Serdica Mathematical Journal
Volume 38, Numbers 13, 2012
This issue contains the Proceedings
of the International Workshop
Polynomial Identities in Algebras. II
which was held on September 26, 2011,
at the Memorial University of Newfoundland in St. John's, NL, Canada.
It is also dedicated to the 65th birthday of
Professor Yuri Bahturin.
C O N T E N T S
·
Preface (pp. iii)
·
Scientific Program (pp. iiiiv)
·
List of Participants (pp. vvii)
·
Professor Yuri Bahturin (on his 65th birthday) (pp. viiviii)
·
Students of Yuri Bahturin (pp. ixx)
·
List of Books, Papers, Translated Books and Edited Volumes
of Yuri Bahturin (pp. xixxii)
·
Bahturin, Yu.
Group gradings on free algebras of nilpotent varieties of algebras
(pp. 16)
·
Elduque, A., M. Kochetov.
Weyl groups of fine gradings on simple Lie algebras of types A, B, C and D
(pp. 736)
·
Parmenter, M. M.
Pancyclic Cayley graphs
(pp. 3742)
·
Centrone, L.
On some recent results about the graded GelfandKirillov dimension of graded PIalgebras
(pp. 4368)
·
Berele, A.
Some questions about products of verbally prime Tideals
(pp. 6978)
·
Rashkova, Ts.
On the nilpotency in matrix algebras with Grassmann entries
(pp. 7990)
·
Bremner, M. R.
Algebras, dialgebras, and polynomial identities
(pp. 91136)
·
Benanti, F., S. Boumova, V. Drensky, G. K. Genov, P. Koev.
Computing with rational symmetric functions and applications to invariant theory and PIalgebras
(pp. 137188)
·
Koshlukov, P., J. César dos Reis.
Gradings and graded identities for the matrix algebra of order two in characteristic 2
(pp. 189198)
·
Bresar, M.
A unified approach to the structure theory of PIrings and GPIrings
(pp. 199210)
·
La Mattina, D.
Varieties of superalgebras of polynomial growth
(pp. 237258)
·
Gordienko, A. S.
Asymptotic behaviour of functional identities
(pp. 259272)
·
Findik, S.
Outer automorphisms of Lie algebras related with generic 2×2 matrices
(pp. 273296)
·
Brandão, A. P. Jr., D. José Gonçalves.
Central Apolynomials for the Grassmann algebra
(pp. 297312)
·
BelovKanel, A., L. Rowen, U. Vishne.
Full exposition of Specht's problem
(pp. 313370)
·
Giambruno, A., S. Mishchenko, M. Zaicev.
Some numerical invariants of multilinear identities
(pp. 371394)
·
Budrevich, M. V., A. E. Guterman.
On the Gibson bounds over finite fields
(pp. 395416)
·
da Silva, V. R. T.
On ordinary and Z_{2}graded polynomial identities of the Grassmann algebra
(pp. 417432)
·
Tvalavadze, M.
Universal enveloping algebras of nonassociative structures
(pp. 433462)
·
Lomond, J.
Growth functions of F_{r}sets
(pp. 463472)
·
Finogenova, O.
Characterizing nonmatrix properties of varieties of algebras in the language of forbidden objects
(pp. 473496)
·
Koshlukov, P., A. Krasilnikov.
A basis for the graded identities of the pair (M_{2}(K), gl_{2}(K))
(pp. 497506)
A B S T R A C T S
GROUP GRADINGS ON FREE ALGEBRAS OF NILPOTENT VARIETIES OF ALGEBRAS
Yuri Bahturin
bahturin@mun.ca
2010 Mathematics Subject Classification:
Primary 16W50, 17B70; Secondary 16R10.
Key words:
graded algebra, nilpotent Lie algebra, grading.
The main result is the classification, up to isomorphism, of all gradings by arbitrary abelian groups on the finitely generated algebras that are free in a nilpotent variety of algebras over an algebraically closed field of characteristic zero.
WEYL GROUPS OF FINE GRADINGS ON SIMPLE LIE ALGEBRAS OF TYPES A, B, C AND D
Alberto Elduque
elduque@unizar.es,
Mikhail Kochetov
mikhail@mun.ca
2010 Mathematics Subject Classification:
Primary 17B70, secondary 17B40, 16W50.
Key words:
Graded algebra, fine grading, Weyl group, simple Lie algebra.
Given a grading G:L = Å_{g Î G}L_{g} on a nonassociative algebra \cL by an abelian group G, we have two subgroups of Aut(L): the automorphisms that stabilize each component L_{g} (as a subspace) and the automorphisms that permute the components. By the Weyl group of G we mean the quotient of the latter subgroup by the former. In the case of a Cartan decomposition of a semisimple complex Lie algebra, this is the automorphism group of the root system, i.e., the socalled extended Weyl group. A grading is called fine if it cannot be refined. We compute the Weyl groups of all fine gradings on simple Lie algebras of types A, B, C and D (except D_{4}) over an algebraically closed field of characteristic different from 2.
PANCYCLIC CAYLEY GRAPHS
M. M. Parmenter
mparmen@mun.ca
2010 Mathematics Subject Classification:
Primary
05C25. Secondary 20K01, 05C45.
Key words:
Cayley graph, Pancyclic, Abelian group.
Let Cay(G;S) denote the Cayley graph
on a finite group G with connection set S. We extend two
results about the existence of cycles in Cay(G;S) from cyclic
groups to arbitrary finite Abelian groups when S is a ``natural''
set of generators for G.
ON SOME RECENT RESULTS ABOUT THE GRADED GELFANDKIRILLOV DIMENSION OF GRADED PIALGEBRAS
Lucio Centrone
centrone@ime.unicamp.br
2010 Mathematics Subject Classification:
16R10, 16W55, 15A75.
Key words:
graded PIalgebras, GelfandKirillov dimension.
We survey some recent results on graded GelfandKirillov dimension of PIalgebras over a field F of characteristic 0. In particular, we focus on verbally prime algebras with the grading inherited by that of Vasilovsky and upper triangular matrices, i.e., UT_{n}(F), UT_{n}(E) and UT_{a,b}(E), where E is the infinite dimensional Grassmann algebra.
SOME QUESTIONS ABOUT PRODUCTS OF VERBALLY PRIME TIDEALS
Allan Berele
aberele@condor.depaul.edu
2010 Mathematics Subject Classification:
16R10.
Key words:
incidence algebras, polynomial identities, verbally prime Tideals.
In [1] we studied identities of finite dimensional incidence algebras and showed how they were gotten by products and intersections of identities of matrices and we left open the question of when two incidence algebras satisfy the same identities, a problem which is still open. In the current paper we revisit this problem: We describe it, give some partial results and some related problems based on the work of Kemer.
ON THE NILPOTENCY IN MATRIX ALGEBRAS WITH GRASSMANN ENTRIES
Tsetska Rashkova
tsrashkova@uniruse.bg
2010 Mathematics Subject Classification:
16R10, 15A75, 16S50.
Key words:
Polynomial identity, upper triangular matrices, nilpotent
commutators, index of nilpotency.
In the paper we consider some classes of subalgebras of M_{n}(E) (for a given n and any n) for E being the Grassmann algebra. We give an estimation of the index of nilpotency of the commutators of length 2 for these classes of matrix algebras.
ALGEBRAS, DIALGEBRAS, AND POLYNOMIAL IDENTITIES
Murray R. Bremner
bremner@math.usask.ca
2010 Mathematics Subject Classification:
Primary 17A30. Secondary 16R10, 1708, 17A32, 17A40, 17A50, 17B60, 17C05, 17D05, 17D10.
Key words:
Algebras, triple systems, dialgebras, triple disystems,
polynomial identities, multilinear operations, computer algebra.
This is a survey of some recent developments in the theory of associative and nonassociative dialgebras, with an emphasis on polynomial identities and multilinear operations. We discuss associative, Lie, Jordan, and alternative algebras, and the corresponding dialgebras; the KP algorithm for converting identities for algebras into identities for dialgebras; the BSO algorithm for converting operations in algebras into operations in dialgebras; Lie and Jordan triple systems, and the corresponding disystems; and a noncommutative version of Lie triple systems based on the trilinear operation abcbca. The paper concludes with a conjecture relating the KP and BSO algorithms, and some suggestions for further research. Most of the original results are joint work with Raúl Felipe, Luiz A. Peresi, and Juana SánchezOrtega.
COMPUTING WITH RATIONAL SYMMETRIC FUNCTIONS AND APPLICATIONS TO INVARIANT THEORY
AND PIALGEBRAS
Francesca Benanti
fbenanti@math.unipa.it,
Silvia Boumova
silvi@math.bas.bg,
Vesselin Drensky
drensky@math.bas.bg,
Georgi K. Genov
guenovg@mail.bg,
Plamen Koev
plamen.koev@sjsu.edu
2010 Mathematics Subject Classification:
05A15, 05E05, 05E10, 13A50, 15A72, 16R10, 16R30, 20G05.
Key words:
Rational symmetric functions, MacMahon partition analysis, Hilbert series, classical invariant theory, noncommutative invariant theory, algebras with polynomial identity, cocharacter sequence.
Let K be a field of any characteristic. Let the formal power series
f(x_{1},¼,x_{d}) = 
å
 a_{n}x_{1}^{n1}¼x_{d}^{nd} = 
å
 m(l)S_{l}(x_{1},¼,x_{d}), a_{n},m(l) Î 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 1x_{1}^{a1}¼x_{d}^{ad},
we use a classical combinatorial method of Elliott of 1903
further developed in the Wcalculus (or Partition Analysis)
of MacMahon in 1916 to compute the generating function
M(f;x_{1},¼,x_{d}) = 
å
 m(l)x_{1}^{l1}¼x_{d}^{ld}, l = (l_{1},¼,l_{d}). 

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.
GRADINGS AND GRADED IDENTITIES FOR THE MATRIX ALGEBRA OF ORDER TWO IN CHARACTERISTIC 2
Plamen Koshlukov
plamen@ime.unicamp.br,
Júlio César dos Reis
jcreis@ime.unicamp.br
2010 Mathematics Subject Classification:
16R10, 16R99, 16W50.
Key words:
gradings on matrix algebras, graded identities, polynomial identities in characteristic two.
Let K be an infinite field and let M_{2}(K) be the matrix algebra of
order two over K. The polynomial identities of M_{2}(K) are known
whenever the characteristic of K is different from 2. The algebra
M_{2}(K) admits a natural grading by the cyclic group of order 2; the
graded identities for this grading are known as well. But M_{2}(K)
admits other gradings that depend on the field and on its
characteristic. Here we describe the graded identities for all
nontrivial gradings by the cyclic group of order 2 when the
characteristic of K
equals 2. It turns out that there is only one grading to
consider. This grading is not elementary. On the other hand
the graded identities are the same as for the elementary grading.
A UNIFIED APPROACH TO THE STRUCTURE THEORY OF PIRINGS AND GPIRINGS
Matej Bresar
matej.bresar@fmf.unilj.si
2010 Mathematics Subject Classification:
16R20, 16R50, 16R60, 16N60.
Key words:
polynomial identity, generalized polynomial identity, functional identity, prime ring, extended centroid, symmetric Martindale ring of quotients.
We give short proofs, based only on basic properties of the extended centroid of a prime ring, of Martindale's theorem on prime GPIrings and (a strengthened version of) Posner's theorem on prime PIrings.
CHARACTERIZATION OF CERTAIN TIDEALS
FROM THE VIEW POINT OF REPRESENTATION THEORY
OF THE SYMMETRIC GROUPS
A. E. Zalesskii
alexandre.zalesski@gmail.com
2010 Mathematics Subject Classification:
08B20, 16R10, 16R40, 20C30.
Key words:
Tideals, Free associative algebras.
Let K[X] be a free associative algebra (without
identity) over a field K of characteristic 0 with free
generators X = (X_{1},X_{2},¼), and let P_{n} be the set of all
multilinear elements of degree n in K[X]. Then P_{n} is a
KS_{n}module, where KS_{n} is the group algebra of the symmetric
group S_{n}. An ideal of K[X] stable under all endomorphisms of
K[X] is called a Tideal. If L is an arbitrary Tideal of
K[X] then L_{n}: = P_{n}ÇL is a KS_{n}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 Tideals A. In certain cases, given a property P,
say, of the sequence, one can find a Tideal L(P) such that a
Tideal L¢ satisfies P L¢ contains L(P).
The results of this paper have to be regarded from this point
of view.
Let m be a natural number, and let R_{n}^{(m)} (respectively,
R_{n}^{(m)}), n > m, be the set of all KS_{n}modules whose
restriction to the subgroup S_{nm} contains an KS_{nm}module labeled by the partition [nm] (respectively,
[1^{nm}]) of nm. We define the property P^{m}
(respectively, P^{m}) by the condition that L_{n} contains
no submodule isomorphic to a module in the set R_{n}^{(m)} (respectively,
R_{n}^{(m)}). Set [a,b] = abba and [a,b,c] = [a,[b,c]] for
a,b,c Î K[X]. We proof that the Tideal L(P^{m})
(respectively, L(P^{m})) coincides with the Tideal
generated by the polynomial
d_{m+1}(X): = [X_{1},X_{2}]¼[X_{2m+1},X_{2m+2}],
(respectively
t_{m+1}(X) = [X_{1},X_{2},X_{3}]¼[X_{3m+1},X_{3m+2},X_{3m+3}]. One can interpret the result as a
characterization of the Tideal generated by d_{m+1}(X) (respectively
t_{m+1}(X)) by the property P^{m} (respectively,
P^{m}).
VARIETIES OF SUPERALGEBRAS OF POLYNOMIAL GROWTH
Daniela La Mattina
daniela.lamattina@unipa.it
2010 Mathematics Subject Classification:
16R10, 16W55, 16P90.
Key words:
polynomial identity, growth, superalgebra.
Let V^{gr} be a variety of associative superalgebras
over a field F of characteristic zero. It is wellknown 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.
ASYMPTOTIC BEHAVIOUR OF FUNCTIONAL IDENTITIES
A. S. Gordienko
asgordienko@mun.ca
2010 Mathematics Subject Classification:
Primary 16R60, Secondary 16R10, 15A03, 15A69.
Key words:
Functional identity, generalized functional identity, codimension, growth, algebra, Amitsur's conjecture, Regev's conjecture.
We calculate the asymptotics of functional codimensions fc_{n}(A) and generalized functional codimensions gfc_{n}(A) of an arbitrary not necessarily associative algebra A over a field F of any characteristic. Namely, fc_{n}(A) ~ gfc_{n}(A) ~ dim(A^{2}) ·(dimA)^{n} as n® ¥ for any finitedimensional 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 fc_{n}(UT_{2}(F)) = gfc_{n}(UT_{2}(F)) = 3^{n+1}2^{n+1}.
OUTER AUTOMORPHISMS OF LIE ALGEBRAS RELATED WITH GENERIC 2×2 MATRICES
Sehmus Findik
sfindik@cu.edu.tr
2010 Mathematics Subject Classification:
17B01, 17B30, 17B40, 16R30.
Key words:
free Lie algebras, generic matrices, inner automorphisms, outer automorphisms.
Let F_{m} = F_{m}(var (sl_{2}(K))) be the relatively free algebra of rank m in the variety of Lie algebras generated by the algebra sl_{2}(K) over a field K of characteristic 0. Our results are more precise for m = 2 when F_{2} 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 F_{2}/F_{2}^{c+1} = F_{2}(var (sl_{2}(K))ÇN_{c})
in the subvariety of var (sl_{2}(K)) consisting of all
nilpotent algebras of class at most c in var (sl_{2}(K)) and for W/W^{c+1}. We show that such automorphisms are Z_{2}graded, i.e., they map the linear combinations of elements of odd, respectively even degree to linear combinations of the same kind.
CENTRAL APOLYNOMIALS FOR THE GRASSMANN ALGEBRA
Antônio Pereira Brandão Jr.
brandao@dme.ufcg.edu.br,
Dimas José Gonçalves
dimasjog@gmail.com
2010 Mathematics Subject Classification:
16R10, 16R40, 16R50.
Key words:
Aidentity, central Apolynomial, Grassmann algebra.
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 Aidentities. They described the Acodimensions of G and conjectured a finite generating set of the Aidentities for G. In 2008 D. Gonçalves and P. Koshlukov answered in the affirmative their conjecture. In this paper we describe the central Apolynomials for G.
FULL EXPOSITION OF SPECHT'S PROBLEM
Alexei BelovKanel
belova@macs.biu.ac.il,
Louis Rowen
rowen@macs.biu.ac.il,
Uzi Vishne
vishne@macs.biu.ac.il
2010 Mathematics Subject Classification:
Primary: 16R10; Secondary: 16R30, 17A01, 17B01, 17C05.
Key words:
Specht's question, polynomial identities, Tideal,
affine algebra, representable algebra, torsion, Noetherian.
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 Tideals. Some
illustrative examples are collected along the way.
SOME NUMERICAL INVARIANTS OF MULTILINEAR IDENTITIES
Antonio Giambruno antonio.giambruno@unipa.it,
Sergey Mishchenko mishchenkosp@ulsu.ru,
Mikhail Zaicev
zaicevmv@mail.ru
2010 Mathematics Subject Classification:
Primary 16R10, 16A30, 16A50, 17B01, 17C05, 17D05, 16P90, 17A, 17D.
Key words:
Polynomial identity, codimensions, colengths.
We consider nonnecessarily 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.
ON THE GIBSON BOUNDS OVER FINITE FIELDS
Mikhail V. Budrevich mbudrevich@yandex.ru,
Alexander E. Guterman guterman@list.ru
2010 Mathematics Subject Classification:
15A15, 15A04.
Key words:
permanent, determinant, finite fields, Pólya problem.
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 nonexistence of singconversion. In addition we show that F_{3} 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.
ON ORDINARY AND $\Z_2$GRADED POLYNOMIAL IDENTITIES OF THE GRASSMANN ALGEBRA
Viviane Ribeiro Tomaz da Silva viviane@mat.ufmg.br
2010 Mathematics Subject Classification:
Primary: 16R10, Secondary: 16W55.
Key words:
Ordinary polynomial identities, graded polynomial identities, cocharacters,
codimensions, Grassmann algebra.
The main purpose of this paper is to provide a survey of results
concerning the ordinary and Z_{2}graded polynomial identities of
the infinite dimensional Grassmann algebra over a field of
characteristic zero, as well as of its sequences of ordinary and
Z_{2}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 PItheory.
UNIVERSAL ENVELOPING ALGEBRAS OF NONASSOCIATIVE STRUCTURES
Marina Tvalavadze mtvalava@fields.utoronto.ca
2010 Mathematics Subject Classification:
Primary 17D15. Secondary 17D05, 17B35, 17A99.
Key words:
Malcev algebras, Leibniz algebras, triple systems, universal
enveloping
algebras, PBWbases.
This is a survey paper to summarize the latest results on the universal enveloping algebras of Malcev algebras, triple systems and Leibniz nary algebras.
GROWTH FUNCTIONS OF F_{r}SETS
Jonny Lomond jjl831@mun.ca
2010 Mathematics Subject Classification:
05C30, 20E08, 20F65.
Key words:
Group action, growth function.
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.
CHARACTERIZING NONMATRIX PROPERTIES OF VARIETIES OF ALGEBRAS IN THE LANGUAGE OF FORBIDDEN OBJECTS
Olga Finogenova olgafinogenova@gmail.com
2010 Mathematics Subject Classification:
16R10, 16R40.
Key words:
algebras with polynomial identity, nonmatrix varieties, Engel property, Lie nilpotency, semigroup identities,
forbidden algebras, varieties of associative algebras.
We discuss characterizations of some nonmatrix 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.
A BASIS FOR THE GRADED IDENTITIES OF THE PAIR (M_{2}(K), gl_{2}(K))
Plamen Koshlukov plamen@ime.unicamp.br,
Alexei Krasilnikov alexei@unb.br
2010 Mathematics Subject Classification:
16R10, 17B01.
Key words:
Graded identities, weak identities, basis of graded identities.
Let M_{2}(K) be the algebra of 2 ×2 matrices over an
infinite integral domain K. In this note we describe a basis for
the \mathbbZ_{2}graded identities of the pair
(M_{2}(K),gl_{2}(K)).
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