DSpace Collection: Volume 38, Number 1-3
http://hdl.handle.net/10525/2782
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A Basis for the Graded Identities of the Pair (M2(K), gl2(K))
http://hdl.handle.net/10525/2807
Title: A Basis for the Graded Identities of the Pair (M2(K), gl2(K))<br/><br/>Authors: Koshlukov, Plamen; Krasilnikov, Alexei<br/><br/>Abstract: Let M2(K) be the algebra of 2×2 matrices over an infinite integral domain K. In this note we describe a basis for the Z2-graded identities of the pair (M2(K),gl2(K)).<br/><br/>Description: 2010 Mathematics Subject Classification: 16R10, 17B01.Characterizing Non-Matrix Properties of Varieties of Algebras in the Language of Forbidden Objects
http://hdl.handle.net/10525/2806
Title: Characterizing Non-Matrix Properties of Varieties of Algebras in the Language of Forbidden Objects<br/><br/>Authors: Finogenova, Olga<br/><br/>Abstract: We discuss characterizations of some non-matrix properties of varieties of associative algebras in the language of forbidden objects. Properties under consideration include the Engel property, Lie nilpotency, permutativity. We formulate a few open problems.<br/><br/>Description: 2010 Mathematics Subject Classification: 16R10, 16R40.Growth Functions of Fr-sets
http://hdl.handle.net/10525/2805
Title: Growth Functions of Fr-sets<br/><br/>Authors: Lomond, Jonny<br/><br/>Abstract: In this paper we consider an open problem from [1], regarding the description of the growth functions of the free group acts. Using the language of graphs, we solve this problem by providing the necessary and sufficient conditions for a function to be a growth function for a free group act.<br/><br/>Description: 2010 Mathematics Subject Classification: 05C30, 20E08, 20F65.Universal Enveloping Algebras of Nonassociative Structures
http://hdl.handle.net/10525/2804
Title: Universal Enveloping Algebras of Nonassociative Structures<br/><br/>Authors: Tvalavadze, Marina<br/><br/>Abstract: This is a survey paper to summarize the latest results on the universal enveloping algebras of Malcev algebras, triple systems and Leibniz n-ary algebras.<br/><br/>Description: 2010 Mathematics Subject Classification: Primary 17D15. Secondary 17D05, 17B35, 17A99.On Ordinary and Z2-graded Polynomial Identities of the Grassmann Algebra
http://hdl.handle.net/10525/2803
Title: On Ordinary and Z2-graded Polynomial Identities of the Grassmann Algebra<br/><br/>Authors: Ribeiro Tomaz da Silva, Viviane<br/><br/>Abstract: The main purpose of this paper is to provide a survey of results concerning the ordinary and Z2-graded polynomial identities of the infinite dimensional Grassmann algebra over a field of characteristic zero, as well as of its sequences of ordinary and Z2-graded codimensions and cocharacters. We also intend to describe briefly the techniques used by the authors in order to illustrate some important methods used in PI-theory.<br/><br/>Description: 2010 Mathematics Subject Classification: Primary: 16R10, Secondary: 16W55.On the Gibson Bounds over Finite Fields
http://hdl.handle.net/10525/2802
Title: On the Gibson Bounds over Finite Fields<br/><br/>Authors: V. Budrevich, Mikhail; E. Guterman, Alexander<br/><br/>Abstract: We investigate the Pólya problem on the sign conversion between the permanent and the determinant over finite fields. The main attention is given to the sufficient conditions which guarantee non-existence of sing-conversion. In addition we show that F3 is the only field with the property that any matrix with the entries from the field is convertible. As a result we obtain that over finite fields there are no analogs of the upper Gibson barrier for the conversion and establish the lower convertibility barrier.<br/><br/>Description: 2010 Mathematics Subject Classification: 15A15, 15A04.Some Numerical Invariants of Multilinear Identities
http://hdl.handle.net/10525/2801
Title: Some Numerical Invariants of Multilinear Identities<br/><br/>Authors: Giambruno, Antonio; Mishchenko, Sergey; Zaicev, Mikhail<br/><br/>Abstract: We consider non-necessarily associative algebras over a field of characteristic zero and their polynomial identities. Here we describe most of the results obtained in recent years on two numerical sequences that can be attached to the multilinear identities satisfied by an algebra: the sequence of codimensions and the sequence of colengths.<br/><br/>Description: 2010 Mathematics Subject Classification: Primary 16R10, 16A30, 16A50, 17B01, 17C05, 17D05, 16P90, 17A, 17D.Full Exposition of Specht's Problem
http://hdl.handle.net/10525/2800
Title: Full Exposition of Specht's Problem<br/><br/>Authors: Belov-Kanel, Alexei; Rowen, Louis; Vishne, Uzi<br/><br/>Abstract: This paper combines [15], [16], [17], and [18] to provide a detailed sketch of Belov’s solution of Specht’s problem for affine algebras over an arbitrary commutative Noetherian ring, together with a discussion of the general setting of Specht’s problem in universal algebra and some applications to the structure of T-ideals. Some illustrative examples are collected along the way.<br/><br/>Description: 2010 Mathematics Subject Classification: Primary: 16R10; Secondary: 16R30, 17A01, 17B01, 17C05.Central A-polynomials for the Grassmann Algebra
http://hdl.handle.net/10525/2799
Title: Central A-polynomials for the Grassmann Algebra<br/><br/>Authors: Pereira Brandão Jr., Antônio; José Gonçalves, Dimas<br/><br/>Abstract: Let F be an algebraically closed field of characteristic 0, and let G be the infinite dimensional Grassmann (or exterior) algebra over F. In 2003 A. Henke and A. Regev started the study of the A-identities. They described the A-codimensions of G and conjectured a finite generating set of the A-identities for G. In 2008 D. Gonçalves and P. Koshlukov answered in the affirmative their conjecture. In this paper we describe the central A-polynomials for G.<br/><br/>Description: 2010 Mathematics Subject Classification: 16R10, 16R40, 16R50.Outer Automorphisms of Lie Algebras related with Generic 2×2 Matrices
http://hdl.handle.net/10525/2798
Title: Outer Automorphisms of Lie Algebras related with Generic 2×2 Matrices<br/><br/>Authors: Fındık, Şehmus<br/><br/>Abstract: Let Fm = Fm(var(sl2(K))) be the relatively free algebra of rank m in the variety of Lie algebras generated by the algebra sl2(K) over a field K of characteristic 0. Our results are more precise for m = 2 when F2 is isomorphic to the Lie algebra L generated by two generic traceless 2 × 2 matrices. We give a complete description of the group of outer automorphisms of the completion L^ of L with respect to the formal power series topology and of the related associative algebra W^. As a consequence we obtain similar results for the automorphisms of the relatively free algebra F2/F2^(c+1) = F2(var(sl2(K)) ∩ Nc) in the subvariety of var(sl2(K)) consisting of all nilpotent algebras of class at most c in var(sl2(K)) and for W/W^(c+1). We show that such automorphisms are Z2-graded, i.e., they map the linear combinations of elements of odd, respectively even degree to linear combinations of the same kind.<br/><br/>Description: 2010 Mathematics Subject Classification: 17B01, 17B30, 17B40, 16R30.Asymptotic behaviour of Functional Identities
http://hdl.handle.net/10525/2797
Title: Asymptotic behaviour of Functional Identities<br/><br/>Authors: Gordienko, A. S.<br/><br/>Abstract: We calculate the asymptotics of functional codimensions fcn(A) and generalized functional codimensions gfc n (A) of an arbitrary not necessarily associative algebra A over a field F of any characteristic. Namely, fcn(A) ∼ gfcn(A) ∼ dim(A^2) · (dim A^n) as n → ∞ for any finite-dimensional algebra A. In particular, codimensions of functional and generalized functional identities satisfy the analogs of Amitsur’s and Regev’s conjectures. Also we precisely evaluate fcn(UT2(F)) = gfcn(UT2(F)) = 3^(n+1) − 2^(n+1).<br/><br/>Description: 2010 Mathematics Subject Classification: Primary 16R60, Secondary 16R10, 15A03, 15A69.Varieties of Superalgebras of Polynomial Growth
http://hdl.handle.net/10525/2796
Title: Varieties of Superalgebras of Polynomial Growth<br/><br/>Authors: La Mattina, Daniela<br/><br/>Abstract: Let V^gr be a variety of associative superalgebras over a field F of characteristic zero. It is well-known that V gr can have polynomial or exponential growth. Here we present some classification results on varieties of polynomial growth. In particular we classify the varieties of at most linear growth and all subvarieties of the varieties of almost polynomial growth.<br/><br/>Description: 2010 Mathematics Subject Classification: 16R10, 16W55, 16P90.Characterization of Certain T-ideals from the view point of representation theory of the Symmetric Groups
http://hdl.handle.net/10525/2795
Title: Characterization of Certain T-ideals from the view point of representation theory of the Symmetric Groups<br/><br/>Authors: Volichenko, I. B.; Zalesskii, A. E.<br/><br/>Abstract: Let K[X] be a free associative algebra (without identity) over a field K of characteristic 0 with free generators X = (X1, X2, ...), and let Pn be the set of all multilinear elements of degree n in K[X]. Then Pn is a KSn-module, where KSn is the group algebra of the symmetric group Sn. An ideal of K[X] stable under all endomorphisms of K[X] is called a T-ideal. If L is an arbitrary T-ideal of K[X] then Ln := Pn ∩ L is a KSn-module too. An important task in the theory of varieties of algebras is to reveal general regularities in the behavior of the sequence A n for various T-ideals A. In certain cases, given a property P, say, of the sequence, one can find a T-ideal L(P) such that a T-ideal L′ satisfies P if and only if L′ contains L(P). The results of this paper have to be regarded from this point of view.<br/><br/>Description: 2010 Mathematics Subject Classification: 08B20, 16R10, 16R40, 20C30.A Unified approach to the Structure Theory of PI-Rings and GPI-Rings
http://hdl.handle.net/10525/2794
Title: A Unified approach to the Structure Theory of PI-Rings and GPI-Rings<br/><br/>Authors: Brešar, Matej<br/><br/>Abstract: We give short proofs, based only on basic properties of the extended centroid of a prime ring, of Martindale’s theorem on prime GPI-rings and (a strengthened version of) Posner’s theorem on prime PI-rings.<br/><br/>Description: 2010 Mathematics Subject Classification: 16R20, 16R50, 16R60, 16N60.Gradings and Graded Identities for the Matrix Algebra of Order Two in Characteristic 2
http://hdl.handle.net/10525/2793
Title: Gradings and Graded Identities for the Matrix Algebra of Order Two in Characteristic 2<br/><br/>Authors: Koshlukov, Plamen; César dos Reis, Júlio<br/><br/>Abstract: Let K be an infinite field and let M2(K) be the matrix algebra oforder two over K. The polynomial identities of M2(K) are known wheneverthe characteristic of K is different from 2. The algebra M2(K) admits anatural grading by the cyclic group of order 2; the graded identities forthis grading are known as well. But M2(K) admits other gradings thatdepend on the field and on its characteristic. Here we describe the gradedidentities for all nontrivial gradings by the cyclic group of order 2 when thecharacteristic of K equals 2. It turns out that there is only one grading toconsider. This grading is not elementary. On the other hand the gradedidentities are the same as for the elementary grading.<br/><br/>Description: 2010 Mathematics Subject Classification: 16R10, 16R99, 16W50.Computing with Rational Symmetric Functions and Applications to Invariant Theory and PI-algebras
http://hdl.handle.net/10525/2792
Title: Computing with Rational Symmetric Functions and Applications to Invariant Theory and PI-algebras<br/><br/>Authors: Benanti, Francesca; Boumova, Silvia; Drensky, Vesselin; K. Genov, Georgi; Koev, Plamen<br/><br/>Abstract: Let K be a field of any characteristic. Let the formal power series f(x1, ..., xd) = ∑ αnx1^n1 ··· xd^nd = ∑ m(λ)Sλ(x1, ..., xd), αn, m(λ) ∈ K, be a symmetric function decomposed as a series of Schur functions. When f is a rational function whose denominator is a product of binomials of the form 1−x1^a1 ··· xd^ad, we use a classical combinatorial method of Elliott of 1903further developed in the Ω-calculus (or Partition Analysis) of MacMahon in 1916 to compute the generating function XM(f;x1, ..., xd ) = ∑ m(λ)x1^λ1 ··· xd^λd, λ = (λ1, ..., λd). M is a rational function with denominator of a similar form as f. We apply the method to several problems on symmetric algebras, as well as problems in classical invariant theory, algebras with polynomial identities, and noncommutative invariant theory.<br/><br/>Description: 2010 Mathematics Subject Classification: 05A15, 05E05, 05E10, 13A50, 15A72, 16R10, 16R30, 20G05