polynomial identities in algebras

POLYNOMIAL IDENTITIES IN ALGEBRAS

POLYNOMIAL IDENTITIES IN ALGEBRASAbstracts of International Workshop at Memo-rial University of Newfoundland, August29, 2002 - September 3, 2002

Edited byYURI BAHTURINMemorial University of NewfoundlandSt. John’s, CanadaandMoscow State UniversityMoscow, Russia

Kluwer Academic PublishersBoston/Dordrecht/London

Contents

Preface vii

Generalized Lie Solvability of Associative Algebras 1Yuri Bahturin

Applications of the group identities theory to the group of unitsof group algebras 3

Adalbert Bovdi

Symmetric units and group identities 5Victor Bovdi

Quantization of Lie and Jordan triple systems 7Murray Bremner

The Ring of Generic Matrices 9Edward Formanek

Standard polynomials, Capelli polynomials and asymptotics 11Antonio Giambruno

Conjecture of Procesi 13Alexander Kemer

Identities of the smash product of a group algebra and the universal envelopeof a Lie superalgebra 15

M.V.Kochetov

Identities in certain algebras in positive characteristic 17Plamen Koshlukov

Quantum hyperalgebras of rank � 19Len Krop

Knuth-Robinson-Schensted correspondence and Weak Polynomial Identitiesof�� 21

Roberto La Scala

Specht’ Problem In Positive Characteristic 23Viktor N. Latyshev

Almost Polynomial Growth Varieties of Linear Algebras 25S.P. Mishchenko

Proper subvarieties of�� 27Vincenzo Nardozza

Polynomial Identities and Reducibility of Semigroups 29Heydar Radjavi

A polynomial problem arising from �-designs 31Rolf Rees

Explicit decomposition of the group algebra �� of the alternating group�� 33Amitai Regev

Computing with matrices over the Grassmann algebra 35Jeno Szigeti

Gradings and graded identities of the algebra of �� upper triangular matrices 37

Angela Valenti

Numerical characteristics and extremal varieetis 39

Mikhail Zaicev

Preface

Dear guests! It is a pleasure to see you all in St. John’s, the capital of Cana-dian province of Newfoundland and Labrador. The International Workshop“Polynomial Identities in Algebras” is hosted by the Department of Mathe-matics and Statistics of Memorial University of Newfoundland, the east mostUniversity of North America. In this package you will find the the schedule oftalks and their abstracts, the list of participants and other useful materials.

THE EDITOR

��

��

9:30 - 10:30 �� Standard Polynomials, CapelliPolynomials And Asymptotics.

��

11:00 - 12:00 �� -Codimensions And Cocharacters.I

��

2:15 - 3:15 �� Behavior Of MultiplicitiesIn Cocharacters Of PI-algebras. I

3:30 - 4:00 !�� "#�� Computing With MatricesOver The Grassmann Algebra.

��

4:15 - 4:45 �� $�� -Polynomial Identities Of Matrices:The Low Degrees.

4:50 - 5:20 %� &�� &'�� Polynomial IdentitiesAnd Reducibility of Semigroups.

(�� & �� )�� * + , )-�- �� $

�� .� ��

9:30 - 10:30 ��/�&�� 0�� Conjecture of Procesi

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11:00 - 12:00 1& 2�� The ring of generic matrices.I

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2:15 - 3:15 �� Specht’s Problemin Positive Characteristic

3:30 - 4:00 �&�� &�� Applications of the groupidentities theory to the groupof units of group algebras.

��

4:15 - 4:45 �� Gradings and graded identitiesof the algebra of �� uppertriangular matrices.

4:50 - 5:20 �� A polynomial problem arisingfrom �-designs.

5:25 - 5:55 �� "�� Knuth-Robinson-Schenstedcorrespondence and Weak PolynomialIdentities of ��

(��) �� *�. )-�- � �� 2��

�� .3� ��

�� "��& "�� )� "�- 4�� -"�� 3� �� 5�� 6 � 4�� &-

"�)�� 3� ��

9:30 - 10:30 4�� 7�� Numerical characteristicsand extremal varieties. I.

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11:00 - 12:00 �� -codimensions and cocharacters.II

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2:15 - 3:15 �� Combinatorics of Wordsand Constructions in Ring Theory.II.

3:30 - 4:00 �� &�� Symmetric units and group identities

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4:15 - 4:45 ��& 0��)� Quantum Hyperalgebra of Rank 1.

4:50 - 5:20 4�� Quantization of Lieand Jordan triple systems.

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9:30 - 10:30 8�� Generalized Lie Solvabilityof Associative Algebras. I.

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11:00 - 12:00 1& 2�� The Ring Of Generic Matrices.II

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(�� & �� 5�� * + , )-�- �� -

"�)�� .� ��

9:30 - 10:30 4�� 7�� Numerical characteristicsand extremal varieties. II.

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11:00 - 12:00 "�� 4�� Almost Polynomial GrowthVarieties of Linear Algebras.

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2:15 - 3:15 8�� Generalized Lie Solvabilityof Associative Algebras. II.

3:30 - 4:00 4�� 0�� Identities of the smash productof a group algebra and the universalenvelope of a Lie superalgebra.

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4:15 - 4:45 ��#� 9��&�##�� Proper subvarieties of��.

4:50 - 5:20 5�� 0�� Identities in certain algebrasin positive characteristic.

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GENERALIZED LIE SOLVABILITYOF ASSOCIATIVE ALGEBRAS

Yuri BahturinDepartment of Mathematics and Statistics

Memorial University of Newfoundland

St. John’s, NF, A1C 5S7, Canada

[email protected]

These lectures will be devoted to discussing some new methods and ap-proaches to polynomial identities in associative algebras arising from recentstudies on bicharacters and twisting in algebras with actions and coactions ofHopf algebras, in the spirit of the works of M. Scheunert (Bahturin - Fischman- Montgomery, Etingof - Zhelaki, etc.), and the classification results aboutgroup gradings of matrix algebras (Bahturin - Sehgal - Zaicev). Combiningthese results, in two joint papers, one with M. Parmenter and the other withS. Montgomery, and M. Zaicev, we can give answers to the questions aboutthe generalized Lie structure of certain associative algebras. Given a skew-symmetric bichacrater � � � �� of a Hopf algebra � over a field and an associative algebra� in the category�� of right�-comodule algebrasone can define a new operation, called the �-bracket, as follows:

��

��

where we use standard notation for the comodule map � � �� :

��

�� where � ��

In the case where� is a cotriangular Hopf algebra we have that �� is a gen-eralized Lie algebras in the sense that it satisfies generalized anti-commutativity(�-anti-commutativity)

��

��

and �-Jacobi identity

��

� ��

In this lecture we consider generalized Lie algebra structures on graded asso-ciative algebras, that is when� � ��,� an abelian group. We are interestedin the situation where such structures are solvable and even commutative. Wefirst formulate some theorems in the case of solvable structures on semi prime,prime or simple associative algebras. Then we describe finite-dimensional as-sociative algebras over an algebraically closed field of characteristic zero gradedby a finite elementary abelian group which are generalized commutative undera skew-symmetric bicharacter on the grading group. A special case is where �is a group algebra (of some other group) itself and we give some informationabout this case, as well.

References

[1] Bahturin, Y., Fischman, D., Montgomery, S., Bicharacters, twistings, and Scheunert’stheorem for Hopf Algeras, J. Algebra, 236 (2001), 246 - 276.

[2] Bahturin, Y., Mikhalev, A., Petrogradsky, V., Zaicev, M., Infinite-Dimensional Lie Super-algebras. Walter De Gruyter Expositions in Math., 9 (1992).

[3] Bahturin. Y., Montgomery, S.,��-envelopes of Lie superalgebras, Proc. AMS, 127 (1999),2829 - 2839.

[4] Bahturin. Y., Montgomery, S., Zaicev M., Generalized Lie Solvability of Associative Alge-bras, In: “Groups, Rings, Lie and Hopf Algebras”. Proc. Int. Workshop held in St. John’s,Maay 28 - June 1, 2001, to appear.

[5] Bahturin,Y. and Parmenter, M., Generalized commutativity in group algebras, CanadianMath. Bulletin, to appear.

[6] Bahturin, Y. and Zaicev, M. Group gradings on matrix algebras, Canadian Math. Bulletin,to appear.

[7] Bahturin Y., Sehgal S., Zaicev M. Group gradings on associative algebras, J. Algebra,241(2001), 677 - 698.

[8] Scheunert, M., Generalized Lie algebras, J. Math. Physics, 20 (1979), 712 - 720.

APPLICATIONSOF THE GROUP IDENTITIES THEORYTOTHEGROUPOFUNITSOFGROUPALGEBRAS

Adalbert BovdiUniversity of Debrecen

Hungary

The interest for modular group algebra theory has appeared in 70-th due toresults of the D.Passman and some Lie properties of the group algebras weredescribed. In that time we find close connection between Lie properties of thegroup algebra and its group of units, and jointly with I.Khripta we described thegroup algebra of torsion groups with solvable group of units. The Lie methodsplayed a decisive role in the investigation group of units, as shows the researchdone by A.Shalev. The theory group of units has undergone significant changesin the past � year. The new progress in the theory of group identities, devotedby several authors, gives new results in the theory of group of units.

Using the theory of the group identities, we have obtained complete descrip-tion of the group algebras of arbitrary groups with either solvable or �-Engelcondition for the group of units. For example:

Let � be a group with infinite Sylow �-subgroup and � be the field ofcharacteristic �. The group of units �� is �-Engel if and only if the algebra�� is �-Engel i.e. � is a nilpotent group and� has a normal subgroup � suchthat the commutator subgroup �� is a finite �-group.

SYMMETRIC UNITS AND GROUP IDENTITIES

Victor BovdiUniversity of Debrecen

Hungary

Let �� be the group of units of a ring � with involution � and �� be the set of the symmetric units of �. A naturalquestion is to characterize those rings � such that the set �� of the sym-metric units satisfies a group identity. In our talk we discuss the propertiesof the group algebras �� of the groups � over the field � of characteristic�, whose symmetric units satisfies a group identity. In particular, we obtain ageneralization of results Giambruno-Sehgal-Valenti (see [1]) in case when ��is a either modular or semiprime group algebra of a non-torsion group � overa field � of characteristic �.

References

[1] Giambruno, A.; Sehgal, S. K.; Valenti, A., Symmetric units and group identities,Manuscripta Math. 96(1998), 443–461.

QUANTIZATION OF LIE AND JORDAN TRIPLESYSTEMS

Murray BremnerDepartment of Mathematics and Statistics

University of Saskatchewan

This talk will show how the decomposition of the group ring of the symmetricgroup into a direct sum of full matrix subrings can be used to give a completeclassification of n-ary operations. Roughly speaking, row equivalence of ma-trices corresponds to quasi-equivalence of operations. In particular, the Lieand Jordan products represent the two non-trivial quasi-equivalence classesof binary operations. For ternary operations, there are infinitely many quasi-equivalence classes, which divide into eight classes, and four infinite familiesof classes each with a single parameter. The Lie triple product is contained inone of the infinite classes, and the other operations in the class can be regardedas quantizations of that product. Similar remarks apply to the Jordan tripleproduct. For special values of the parameter, the operation satisfies an identityof degree 5. This identifies some new ternary operations which define varietiesof triple systems, similar to Lie and Jordan triple systems, which seem to be aninteresting direction for further research.

THE RING OF GENERIC MATRICES

Edward FormanekPennState University

The ring of � � � generic matrices and the generic division ring were firstnamed by Procesi in 1966, but they had already been studied by Amitsur in 1955as a relatively free ring and its Ore ring of fractions. They play an importantrole in the theory of PI-rings, the study of finite-dimensional division algebras,and the invariant theory of � � � matrices. My first talk will be a historicalsurvey. Although most of what is known about the ring of generic matriceswas established in the last fifty years, work of Sylvester, Fricke and Klein on� matrices in the nineteenth century anticipated some later developments.In the twentieth century, there were early relevant works by Dubnov in Russiaand a few physicists in the in the United States. My second talk will be aboutwhat I consider the most significant open quesion about generic matrices: Is thecenter of the �� generic division ring a rational function field? An answer,positive or negative, would have important consequences. A positive answerwould imply the Mercuriev-Suslin theorem that the Brauer group is generatedby cyclics in the presence of enough roots of unity. A negative answer wouldgive naturally occurring examples of unirational fields which are not rational.Progress on this question has been slow.

STANDARD POLYNOMIALS,CAPELLI POLYNOMIALS AND ASYMPTOTICS

Antonio GiambrunoDepartment of Mathematics and Applications

University of Palermo

Let�� be the algebra of � matrices over a field � of characteristiczero. It is well known that �� satisfies ��, the standard polynomial ofdegree , and ��, the �� -Capelli polynomial. For a polynomial� , let �� be the T-ideal of the free algebra generated by � . Even if ��and �� do not generate �� , the T-ideal of identities of �� ,there is a general feeling that �� and �� should be quite closeto �� .

We study the asymptotic behavior of the sequence of codimensions of theT-ideal generated by a standard or a Capelli polynomial and we compare it withthe corresponding asymptotics of �� . In this study two more algebrascome into play: the algebra of �� matrices having the last � rowsand the last columns equal to zero and the algebras of upper block-triangularmatrices.

CONJECTURE OF PROCESI

Alexander KemerUlyanovsk University

Let � � � are generated algebras of generic matrices of order � overthe ring of integers � and over an infinite field � of characteristic � respec-tively. The conjecture of C.Procesi [1]is well-known: A kernel of the canonicalepimorphism � � � is equal to �� .

Let � �� are given. Denote by � the natural homomorphism

��

The conjecture of C. Procesi can be also formulated in the following form: Ifthe algebra�� satisfies identity �� then � � �� modulo � �� for some polynomial � � �� .

In [2] C. Procesi has proved his conjecture for � � � � ! and every . In1985 W. Schelter [3] gave a negative answer to this conjecture for � � � � � .

Recently the authur gave a negative answer in every characteric.

Theorem 1 For every prime p there exist � � and multilinear polynomial� � ��"�such that the algebra�� satisfies identity �� but � �� modulo � �� for every polynomial � � ��"�.

We also proved some positive results in the case � � �.

Theorem 2 For � � �� and for every sufficiently large prime � theconjecture of Procesi has a positive answer.

Theorem 3 For every and for every sufficiently large prime � the followingstatement is true: If the algebra �� satisfies identity �� then thereexist some polynomial � with integer coefficients such that �� satisfies aweak identity � � ��.

References

[1] Procesi C. (1976) The invariant theory of � � � matrices, Adv. in Math., Vol 19, pp306-381.

[2] Procesi C. (1984) Computing with �� matrices, J. Algebra, Vol. 87, no 2, pp 342-359.

[3] Schelter W. F. (1985) On question concerning generic matrices over the integers, J.Algebra, Vol. 96, pp 48-53.

IDENTITIES OF THE SMASH PRODUCTOF A GROUP ALGEBRAAND THE UNIVERSAL ENVELOPEOF A LIE SUPERALGEBRA

M.V.KochetovMemorial University of Newfoundland

[email protected]

Suppose � is a Hopf algebra, � an �-module algebra. Then we can formthe smash product �� , which can be viewed as a “deformation” of theusual tensor product � � � (the latter arises when the action of � on � istrivial). In particular, if � is the group algebra of a group � that acts on �by automorphisms, then �� is the so called skew group ring of � withcoefficients in �.

We study the question when �� satisfies a polynomial identity. An obvi-ous necessary condition is that both � and� must be #� . In the case of tensorproduct, this is also sufficient (by a theorem of A.Regev). In general, we haveto take into account the structure of the action of� on�. To this end, we definean appropriate kind of delta-sets and obtain some necessary conditions on theaction of � on � in terms of these delta-sets for a certain class of algebras.

Then we specialize to the case when � is a group algebra, i.e. �� isa skew group ring. The existence of a polynomial identity for such �� inthe case of semiprime � was considered by D.Passman. Recently, Yu.Bahturinand V.Petrogradsky gave explicit necessary and sufficient conditions for ��to be #� , where � is the restricted envelope of a �-Lie algebra (hence notsemiprime in general). We apply our results on delta-sets, together with knownfacts about group algebras and universal envelopes, to find explicit necessaryand sufficient conditions for�� to be #� , where� is the universal envelopeof a Lie superalgebra.

IDENTITIES IN CERTAIN ALGEBRASIN POSITIVE CHARACTERISTIC

Plamen Koshlukov

We discuss identities in associative algebras over infinite fields of charac-teristic � �� . First, using descriptions of the 2-graded identities satisfied bythe algebras �� and � ��, we show that their T-ideals do not coincideif � �� . Here � is the unitary Grassmann algebra of an infinite dimensionalvector space. If one considers nonunitary Grassmann algebras, the above twoalgebras share the same T-ideal as it is the case in characteristic 0. Furthermorewe study graded identities of the algebras�� and��, and showthat these satisfy different graded identities if one considers certain natural grad-ings by the cyclic group of order 4. This fact holds in positive characteristiconly.

QUANTUM HYPERALGEBRAS OF RANK �

Len KropDePaul University

Chicago,Illinois 60614

[email protected]

We study the algebra � obtained via Lusztig’s "integral" form of the genericquantum algebar for the Lie algebra � � $�� modulo the two-sided ideal gener-ated by%��. We show that � is a smach product of the quantum deformationof the restricted universal enveloping algebra � of � modulo %� � � and theordinary universal enveloping algebra � of �. We show that the above smashproduct reduces to the tensor product modulo the nilpotent radical of �, and wecompute the primitive (� prime) ideals of� . Next we describe a decompositionof � into the simple � -submodules.For a primitive central idempotent & of �we show that &� is an indecomposabletwo sided ideal of � . The center of � is explicitely computed.Some sublattices of ideals of � are classified, notably the sublattice of cofiniteideals.

KNUTH-ROBINSON-SCHENSTEDCORRESPONDENCE AND WEAKPOLYNOMIAL IDENTITIES OF��

Roberto La ScalaUniversity of Bari

[email protected]

In this talk we show that the ideal �� of the weak polynomial identities of thesuperalgebra �� is generated by the proper polynomials �� and�� . This is proved for any infinite field � of characteristicdifferent from 2. Precisely, if ' is the subalgebra of the proper polynomialsof the free algebra � �"�, we determine a basis and the dimension of any mul-tihomogeneous component of the quotient algebra '�� '(' � ��. Wecompute also the Hilbert series of this algebra. The arguments we use for prov-ing such results are essentially the following. The first step consists in provingthat, by means of the identities �� and �� , a gen-erating set for '�� is given by products of commutators of length 2 whichsatisfy some combinatorial conditions. We call such “standard products” c-arrays. We develop a variant of the Knuth-Robinson-Schensted algorithmsthat establishes a correspondence between c-arrays and semistandard tableauxof double shape. We prove then that the multilinear components of the algebra'�� are isomorphic to the multilinear components of the invariant algebra ofthe orthogonal group )�� with respect to a suitable (multi)graduation. Forsuch algebra and �� , De Concini & Procesi proved in fact that thereexists a “standard basis” parametrized by semistandard tableaux of double shape(the standardity is here according to the french notation). By this result andour variant of the KRS- correspondence we are able to infer the linear indepen-dence of the c-arrays modulo �� in the multilinear case. Further combinatorialarguments allow to extend the independence to the multihomogeneous case.All computations and algorithms have been implemented by Maple.

SPECHT’ PROBLEMIN POSITIVE CHARACTERISTIC

Viktor N. LatyshevMoscow State University

Russia

[email protected]

In this lecture we are going to treat various aspects of the famous Specht’sProblem in zero and positive charcteristic. We will discuss both the classicalformulation of the problem in the language of � -ideals and its generalizationin the modern language of � -spaces. Our aim is the most recent results onSpecht’s Problem achieved by a number of mathematicians.

ALMOST POLYNOMIAL GROWTH VARIETIESOF LINEAR ALGEBRAS

S.P. MishchenkoDepartment of Algebra and Geometric Computations,

Faculty of Mathematics and Mechanics

Ul’yanovsk State University,

Ul’yanovsk, 432700, Russia

[email protected]

We study the behaviour of codimension sequence of polynomial identitiesof linear algebras over a field � of characteristic 0. We give known and newresults about linear algebras varieties with almost polynomial growth in thecase of associative (with or without involution, ��-graded), Lie and Leibnizalgebras. In the theory of associative algebras the basic role in this aria play theinfinite dimensional Grassmann algebra� and the algebra of � upper trian-gular matrices ��. Well known that there are only two varieties of associativealgebras with almost polynomial growth. The generate by algebras � and ��respectively. The name of author is A.Kemer. In the theory of associative alge-bras with involution the situation is the similar. There exist only two varietieswith almost polynomial growth. The first is the algebra �� with ex-change involution, that plays a role analogous to the role played by the infinitedimensional Grassmann algebra in the ordinary theory. The next herouz is thesome four dimensional algebra that plays a role analogous to the role playedby the algebra of � upper triangular matrices in the ordinary theory. Thenames of authors are A.Giambruno, S.Mishchenko and A.Valenty. In the the-ory of associative ��-graded algebras there exist only five varieties with almostpolynomial growth. Basic role play the infinite dimensional Grassmann algebrawith trivial or natural grading; the algebra � upper triangular matrices withtrivial or natural grading. The fifth variety generated by commutative algebra� � � � �� where �� with grading �� , �� . The namesof authors are A.Gambruno, S.Mishchenko, A.Valenty and M.Zaicev. In thetheory of varieties of Lie algebras there exist only four solvable varieties withalmost polynomial growth. So far only one example of a non solvable varietywith almost polynomial growth has been constructed: this is the variety gener-

ated by the 3-dimensional simple Lie algebra. The existence of new examplesof non soluble varieties with almost polynomial growth seems an interestingand hard problem in the theory of varieties of Lie algebras is very hard prob-lem. The names of authors are V.Drensky, S.Mishchenko, Yu.Razmuslov andI.Volichenko. By definition an algebra is Leibniz algebra if the product fromright side is the derivation. The correspond identity has the form

��*�+ � ��+�* � ��*+��

Similar to Lie case the author and L.Abanina (Ulyanovsk state university, Rus-sia) construct three new examples of the varieties with almost polynomialgrowth. These three varieties are the similar to the four solvable Lie algebrasvarieties but one pair from the Lie case give only one new example of Leibnizalgebras variety with almost polynomial growth. Give more information aboutthe last one. Denote by �� ,� -,- � , two-dimensional Leibniz algebra(the other product of basis elements equal to zero. Let � be an associativecommumative ring of polynomials in the variable � consider as algebra withzero table of multiplication. We turn � into an ��-module by setting

��, � �� - � �� ,�� -��

where the prime on the polynomial denomes the taking of the derivative. Let�� be direct product of vector spaces � and ��. Define the product as

�.� � ��.� � ��

where .��.� � ��, �� . The variety /0�� generated by algebra ��has almost polynomial growth.

PROPER SUBVARIETIES OF��

Vincenzo NardozzaItaly

Let � be a field of characteristic zero, and �� the variety of associativeunitary algebras generated by the polynomial identity �� . Wedescribe asymptotically its lattice of the proper subvarieties. More precisely,we define certain algebras �� for any � � and show that if � is a propersubvariety of ��, then its � -ideal is asymptotically equivalent to the � -idealof � , �, �� or �� , for a suitable � � .

POLYNOMIAL IDENTITIES AND REDUCIBILITYOF SEMIGROUPS

Heydar RadjaviDalhousie University

Halifax

Canada

We are interested in the effect of polynomial identities on > reducibilityand triangulaizability of multiplicative semigroups of linear operators. Weshall confine ourselves to finite dimensions, although our questions can also beasked and answered for compact operators on banach spaces. For a specificcase, let S denote a semigroup of operators on a finite-dimensional space andconsider the question: What polynomials � in two noncommuting variableshave the property that if some power of � is identically zero on �, then � issimultaneously triangularizable? A known example of such a polynomial, givenby a theorem of Guralnick, is �� *� � �* � *�. We discuss several results,including generalizations of this theorem, and some examples of interest.

A POLYNOMIAL PROBLEM ARISING FROM � -DESIGNS

Rolf ReesDept. of Mathematics and Statistics

Memorial University of Newfoundland

A �-wise balanced design of type �� /�%� 1� is a pair �"�'� where" is a/-element set of points and' is a collection of sub-sets of" called blocks suchthat every �-element subset of" is contained in exactly1 blocks and the numberof points in any given block is a member of % . In consideration of a problemconcerning the existence of a certain class of �-designs there arises a sequence ofpolynomials in two variables for which we can determine a recurrence relationbut not a closed-form formula. Our interest is in the behavior of this class ofpolynomials; in particular, we would like to determine for which �� * � thesepolynomials take on non-negative values. This talk constitutes a desperate pleafor help!

EXPLICIT DECOMPOSITIONOF THE GROUP ALGEBRA ��

OF THE ALTERNATING GROUP��

Amitai RegevWeizmann Institute of Science

Rehovot

Israel

In talk 1 we describe the decomposition of the group-algebra of the alternatinggroup into a direct sum of simple two sided ideals, then the decomposition ofeach such ideal into a direct sum of simple left ideals. The description is in termsof Young tableaux, and is similar to the well known analogue decompositionof the group-algebra �� of the symmetric group ��.Talk 2. ��-codimensions and cocharacters. The fact that �� is a sub-spaceof �� gives rise to �� codimensions and cocharacters. We study theseinvariants for various p.i. algebras, in particular for the Grassmann algebra.Joint work with A. Henke.

COMPUTING WITH MATRICESOVER THE GRASSMANN ALGEBRA

Jeno SzigetiUniversity of Miscolc

Hungary)

We shall consider ��matrices over a Grassmann algebrapart of�. Let / be a generator of� not occurring in the elements of�, then� can be completelyread off the so called companion matrix �� / and �� / ��lies in a commutative environment. The main aim of the talk is to show, howcan we use �� / instead of � in different calculations. More generally,for ��matrices, over a ��-graded ring � � �� with �� , ourcomputational technique leads to a new concept of determinant, which can beused to derive an invariant Cayley-Hamilton identity. This talk reports a jointwork with Sudarshan Sehgal.

GRADINGSAND GRADED IDENTITIES OF THE ALGEBRAOF� �� UPPER TRIANGULAR MATRICES

Angela ValentiUniversity of Palermo

We study the graded identities of the algebra �� of �� upper triangularmatrices over a field � endowed with an elementary grading. Such gradingsappear quite naturally. For instance it has been recently proved that every finiteabelian grading on the algebra �� is elementary provided � is algebraicallyclosed of characteristic zero.

In case � is infinite and �� has its usual ��-grading, we describe a basisof the ideal of the graded identities for this algebra and we compute some of thenumerical invariants of this ideal. In general we show that one can distinguishthe elementary gradings by means of the graded identities they satisfy.

NUMERICAL CHARACTERISTICSAND EXTREMAL VARIETIES

Mikhail ZaicevMoscow State University

We consider varieties of associative algebras over a field of characteristiczero. Dimension of multilinear component in N variables of the relatively freealgebra of the variety is usually called its N-th codimension. An importantnumerical characteristics of a variety is the sequence of codimensions and itsasymptotic behaviour. It is well-known that any variety of associative alge-bras over a field of characteristic zero has exponentially bounded codimensiongrowth. Recently it was proved that the exponent of any variety exists and isan integer. We say that V is a minimal variety of given exponent n if exp(V)=nand exp(W)<n for any proper subvariety W of V. We describe all minimal va-rieties of non-polynomial growth in terms of generating algebras and T-ideals.For example, any minimal variety of a finite basic rank can be generated by anupper block-triangular matrix algebra. In particular, we prove in the positive aconjecture of Drensky that V is a minimal variety if and only if its T-ideal is theproduct of verbally prime T-ideals. As a corollary we obtain that there is onlya finite number of minimal varieties for any given exponent.

polynomial identities in algebras

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