Primitive Algebras Research Articles

Nil algebras, Lie algebras and wreath products with intermediate and oscillating growth

We construct finitely generated nil algebras with prescribed growth rates. In particular, any increasing submultiplicative function is realized as the growth function of a nil algebra up to a polynomial error term and an arbitrarily slow distortion.We then move on to examples of nil algebras and domains with strongly oscillating growth functions and construct primitive algebras for which the Gel'fand-Kirillov dimension is strictly sub-additive with respect to tensor products, thus answering a question from [16,17].

Algebraic Characteristics of the Matrix Conversions Class and Its Hardware Implementation

This paper derives the algebraic characteristic of the matrix transformations class by the method of isomorphic mappings on the algebraic characteristic of the class of vector transformations using the primitive program algebras. The paper also describes the hardware implementation of the matrix operations accelerator based on the obtained results. The urgency of the work is caused by the fact that today there is a rapid integration of computer technology in all spheres of society and, as a consequence, the amount of data that needs to be processed per unit time is constantly increasing. Many problems involving large amounts of complex computation are solved by methods based on matrix operations. Therefore, the study of matrix calculations and their acceleration is a very important task. In this paper, as a contribution in this direction, we propose a study of the matrix transformations class using signature operations of primitive program algebra such as multi place superposition, branching, cycling, which are refinements of the most common control structures in most high-level programming languages, and also isomorphic mapping. Signature operations of primitive program algebra in combination with basic partial-recursive matrix functions and predicates allow to realize the set of all partial-recursive matrix functions and predicates. Obtained the result on the basis of matrix primitive program algebra. Isomorphism provides the reproduction of partially recursive functions and predicates for matrix transformations as a map of partially recursive vector functions and predicates. The completeness of the algebraic system of matrix transformations is ensured due to the available results on the derivation of the algebraic system completeness for vector transformations. A name model of matrix data has been created and optimized for the development of hardware implementation. The hardware implementation provides support for signature operations of primitive software algebra and for isomorphic mapping. Hardware support for the functions of sum, multiplication and transposition of matrices, as well as the predicate of equality of two matrices is implemented. Support for signature operations of primitive software algebra is provided by the design of the control part of the matrix computer based on the RISC architecture. The hardware support of isomorphism is based on counters, they allow to intuitively implement cycling in the functions of isomorphic mappings. Fast execution of vector operations is provided by the principle of computer calculations SIMD.

On a locally nilpotent radical Jacobson for special Lie algebras

In the paper investigates the possibility of homological description of Jacobson radical and locally nilpotent radical for Lie algebras, and their relation with a $$PI$$ - irreducibly represented radical, and some properties of primitive Lie algebras are studied. We prove an analog of The F. Kubo theorem for almost locally solvable Lie algebras with a zero Jacobson radical. It is shown that the Jacobson radical of a special almost locally solvable Lie algebra $$L$$ over a field $$F$$ of characteristic zero is zero if and only if the Lie algebra $$L$$ has a Levi decomposition $$L=S\oplus Z(L)$$, where $$Z(L)$$ is the center of the algebra $$L$$, $$S$$ is a finite-dimensional subalgebra $$L$$ such that $$J(L)=0$$. For an arbitrary special Lie algebra $$L$$, the inclusion of $$IrrPI(L)\subset J(L)$$ is shown, which is generally strict. An example of a Lie algebra $$L$$ with strict inclusion $$J(L)\subset IrrPI(L)$$ is given. It is shown that for an arbitrary special Lie algebra $$L$$ over the field $$F$$ of characteristic zero, the inclusion of $$N (L)\subset IrrPI(L)$$, which is generally strict. It is shown that most Lie algebras over a field are primitive. An example of an Abelian Lie algebra over an algebraically closed field that is not primitive is given. Examples are given showing that infinite-dimensional commutative Lie algebras are primitive over any fields; a finite-dimensional Abelian algebra of dimension greater than 1 over an algebraically closed field is not primitive; an example of a non-Cartesian noncommutative Lie algebra is primitive. It is shown that for special Lie algebras over a field of characteristic zero $$PI$$-an irreducibly represented radical coincides with a locally nilpotent one. An example of a Lie algebra whose locally nilpotent radical is neither locally nilpotent nor locally solvable is given. Sufficient conditions for the primitiveness of a Lie algebra are given, and examples of primitive Lie algebras and non-primitive Lie algebras are given.

О почти локально разрешимых алгебрах Ли с нулевым радикалом Джекобсона и локально нильпотентном радикале для алгебр Ли

В статье доказывается аналог теоремы Ф. Кубо [1] для почти локально разрешимых алгебр Ли с нулевым радикалом Джекобсона. Первый раздел направлен на выяснение некоторых аспектов гомологического описания радикала Джекобсона. Доказана теорема, обобщающая теорему Е. Маршалла на случай почти локально разрешимых алгебр Ли, следствием которой и является аналог теоремы Кубо. Во втором разделе исследуются некоторые свойства локально нильпотентного радикала алгебры Ли. Рассматриваются примитивные алгебры Ли. Приведены примеры, показывающие, что бесконечномерные коммутативные алгебры Ли являются примитивными над любыми полями; конечномерная абелева алгебра, размерности больше 1, над алгебраически замкнутым полем не является примитивной; пример неартиновой некоммутативной алгебры Ли являющейся примитивной. Показано, что для специальных алгебр Ли над полем характеристики нуль PI-неприводимо представленный радикал совпадает с локально нильпотентным. Приведен пример алгебры Ли, локально нильпотентный радикал которой не является ни локально нильпотентным, ни локально разрешимым. Даются достаточные условия примитивности алгебры Ли, приводятся примеры примитивных алгебр Ли и алгебры Ли не являющейся примитивной.

Matrix wreath products of algebras and embedding theorems

We introduce a new construction of matrix wreath products of algebras that is similar to wreath products of groups. We then use it to prove embedding theorems for Jacobson radical, nil, and primitive algebras. In §6, we construct finitely generated nil algebras of arbitrary Gelfand-Kirillov dimension ≥ 8 \geq 8 over a countable field which answers a question from [New trends in noncommutative algebra, Amer. Math. Soc., Providence, RI, 2012, pp. 41–52].

Prime and primitive algebras with prescribed growth types

Bartholdi and Smoktunowicz constructed in 2014 finitely generated monomial algebras with prescribed sufficiently fast growth types. We show that their construction need not result in a prime algebra, but it can be modified to provide prime algebras without further limitations on the growth type.

Free skew Boolean algebras

We study the structure and properties of free skew Boolean algebras (SBAs). For finite generating sets, these free algebras are finite and we give their representation as a product of primitive algebras and provide formulas for calculating their cardinality. We also characterize atomic elements and central elements, and calculate the number of such elements. These results are used to study minimal generating sets of finite SBAs. We also prove that the center of the free infinitely generated algebra is trivial and show that all free algebras have intersections.

A CHARACTERIZATION OF PRIMITIVE ALGEBRAS VIA B*- SEMINORMS

Primitive algebras are characterized in terms of sub-seminorms. We prove a necessary and a sufficient condition for aB � − seminorm to be minimal. The results of this note are also valid for M � − seminorms introduced

On primitive lie algebras

Sufficient conditions for Lie algebra primitiveness and examples of primitive Lie algebras and nonprimitive Lie algebras are given.

Commuting holomorphic maps on the spectral unit ball

We prove that if F is a holomorphic map from the open spectral unit ball of a primitive Banach algebra into itself satisfying F (0) = 0, F ′ (0) = I and F (x) x = xF (x) for every x, then F is the identity map. Using this, we prove that if A is a semisimple Banach algebra and B is a primitive Banach algebra, then any unital spectral isometry from A onto B which locally preserves commutativity is a Jordan morphism. The same is true when A and B are both assumed to be von Neumann algebras.

On Primitive Cellular Algebras

First we define and study the exponentiation of a cellular algebra by a permutation group which is analogous to the corresponding operation (the wreath product in primitive action) in permutation group theory. Necessary and sufficient conditions for the resulting cellular algebra to be primitive and Schurian are given. This enables us to construct infinite series of primitive non-Schurian algebras. Also we define and study for cellular algebras the notion of a base which is similar to that for permutation groups. We present an upper bound for the size of an irredundant base of a primitive cellular algebra in terms of the parameters of its standard representation. This produces new upper bounds for the order of the automorphism group of such an algebra and in particular for the order of a primitive permutation group. Finally we generalize to 2-closed primitive algebras some classical theorems for primitive groups and show that the hypothesis for a primitive algebra to be 2-closed is essential.

Primitive algebras with pi primitizers1

We show that a primitive (resp*-primitive) associative algebra having a nonzero PI primitizer (resp. *-primitizer) is necessarily PI Similarly, a primitive Jordan algebra having a PI primitizer with nonzero square is PI.

Two inequalities for parameters of a cellular algebra

Two inequalities are proved. The first is a generalization for cellular algebras of a well- known theorem about the coincidence of the degree and the multiplicity of an irreducible representation of a finite group in its regular representation. The second inequality that is proved for primitive cellular algebras gives an upper bound for the minimal subdegree of a primitive permutation group in terms of the degrees of its irreducible representations in the permutation representation.

Primitive Algebras with Arbitrary Gelfand-Kirillov Dimension

We construct, for every real β ≥ 2, a primitive affine algebra with Gelfand-Kirillov dimension β. Unlike earlier constructions, there are no assumptions on the base field. In particular, this is the first construction over ℝ or L. Given a recursive sequence {v n} of elements in a free monoid, we investigate the quotient of the free associative algebra by the ideal generated by all nonsubwords in {v n}. We bound the dimension of the resulting algebra in terms of the growth of {v n}. In particular, if ⋎ν n⋎ is less than doubly exponential, then the dimension is 2. This also answers affirmatively a conjecture of Salwa (1997, Comm. Algebra 25, 3965–3972).

Local-to-global inheritance of primitivity in Jordan algebras

We show that a strongly prime Jordan algebra having a primitive local algebra is primitive. As a tool, similar results concerning one-sided primitivity and \( \ast \)-primitivity of associative algebras are established.

Reduction of the adjoint representation of sl(2,1): generators of primitive ideals

We present a reduction of the adjoint representation of the Lie superalge-bra,sl(2,1) and a study of the quotient algebra B(c,k)= u/u(C−c)+u(D−kc), where c,k are two complex numbers. Under some additional conditions, we prove that every irreducible infinite dimensional representation of B(c,k) is faithful, and that B(C,K) is a primitive algebra. We give explicitly a set of generators of primitive degenerate ideal of infinite codimension. Essentially we prove that any minimal primitive ideal of u(sl(2,1)) is generated, as a 2-sided ideal, by its intersection with the algebra of gg-iuvariants.

The structure of primitive quadratic Jordan algebras

In this paper we give a complete description of primitive quadratic Jordan algebras following the classification of strongly prime quadratic Jordan algebras (K. McCrimmon and E. I. Zel'manov, Adv. Math. 69, No. 2 (1988), 133–222). The proof is based in two results of independent interest: we show that every P. I. primitive Jordan algebra is simple and unital with nonzero socle and prove that associative tight (*-tight) envelopes of special primitive Jordan algebras are also primitive (*-primitive). As a consequence of the latter fact we see that an associative algebra A is one-sided primitive if and only if A + is primitive, and an associative algebra A with involution * is *-primitive if and only if an ample subspace H 0 ( A , *) is primitive.

On maximal modular inner ideals in jordan algebras

In this paper we give a complete description of maximal modular inner ideals of Jordan algebras by determining primitizers of primitive Jordan algebras, as a consequence we answer the question of Hogben and McCrimmon about maximality of such objects.

A characterization of primitive Boolean algebras

A characterization of primitive Boolean algebras

Extended centroids of power series rings

Prime rings came into prominence when Posner characterized prime rings satisfying a polynomial identity [9]. The scarcity of invertible central elements made it difficult to generalize results from central simple and primitive algebras to prime rings. For example, we do not automatically have tensor products at our disposal. In [5], the first author introduced the Martindale ring of quotients Q(R) of a prime ring R in his theorem characterizing prime rings satisfying a generalized polynomial identity (GPI). Q(R) is a prime ring containing R whose center C is a field called the extended centroid of R. The central closure of R is the subring RC of Q(R) generated by R and C. RC is a closed prime ring since its extended centroid equals its center C. Hence we have a useful procedure for proving results about an arbitrary prime ring R. We first answer the question for closed prime rings and then apply to R the information obtained from RC. It should be noted that simple rings and free algebras of rank at least 2 are closed prime rings. For these reasons, closed prime rings are natural objects to study.