive characteristic at most d then the algebra K⟨Xd⟩Sym(d), taking into account that Koryukin’s action is infinitely generated, describe a minimal generating set.

Free limits of free algebras

Consider a diagram ⋯→F3→F2→F1 of algebraic systems, where Fn denotes the free object on n generators and the connecting maps send the extra generator to some distinguished trivial element. We prove that (a) if the Fi are free associative algebras over a fixed field then the limit in the category of graded algebras is again free on a set of homogeneous generators; (b) on the other hand, the limit in the category of associative (ungraded) algebras is a free formal power series algebra on a set of homogeneous elements, and (c) if the Fi are free Lie algebras then the limit in the category of graded Lie algebras is again free.

Demi-shuffle duals of Magnus polynomials in a free associative algebra

Demi-shuffle duals of Magnus polynomials in a free associative algebra

Using a q-shuffle algebra to describe the basic module V(Λ_0) for the quantized enveloping algebra Uq(sl^2)

We consider the quantized enveloping algebra Uq(sl^2). and its basic module V(Λ0). This module is infinite-dimensional, irreducible, integrable, and highest-weight. We describe V(Λ0) using a q-shuffle algebra in the following way. Start with the free associative algebra V on two generators x, y. The standard basis for V consists of the words in x, y. In 1995 M. Rosso introduced an associative algebra structure on V, called a q-shuffle algebra. For u, v ∈ {x, y} their q-shuffle product is u ⋆ v = uv + q(u,v)vu, where (u,v) = 2 (resp. (u,v) = − 2) if u = v (resp. u ≠ v). Let U denote the subalgebra of the q-shuffle algebra V that is generated by x, y. Rosso showed that the algebra U is isomorphic to the positive part of Uq(sl^2). In our first main result, we turn U into a Uq(sl^2).-module. Let U denote the Uq(sl^2).-submodule of U generated by the empty word. In our second main result, we show that the Uq(sl^2).-modules U and V(Λ0) are isomorphic. Let V denote the subspace of V spanned by the words that do not begin with y or xx. In our third main result, we show that U = U ∩ V.

Hilbert series of T-spaces

We show that the Hilbert series of a T-space in a free associative algebra A is either a rational function, or differs from the Hilbert series of the commutator [A,A] by a rational function.

On the GL(n)-module structure of Lie nilpotent associative relatively free algebras

Let K〈X〉 denote the free associative algebra generated by a set X={x1,…,xn} over a field K of characteristic 0. Let Ip, for p≥2, denote the two-sided ideal in K〈X〉 generated by all commutators of the form [u1,…,up], where u1,…,up∈K〈X〉. We discuss the GL(n,K)-module structure of the quotient K〈X〉/Ip+1 for all p≥1 under the standard diagonal action. We give a bound on the values of partitions λ such that the irreducible GL(n,K)-module Vλ appears in the decomposition of K〈X〉/Ip+1 as a GL(n,K)-module. As an application, we take K=C and we consider the algebra of invariants (C〈X〉/Ip+1)G for G=SL(n,C), O(n,C), SO(n,C), or Sp(2s,C) (for n=2s). By a theorem of Domokos and Drensky, (C〈X〉/Ip+1)G is finitely generated. We give an upper bound on the degree of generators of (C〈X〉/Ip+1)G in a minimal generating set. In a similar way, we consider also the algebra of invariants (C〈X〉/Ip+1)G, where G=UT(n,C), and give an upper bound on the degree of generators in a minimal generating set. These results provide useful information about the invariants in C〈X〉G from the point of view of Classical Invariant Theory. In particular, for all G as above we give a criterion when a G-invariant of C〈X〉 belongs to Ip.

Centralizers in Free Associative Algebras and Generic Matrices

This paper is concerned with the completion of the proof of the Bergman centralizer theorem using generic matrices based on our previous quantization proof (Kanel-Belov et al. in Commun Algebra 46:2123–2129, 2018). Additionally, we establish that the algebra of generic matrices with characteristic coefficients is integrally closed.

Homogeneous approximation for minimal realizations of series of iterated integrals

In the paper, realizable series of iterated integrals with scalar coefficients are considered and an algebraic approach to the homogeneous approximation problem for nonlinear control systems with output is developed. In the first section we recall the concept of the homogeneous approximation of a nonlinear control system which is linear w.r.t.\ the control and the concept of the series of iterated integrals. In the second section the statement of the realizability problem is given, a criterion for realizability and a method for constructing a minimal realization of the series are recalled. Also we recall some ideas of the algebraic approach to the description of the homogeneous approximation: the free graded associative algebra, which is isomorphic to the algebra of iterated integrals, the free Lie algebra, the Poincar\'{e}-Birkhoff-Witt basis, the dual basis and its construction by use of the shuffle product, the definition of the core Lie subalgebra, which defines the homogeneous approximation of a control system. In the third section we show how to find the core Lie subalgebra of the systems that is a realization of the one-dimensional series of iterated integrals without finding the system itself. The result obtained is illustrated by the example, in which we demonstrate two methods for finding the core Lie subalgebra of the realizing system. In the last section it is shown that for any graded Lie subalgebra of finite codimension there exists a one-dimensional homogeneous series such that this Lie subalgebra is the core Lie subalgebra for its minimal realization. The proof is constructive: we give a method of finding such a series; we use the dual basis to the Poincar\'{e}-Birkhoff-Witt basis of the free associative algebra, which is built by the core Lie subalgebra, and the shuffle product in this algebra. As a consequence, we get a classification of all possible homogeneous approximations of systems that are realizations of one-dimensional series of iterated integrals.

Double Lie algebras of a nonzero weight

We introduce the notion of λ-double Lie algebra, which coincides with usual double Lie algebra when λ=0. We show that every λ-double Lie algebra for λ≠0 provides the structure of modified double Poisson algebra on the free associative algebra. In particular, it confirms the conjecture of S. Arthamonov (2017). We prove that there are no simple finite-dimensional λ-double Lie algebras.

Computing free non-commutative Gröbner bases over [formula omitted] with Singular:Letterplace

With this paper we present an extension of our recent ISSAC paper about computations of Gröbner(-Shirshov) bases over free associative algebras Z〈X〉. We present all the needed proofs in details, add a part on the direct treatment of the ring Z/mZ as well as new examples and applications to e.g. Iwahori-Hecke algebras. The extension of Gröbner bases concept from polynomial algebras over fields to polynomial rings over rings allows to tackle numerous applications, both of theoretical and of practical importance. Gröbner and Gröbner-Shirshov bases can be defined for various non-commutative and even non-associative algebraic structures. We study the case of associative rings and aim at free algebras over principal ideal rings. We concentrate on the case of commutative coefficient rings without zero divisors (i.e. a domain). Even working over Z allows one to do computations, which can be treated as universal for fields of arbitrary characteristic. By using the systematic approach, we revisit the theory and present the algorithms in the implementable form. We show drastic differences in the behaviour of Gröbner bases between free algebras and algebras, which are close to commutative. Even the process of the formation of critical pairs has to be reengineered, together with implementing the criteria for their quick discarding. We present an implementation of algorithms in the Singular subsystem called Letterplace, which internally uses Letterplace techniques (and Letterplace Gröbner bases), due to La Scala and Levandovskyy. Interesting examples and applications accompany our presentation.

Derived categories of coherent sheaves on some zero-dimensional schemes

Let R N = k [ y 1 , … , y N ] / ( y 1 , … , y N ) 2 be the truncated polynomial algebra . Geometrically, it is the algebra of functions on the second infinitesimal neighborhood of a closed point in N -dimensional affine space . In this note we study D b ( R N − mod ) , the bounded derived category of finitely generated R N -modules. We show that for N ≥ 2 the lattice of triangulated subcategories in D b ( R N − mod ) has a rich structure (which is probably wild), in contrast to the case of zero-dimensional complete intersections. Our homological methods produce some applications to universal localizations of free associative algebras . These applications are based on a relation between triangulated subcategories in D b ( R N − mod ) and universal localizations of a free graded associative algebra in N variables.

Central polynomials of graded algebras: Capturing their exponential growth

Let G be a finite abelian group and let A be an associative G-graded algebra over a field of characteristic zero. A central G-polynomial is a polynomial of the free associative G-graded algebra that takes central values for all graded substitutions of homogeneous elements of A. We prove the existence and the integrability of two limits called the central G-exponent and the proper central G-exponent that give a quantitative measure of the growth of the central G-polynomials and the proper central G-polynomials, respectively. Moreover, we compare them with the G-exponent of the algebra.

The Eigenvalues of Hyperoctahedral Descent Operators and Applications to Card-Shuffling


 We extend an algebra of Mantaci and Reutenauer, acting on the free associative algebra, to a vector space of operators acting on all graded connected Hopf algebras. These operators are convolution products of certain involutions, which we view as hyperoctahedral variants of Patras's descent operators. We obtain the eigenvalues and multiplicities of all our new operators, as well as a basis of eigenvectors for a subclass akin to Adams operations. We outline how to apply this eigendata to study Markov chains, and examine in detail the case of card-shuffles with flips or rotations.

Skew Polynomial Rings: the Schreier Technique

Schreier bases are introduced and used to show that skew polynomial rings are free ideal rings, i.e., rings whose one-sided ideals are free of unique rank, as well as to compute a rank of one-sided ideals together with a description of corresponding bases. The latter fact, a so-called Schreier-Lewin formula (Lewin Trans. Am. Math. Soc.145, 455–465 1969), is a basic tool determining a module type of perfect localizations which reveal a close connection between classical Leavitt algebras, skew polynomial rings, and free associative algebras.

Symmetric polynomials in the free metabelian associative algebra of rank 2

Let $F$ be the free metabelian associative algebra generated by $x$ and $y$ over a field of characteristic zero. We call a polynomial $f\in F$ symmetric, if $f(x,y)=f(y,x)$. The set of all symmetric polynomials coincides with the algebra $F^{S_2}$ of invariants of the symmetric group $S_2$. In this paper, we give the full description of the algebra $F^{S_2}$.

Symmetric polynomials in free associative algebras

By a result of Margarete Wolf in 1936, we know that the algebra $K\langle X_d\rangle^{Sym(d)}$ of symmetric polynomials in noncommuting variables is not finitely generated. In 1984, Koryukin proved that if we equip the homogeneous component of degree $n$ with the additional action of $Sym(n)$ by permuting the positions of the variables, then the algebra of invariants $K\langle X_d\rangle^G$ of every reductive group $G$ is finitely generated. First, we make a short comparison between classical invariant theory of finite groups and its noncommutative counterpart. Then, we expose briefly the results of Wolf. Finally, we present the main result of our paper, which is, over a field of characteristic 0 or of characteristic $p>d$, the algebra $K\langle X_d\rangle^{Sym(d)}$ with the action of Koryukin is generated by the elementary symmetric polynomials.

On central polynomials and codimension growth

Let $A$ be an associative algebra over a field of characteristic zero. A central polynomial is a polynomial of the free associative algebra that takes central values of $A.$ In this survey, we present some recent results about the exponential growth of the central codimension sequence and the proper central codimension sequence in the setting of algebras with involution and algebras graded by a finite group.