Showing papers on "Free algebra published in 2009"

Engineering Systems and Free Semi-Algebraic Geometry

Abstract: This article sketches a few of the developments in the recently emerging area of real algebraic geometry (in short RAG) in a free* algebra, in particular on “noncommutative inequalities”. Also we sketch the engineering problems which both motivated them and are expected to provide directions for future developments. The free* algebra is forced on us when we want to manipulate expressions where the unknowns enter naturally as matrices. Conditions requiring positive definite matrices force one to noncommutative inequalities. The theory developed to treat such situations has two main parts, one parallels classical semialgebraic geometry with sums of squares representations (Positivstellensatze) and the other has a new flavor focusing on how noncommutative convexity (similarly, a variety with positive curvature) is very constrained, so few actually exist.

The free ample monoid

Abstract: We show that the free weakly E-ample monoid on a set X is a full submonoid of the free inverse monoid FIM(X) on X. Consequently, it is ample, and so coincides with both the free weakly ample and the free ample monoid FAM(X) on X. We introduce the notion of a semidirect product Y*T of a monoid T acting doubly on a semilattice Y with identity. We argue that the free monoid X* acts doubly on the semilattice of idempotents of FIM(X) and that FAM(X) is embedded in . Finally we show that every weakly E-ample monoid has a proper ample cover.

Wandering vectors and the reflexivity of free semigroup algebras

Abstract: A free semigroup algebra S is the weak-operator-closed (non-self-adjoint) operator algebra generated by n isometries with pairwise orthogonal ranges. A unit vector x is said to be wandering for S if the set of images of x under non-commuting words in the generators of S is orthonormal. We establish the following dichotomy: either a free semigroup algebra has a wandering vector, or it is a von Neumann algebra. Consequences include that every free semigroup algebra is reflexive, and that certain free semigroup algebras are hyper-reflexive with a very small hyper-reflexivity constant.

Automorphisms of Endomorphism Monoids of 1-Simple Free Algebras

Abstract: For any set X and any variety 𝒱 of algebras, let 𝒜 = 𝒜 𝒱(X) be the free algebra in 𝒱 with set X of free generators, and let End(𝒜) be the monoid of endomorphisms of 𝒜. We provide a general approach to the description of the automorphism group of End(𝒜), and some subsemigroups of End(𝒜), provided that 𝒜 is 1-simple. We show that every automorphism of End(𝒜) is a unique extension of an automorphism of the monoid of endomorphisms of rank at most 1.

Algebras for parameterised monads

Abstract: Parameterised monads have the same relationship to adjunctions with parameters as monads do to adjunctions. In this paper, we investigate algebras for parameterised monads.We identify the Eilenberg-Moore category of algebras for parameterised monads and prove a generalisation of Beck's theorem characterising this category. We demonstrate an application of this theory to the semantics of type and effect systems.

On the multiplicative and T-space structure of the relatively free Grassmann algebra

Abstract: We investigate the multiplicative and -space structure of the relatively free algebra with unity corresponding to the identity over an infinite field of characteristic . One of the basic results is the decomposition of quotient -spaces connected with into a direct sum of simple components. Also, the -spaces under consideration are commutative subalgebras of ; thus, the structure of and its subalgebras is described as modules over these commutative subalgebras. Finally, we consider the specifics of the case . Bibliography: 15 titles.

On the Kurosh problem in varieties of algebras

Abstract: We consider a couple of versions of the classical Kurosh problem (whether there is an infinite-dimensional algebraic algebra) for varieties of linear multioperator algebras over a field. We show that, given an arbitrary signature, there is a variety of algebras of this signature such that the free algebra of the variety contains polylinear elements of arbitrarily large degree, while the clone of every such element satisfies some nontrivial identity. If, in addition, the number of binary operations is at least 2, then each such clone may be assumed to be finite-dimensional. Our approach is the following: we cast the problem in the language of operads and then apply the usual homological constructions in order to adopt Golod’s solution to the original Kurosh problem. This paper is expository, so that some proofs are omitted. At the same time, the general relations of operads, algebras, and varieties are widely discussed.

On Kurosh problem in varieties of algebras

Abstract: We consider a couple of versions of classical Kurosh problem (whether there is an infinite-dimensional algebraic algebra?) for varieties of linear multioperator algebras over a field. We show that, given an arbitrary signature, there is a variety of algebras of this signature such that the free algebra of the variety contains multilinear elements of arbitrary large degree, while the clone of every such element satisfies some nontrivial identity. If, in addition, the number of binary operations is at least 2, then one can guarantee that each such clone is finitely-dimensional. Our approach is the following: we translate the problem to the language of operads and then apply usual homological constructions, in order to adopt Golod's solution of the original Kurosh problem. The paper is expository, so that some proofs are omited. At the same time, the general relations of operads, algebras, and varieties are widely discussed.

Free Heyting algebras: revisited

Abstract: We use coalgebraic methods to describe finitely generated free Heyting algebras. Heyting algebras are axiomatized by rank 0-1 axioms. In the process of constructing free Heyting algebras we first apply existing methods to weak Heyting algebras--the rank 1 reducts of Heyting algebras--and then adjust them to the mixed rank 0-1 axioms. On the negative side, our work shows that one cannot use arbitrary axiomatizations in this approach. Also, the adjustments made for the mixed rank axioms are not just purely equational, but rely on properties of implication as a residual. On the other hand, the duality and coalgebra perspectives do allow us, in the case of Heyting algebras, to derive Ghilardi's (Ghilardi, 1992) powerful representation of finitely generated free Heyting algebras in a simple, transparent, and modular way using Birkhoff duality for finite distributive lattices.

Algebraic categories whose projectives are explicitly free

Abstract: Let M = (M;m;u) be a monad and let (MX;m) be the free M-algebra on the object X Consider an M-algebra (A;a), a retraction r : (MX;m)! (A;a) and a section t : (A;a)! (MX;m) of r The retract (A;a) is not free in general We observe that for many monads with a 'combinatorial avor' such a retract is not only a free algebra (MA0;m), but it is also the case that the object A0 of generators is determined in a canonical way by the section t We give a precise form of this property, prove a characterization, and discuss examples from combinatorics, universal algebra, convexity and topos theory

On coproducts in varieties, quasivarieties and prevarieties

Abstract: If the free algebra F on one generator in a variety V of algebras (in the sense of universal algebra) has a subalgebra free on two generators, must it also have a subalgebra free on three generators? In general, no; but yes if F generates the variety V: Generalizing the argument, it is shown that if we are given an algebra and subalgebras, A0 An; in a prevariety (SP -closed class of algebras) P such that An generates P; and also subalgebras Bi Ai 1 (0 0 the subalgebra of Ai 1 generated by Ai and Bi is their coproduct in P; then the subalgebra of A generated by B1;:::;Bn is the coproduct in P of these algebras. Some further results on coproducts are noted: If P satises the amalgamation property, then one has the stronger \transitivity" statement, that if A has a nite family of subalgebras ( Bi)i2I such that the subalgebra of A generated by the Bi is their coproduct, and each Bi has a nite family of subalgebras ( Cij)j2Ji with the same property, then the subalgebra of A generated by all the Cij is their coproduct. For P a residually small prevariety or an arbitrary quasivariety, relationships are proved between the least number of algebras needed to generate P as a prevariety or quasivariety, and behavior of the coproduct operation in P: It is shown by example that for B a subgroup of the group S = Sym() of all permutations of an innite set ; the group S need not have a subgroup isomorphic over B to the coproduct with amalgamation S ' B S: But under various additional hypotheses on B; the question remains open.

Quantum Enveloping Algebras with von Neumann Regular Cartan-like Generators and the Pierce Decomposition

Abstract: Quantum bialgebras derivable from U q (sl 2) which contain idempotents and von Neumann regular Cartan-like generators are introduced and investigated. Various types of antipodes (invertible and von Neumann regular) on these bialgebras are constructed, which leads to a Hopf algebra structure and a von Neumann-Hopf algebra structure, respectively. For them, explicit forms of some particular R-matrices (also, invertible and von Neumann regular) are presented, and the latter respects the Pierce decomposition.

Primitive elements of free nonassociative algebras

Abstract: Improved algorithms to construct complements of primitive systems of elements of free nonassociative algebras with respect to free generating sets and algorithms to realize the rank of a system of elements are constructed and implemented.

On universal Lie nilpotent associative algebras

Abstract: We study the quotient Q i ( A ) of a free algebra A by the ideal M i ( A ) generated by the ith commutator of any elements. 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 ) ).

Theorems on equalization and monomiality in a relatively free Grassmann algebra

Abstract: In this paper, we prove theorems on equalization and monomiality, which are essential for developing the structural theory of T-spaces in a relatively free algebra k〈1, x1,…, xi,…〉/([[x1, x2], x3])T over an infinite field k of characteristic p > 2. Additionally, some specifics of the case p = 2 are considered.

Exactness of universal free products of finite dimensional $C^*$-algebras with amalgamation

Abstract: We investigate free products of finite dimensional $C^*$-algebras with amalgamation over diagonal subalgebras. We look to determine under what circumstances a given free product is exact and/or nuclear. In some cases we find a description of the algebra in terms of a more readily understood algebra.

Schreier rewriting beyond the classical setting

Abstract: Using actions of free monoids and free associative algebras, we establish some Schreiertype formulas involving ranks of actions and 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 a generalization of the Schreier formula in the case of subgroups of infinite index. We also study and apply large modules over free associative and free group algebras.

An algorithm to determine the Hilbert series for graded associative algebras

Abstract: In this paper we present an algorithm to compute the Hilbert series for quotients of the free algebra with homogeneous two-sided ideals. We also give a modified version of the algorithm for quotien ...

On T-spaces and related concepts and results

Abstract: This paper is a brief survey of the T-space concept and related results. A connection between T-spaces and so-called varieties of pairs, analogous to the connection between T-ideals and varieties of algebras, is established. The concepts of A-equivalency and T-equivalency are introduced, after which some applications are considered.

Some Results on dh-Closed Homogeneous Gr\"obner Bases and dh-Closed Graded Ideals

Abstract: Let $K$ be a field and $R=\oplus_{p\in\mathbb{N}}R_p$ an $\mathbb{N}$-graded $K$-algebra, which has an SM $K$-basis (i.e. a skew multiplicative $K$-basis) such that $R$ holds a Grobner basis theory. It is proved that there is a one-to-one correspondence between the set of Grobner bases in $R$ and the set of dh-closed homogeneous Grobner bases in the polynomial algebra $R[t]$; and that the similar result holds true if $R$ and $R[t]$ are replaced respectively by the free algebra $K $ and the free algebra $K $. Moreover, it is shown that dh-closed graded ideals in $R[t]$ and $K $ can be realized by dh-closed homogeneous Grobner bases. The latter result indeed tells us that algebras defined by dh-homogeneous Grobner bases can be studied as Rees algebras effectively via more simpler algebras as demonstrated in ([7], [8]).

Some of A.I. Shirshov’s works

Abstract: In his first published paper “Subalgebras of free Lie algebras” A.I. Shirshov proved for Lie algebras an analog of the famous Nielsen-Schreier theorem: every subalgebra of a free Lie algebra is free. Three years later this theorem was independently proved and extended to restricted Lie algebras by E. Witt [38]. Much later this result was generalized to Lie superalgebras (A.S. Shtern [29]), and to colored Lie superalgebras (A.A. Mikhalev [20, 21, 22]). These results went through further development in the field of quantum algebra as follows. The Shirshov-Witt theorem for Lie algebras over fields of characteristic zero admits an equivalent formulation in terms of a free associative algebra: Every Hopf subalgebra of a free algebra k with the coproduct defined by Δ (y i )=y i ⊗1+1⊗y i is free. If we consider the free algebra as a braided Hopf algebra with a very special braiding (τ(y i ⊗y j )=p ij y j ⊗y i P ij P ji =1), then we get a reformulation of the Mikhalev-Shtern generalization as well. We may consider the free associative algebra k as a braided Hopf algebra provided that V is a braided space with arbitrary braiding (not necessary invertible). In this setting the braided version of the Shirshov-Witt theorem takes the following form [12]: If a subalgebra \( U \subseteq k\left\langle V \right\rangle \) is a right categorical right coideal, that is \( \Delta U \subseteq U\underline \otimes k\left\langle V \right\rangle \), \( \tau \left( {k\left\langle V \right\rangle \otimes U} \right) \subseteq U \otimes k\left\langle V \right\rangle \), then U is a free subalgebra.

Algebras Associated With A Free Inverse Monoid

Abstract: Let S be an ideal of the free inverse monoid on a set X , and let ℬ denote the Banach algebra l 1 ( S ). It is shown that the following statements are equivalent: ℬ is *-primitive; ℬ is prime; X is infinite. A similar result holds if ℬ is replaced by ℂ[ S ], the complex semigroup algebra of S .

A non-commutative homogeneous coordinate ring for the degree six del Pezzo surface

Abstract: Let R be the free algebra on x and y modulo the relations x^5=yxy and y^2=xyx endowed with the grading deg x=1 and deg y=2. Let B_3 denote the blow up of the projective plane at three non-colliear points. The main result in this paper is that the category of quasi-coherent sheaves on B_3 is equivalent to the quotient of the category of graded R-modules modulo the full subcategory of modules M such that for each m in M, $(x,y)^nm=0$ for n sufficiently large. This is proved by showing the R is a twisted homogeneous coordinate ring (in the sense of Artin and Van den Bergh) for B_3. This reduces almost all representation-theoretic questions about R to algebraic geometric questions about the del Pezzo surface B_3. For example, the generic simple R-module has dimension six. Furthermore, the main result combined with results of Artin, Tate, and Van den Bergh, imply that R is a noetherian domain of global dimension three, and has other good homological properties.

Free algebras of a unary variety with Mal’tsev’s operation that satisfies the Pixley conditions

Abstract: In this paper we consider the variety V P of algebras with one unary and one ternary operation p that satisfies the Pixley identities, provided that operations are permutable. We describe the structure of a free algebra of the variety V P and study the structure of unary reducts of free algebras. We prove the solvability of the word problem in free algebras and the uniqueness of a free basis; we also describe groups of automorphisms of free algebras. Similar results are obtained for free algebras of a subvariety of the variety V P defined by the identities p(p(x, y, z), y, z) = p(x, y, z) and p(x, y, p(x, y, z)) = p(x, y, z).

Bernoulli-type Relations in Some Noncommutative Polynomial Ring

Abstract: We find particular relations which we call "Bernoulli-type" in some noncommutative polynomial ring with a single nontrivial relation. More precisely, our ring is isomorphic to the universal enveloping algebra of a two-dimensional non-abelian Lie algebra. From these Bernoulli-type relations in our ring, we can obtain a representation on a certain left ideal with the Bernoulli numbers as structure constants.

Free Groups in Quaternion Algebras

Abstract: In \cite{jpsf} we constructed pairs of units $u,v$ in $\Z$-orders of a quaternion algebra over $\Q (\sqrt{-d})$, $d \equiv 7 \pmod 8$ positive and square free, such that $ $ is free for some $n\in \mathbb{N}$. Here we extend this result to any imaginary quadratic extension of $\ \mathbb{Q}$, thus including matrix algebras. More precisely, we show that $ $ is a free group for all $n\geq 1$ and $d>2$ and for $d=2$ and all $n\geq 2$. The units we use arise from Pell's and Gauss' equations. A criterion for a pair of homeomorphisms to generate a free semigroup is also established and used to prove that two certain units generate a free semigroup but that, in this case, the Ping-Pong Lemma can not be applied to show that the group they generate is free.