Free Associative Algebra Research Articles

Proper central and core polynomials of relatively free associative algebras with identity of Lie nilpotency of degrees 5 and 6

We study the centre of a relatively free associative algebra with the identity of Lie nilpotency of degree over a field of characteristic 0. It is proved that the core of the algebra (the sum of all ideals of contained in its centre) is generated as a -ideal by the weak Hall polynomial . It is also proved that every proper central polynomial of is contained in the sum of and the -space generated by and the commutator of degree 4. This implies that the centre of is contained in the -ideal generated by the commutator of degree 4. Similar results are obtained for ; in particular, it is proved that the core is generated as a -ideal by the commutator of degree 5. Bibliography: 15 titles.

FIXED ELEMENTS OF NONINJECTIVE ENDOMORPHISMS OF POLYNOMIAL ALGEBRAS IN TWO VARIABLES

Let $A_{2}$ be a free associative algebra or polynomial algebra of rank two over a field of characteristic zero. The main results of this paper are the classification of noninjective endomorphisms of $A_{2}$ and an algorithm to determine whether a given noninjective endomorphism of $A_{2}$ has a nontrivial fixed element for a polynomial algebra. The algorithm for a free associative algebra of rank two is valid whenever an element is given and the subalgebra generated by this element contains the image of the given noninjective endomorphism.

Parametrized Gröbner–Shirshov bases

ABSTRACTWe consider the problem of describing Gröbner–Shirshov bases for free associative algebras in finite terms. To this end we consider parametrized elements of an algebra and give methods for working with them which under favorable conditions lead to a basis given by finitely many patterns. On the negative side we show that in general there can be no algorithm. We relate our study to the problem of verifying that a given set of words in certain groups yields Bokut’ normal forms (or groups with a standard basis).

Products of commutators in a Lie nilpotent associative algebra

Let F be a field and let F〈X〉 be the free unital associative algebra over F freely generated by an infinite countable set X={x1,x2,…}. Define a left-normed commutator [a1,a2,…,an] recursively by [a1,a2]=a1a2−a2a1, [a1,…,an−1,an]=[[a1,…,an−1],an] (n≥3). For n≥2, let T(n) be the two-sided ideal in F〈X〉 generated by all commutators [a1,a2,…,an] (ai∈F〈X〉).Let F be a field of characteristic 0. In 2008 Etingof, Kim and Ma conjectured that T(m)T(n)⊂T(m+n−1) if and only if m or n is odd. In 2010 Bapat and Jordan confirmed the “if” direction of the conjecture: if at least one of the numbers m, n is odd then T(m)T(n)⊂T(m+n−1). The aim of the present note is to confirm the “only if” direction of the conjecture. We prove that if m=2m′ and n=2n′ are even then T(m)T(n)⊈T(m+n−1). Our result is valid over any field F.

An explicit basis for the Grassmann T-space

Let S3 denote the Grassmann T-space generated by the commutator [x1,x2,x3] in the free associative algebra K〈x1,x2,…〉 over a field K of characteristic zero. We construct an explicit linear basis for each KSn-module S3∩Pn, where Pn is the space of all multilinear polynomials of degree n in indeterminates x1,…,xn. This provides a solution to the problem of finding a linear basis for S3, which was posed by Latyshev in circa 1990.

Monomial right ideals and the Hilbert series of noncommutative modules

In this paper we present a procedure for computing the rational sum of the Hilbert series of a finitely generated monomial right module N over the free associative algebra K〈x1,…,xn〉. We show that such procedure terminates, that is, the rational sum exists, when all the cyclic submodules decomposing N are annihilated by monomial right ideals whose monomials define regular formal languages. The method is based on the iterative application of the colon right ideal operation to monomial ideals which are given by an eventual infinite basis. By using automata theory, we prove that the number of these iterations is a minimal one. In fact, we have experimented efficient computations with an implementation of the procedure in Maple which is the first general one for noncommutative Hilbert series.

Some factorizations in the twisted group algebra of symmetric groups

In this paper we will give a similar actorization as in [3, 4], where Svrtan and Meljanac examined certain matrix factorizations on Fock-like representation of a multiparametric quon algebra on the free associative algebra of noncommuting polynomials equipped with multiparametric partial derivatives. In order to replace these matrix factorizations (given from the right) by twisted algebra computation, we first consider the natural action of the symmetric group Sn on the polynomial ring Rn in n2 commuting variables Xab and also introduce a twisted group algebra (defined by the action of Sn on Rn) which we denote by A(Sn). Here we consider some factorizations given from the left because they will be more suitable in calculating the constants (= the elements which are annihilated by all multiparametric partial derivatives) in the free algebra of noncommuting polynomials.

A Deformation of the Orlik-Solomon Algebra

A deformation of the Orlik-Solomon algebra of a matroid $\mathfrak{M}$ is defined as a quotient of the free associative algebra over a commutative ring $R$ with $1$. It is shown that the given generators form a Gröbner basis and that after suitable homogenization the deformation and the Orlik-Solomon have the same Hilbert series as $R$-algebras. For supersolvable matroids, equivalently fiber type arrangements, there is a quadratic Gröbner basis and hence the algebra is Koszul.

On Ideals That Are Closed Under Continuous Endomorphisms

Let 𝔽 be the field ℝ or ℂ, and let 𝔽 ⟨ X ⟩ be the free associative algebra generated by the infinite set X. If f ∈ 𝔽 ⟨ X ⟩, define its norm ‖f‖ as the sum of the absolute value of its coefficients. We describe the ideals of 𝔽 ⟨ X ⟩ which are closed under all continuous endomorphisms of 𝔽 ⟨ X ⟩. An element f in the completion is called a series identity for a normed algebra A if f belongs to the intersection of the kernels of all continuous homomorphisms from to A. We describe such identities for a nil Banach algebra.

Algebraic Simplification and Cryptographic Motives

Abstract. Algebraic simplification is considered from the general point of view. The main notion here is the so-called scheme of simplification due to the author. For a corepresented algebra, simplificators are related to the standard basis (Grobner basis, Grobner–Shirshov basis) of the ideal of defining relations. We introduce as a main subject for study the class of algebras that may be obtained from ordered semigroup algebras by “deformation” in some sense. This class contains free associative algebras and universal enveloping algebras of Lie algebras. Primary attention is paid to the latter algebras. Some possible applications to cryptography are given.

Proper central and core polynomials of relatively free associative algebras with identity of Lie nilpotency of degrees 5 and 6

Исследуются центры относительно свободной ассоциативной алгебры $F^{(n)}$ с тождеством $[x_1,…,x_n]=0$ лиевой нильпотентности степени $n=5,6$ над полем характеристики 0. Доказано, что ядро $Z^*(F^{(5)})$ алгебры $F^{(5)}$ (сумма всех идеалов алгебры $F^{(5)}$, содержащихся в ее центре) порождается как $\mathrm T$-идеал слабым многочленом Холла $[[x,y]^{2},y]$. Доказано также, что всякий собственный центральный многочлен алгебры $F^{(5)}$ содержится в сумме $\mathrm T$-пространства, порожденного многочленом Холла $[[x,y]^{2}, z]$ и коммутатором $[x_1,…, x_4]$ степени 4, и его ядра $Z^*(F^{(5)})$. Отсюда следует, что центр алгебры $F^{(5)}$ содержится в $\mathrm T$-идеале, порожденном коммутатором степени 4. Для алгебры $F^{(6)}$ получены аналогичные результаты, доказано, в частности, что ядро $Z^{*}(F^{(6)})$ как $\mathrm T$-идеал порождается коммутатором степени 5. Библиография: 15 названий.

On centres of relatively free associative algebras with a Lie nilpotency identity

We study central polynomials of a relatively free Lie nilpotent algebra of degree . We prove a product theorem, which generalizes the well-known results of Latyshev and Volichenko. We construct generalized Hall polynomials, by using which we prove that the core centre of the algebra is nontrivial for any . We obtain a number of special results when and . Bibliography: 27 titles.

A REMARK ON THE CONJUGATION IN THE STEENROD ALGEBRA

Abstract. We investigate the Hopf algebra conjugation, χ, of the mod2 Steenrod algebra, A 2 , in terms of the Hopf algebra conjugation, χ ′ , ofthe mod 2 Leibniz–Hopf algebra. We also investigate the fixed points ofA 2 under χ and their relationship to the invariants under χ ′ . 1. IntroductionThe mod 2 Steenrod algebra A 2 is the free associative graded algebra gen-erated by the Steenrodsquares Sq n [18] of degree n, n ≥ 1, over F 2 subject tothe AdemrelationsSq a Sq b = ⌊ X a/2⌋s=0 b−1−sa−2sSq a+b−s Sq s for 0 < a < 2b.Conventionally, Sq 0 = 1, the multiplicative identity. Topologically, A 2 is thealgebra of stable cohomology operations for ordinary cohomology H ∗ over F 2 .A monomial in A 2 can be written in the form Sq j 1 Sq j 2 ···Sq j r , which weshall denote by Sq j 1 ,j 2 ,...,j r . Admissible monomials form a vector space ba-sis “admissible basis” for A 2 . Milnor [16] determined the graded connectedHopf algebra structure of A 2 by a cocomutative coproduct given by ∆(Sq

Formal geometry for noncommutative manifolds

This paper develops the tools of formal algebraic geometry in the setting of noncommutative manifolds, roughly ringed spaces locally modeled on the free associative algebra. We define a notion of noncommutative coordinate system, which is a principal bundle for an appropriate group of local coordinate changes. These bundles are shown to carry a natural flat connection with properties analogous to the classical Gelfand–Kazhdan structure.Every noncommutative manifold has an underlying smooth variety given by abelianization. A basic question is existence and uniqueness of noncommutative thickenings of a smooth variety, i.e., finding noncommutative manifolds abelianizing to a given smooth variety. We obtain new results in this direction by showing that noncommutative coordinate systems always arise as reductions of structure group of the commutative bundle of coordinate systems on the underlying smooth variety; this also explains a relationship between D-modules on the commutative variety and sheaves of modules for the noncommutative structure sheaf.

On Tame and Wild Automorphisms of Algebras

Let Bn be a polynomial algebra of n variables over a field F. Considering a free associative algebra An of rank n over F as a polynomial algebra of noncommuting variables, we choose the ideal R of all polynomials with a zero absolute term in Bn and An. The well-known concept of wild automorphisms of the algebras An and Bn is transferred to R; the study of wild automorphisms is reduced to monic automorphisms of the algebra R, i.e., those identical on each factor Rk/Rk+1. In particular, this enables us to study the properties of the known Nagata and Anik automorphisms in detail. For n = 3 we investigate the hypothesis that the Anik automorphism is tame modulo Rk for every given integer k > 1.

A Cacti theoretical interpretation of the axioms of bialgebras and H-module algebras

We establish a dictionary between the Cacti algebra axioms on a Cacti algebra structure with underlying free associative algebra, under suitable good behavior with degrees. Using these ideas, for an associative algebra A and a bialgebra H, we also translate Cacti algebra maps Ω(H)→C•(A) (where Ω(H) stands for the cobar construction on H and C•(A) is the Hochschild cohomology complex) with H-module algebra structures on A, and illustrate with examples of applications.

Extended letterplace correspondence for nongraded noncommutative ideals and related algorithms

Let K〈xi〉 be the free associative algebra generated by a finite or a countable number of variables xi. The notion of "letterplace correspondence" introduced in [R. La Scala and V. Levandovskyy, Letterplace ideals and non-commutative Gröbner bases, J. Symbolic Comput. 44(10) (2009) 1374–1393; R. La Scala and V. Levandovskyy, Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra, J. Symbolic Comput. 48 (2013) 110–131] for the graded (two-sided) ideals of K〈xi〉 is extended in this paper also to the nongraded case. This amounts to the possibility of modelizing nongraded noncommutative presented algebras by means of a class of graded commutative algebras that are invariant under the action of the monoid ℕ of natural numbers. For such purpose we develop the notion of saturation for the graded ideals of K〈xi,t〉, where t is an extra variable and for their letterplace analogues in the commutative polynomial algebra K[xij, tj], where j ranges in ℕ. In particular, one obtains an alternative algorithm for computing inhomogeneous noncommutative Gröbner bases using just homogeneous commutative polynomials. The feasibility of the proposed methods is shown by an experimental implementation developed in the computer algebra system Maple and by using standard routines for the Buchberger algorithm contained in Singular.

Noncommutative Gröbner Bases over Rings

In this work, we propose a method for computing noncommutative Gröbner bases over a noetherian valuation ring. We have generalized the fundamental theorem on normal forms over an arbitrary ring. The classical method of dynamical commutative Gröbner bases is generalized to Buchberger's algorithm over R = 𝒱 ⟨ x 1,…, x m ⟩, a free associative algebra with noncommuting variables, where 𝒱 = ℤ/nℤ and 𝒱 = ℤ. The proposed process generalizes previous known techniques for the computation of commutative Gröbner bases over a nætherian valuation ring and/or noncommutative Gröbner bases over a field.

Weight ideals associated to regular and log-linear arrays

Certain weight-based orders on the free associative algebra R=k〈x1,…,xt〉 can be specified by t×∞ arrays whose entries come from the subring of positive elements in a totally ordered field. If such an array satisfies certain additional conditions, it produces a partial order on R which is an admissible order on the quotient R/I, where the ideal I is a homogeneous binomial ideal called the weight ideal associated to the array. The structure of the weight ideal is determined entirely by the array. This article discusses the structure of the weight ideals associated to two distinct types of arrays which define admissible orders on the associated quotient algebra.

Limit T-subalgebras in free associative algebras

Let F〈X〉 be the free unitary associative algebra over a field F on a free generating set X. An unitary subalgebra R of F〈X〉 is called a T-subalgebra if R is closed under all endomorphisms of F〈X〉. A T-subalgebra R⁎ in F〈X〉 is limit if every larger T-subalgebra W⫌R⁎ is finitely generated (as a T-subalgebra) but R⁎ itself is not. It follows easily from Zorn's lemma that if a T-subalgebra R is not finitely generated then it is contained in some limit T-subalgebra R⁎. In this sense limit T-subalgebras form a “border” between those T-subalgebras which are finitely generated and those which are not. In the present article we give the first example of a limit T-subalgebra in F〈X〉, where F is an infinite field of characteristic p>2 and |X|≥4. Note that, by Shchigolev's result, over a field F of characteristic 0 every T-subalgebra in F〈X〉 is finitely generated; hence, over such a field limit T-subalgebras in F〈X〉 do not exist.