gradings on the algebra of upper triangular matrices of size three abstract let ut 3 f be the algebra of 3 3 upper triangular matrices over a field f on ut 3 f up to isomorphism there are at most five non trivial elementary gradings and we study the graded polynomial identities for such gradings in case f is of characteristic zero we give a complete description of the space of multilinear graded identities in the language of young diagrams through the representation theory of a young subgroup of s n we finally compute the multiplicities in the graded cocharacter sequence for every elementary g grading on ut 3 f

exponential codimension growth of pi algebras an exact estimate abstract letabe an associative pi algebra over a fieldfof characteristic zero by studying the exponential behavior of the sequence of codimensions cn a ofa we prove thatinv a limn c n a always exists and is an integer we also give an explicit way for computing such integer letbbe a finite dimensionalz2 graded algebra whose grassmann envelopeg b satisfies the same identities ofa theninv a inv g b dim c 0 dim c 1 wherec 0 c 1 is a suitablez2 graded semisimple subalgebra ofb

representability and specht problem for g graded algebras abstract let w be an associative pi algebra over a field f of characteristic zero graded by a finite group g let id g w denote the t ideal of g graded identities of w we prove 1 g graded pi equivalence there exists a field extension k of f and a finite dimensional z 2 z g graded algebra a over k such that id g w id g a where a is the grassmann envelope of a 2 g graded specht problem the t ideal id g w is finitely generated as a t ideal 3 g graded pi equivalence for affine algebras let w be a g graded affine algebra over f then there exists a field extension k of f and a finite dimensional algebra a over k such that id g w id g a

group gradings on associative algebras abstract let r g g rg be a g graded ring we describe all types of gradings on r if g is torsion free and r is artinian semisimple if r is a matrix algebra over an algebraically closed field f then we give a description of all g gradings on r provided that g is an abelian group in the case of an abelian group g we also classify all finite dimensional graded simple algebras and finite dimensional graded division algebras over an algebraically closed field of characteristic zero

applications of hook young diagrams to p i algebras abstract a the multiplicities m in the cocharacter n a of any p i algebra a are polynomially bounded b a hook containing n a b is obtained from the hooks containing n a and n b these results are obtained by applying a theory of hook young diagrams to p i algebras and they generalize results known for algebras satisfying capelli identities

codimension growth and minimal superalgebras a celebrated theorem of kemer 1978 states that any algebra satisfying a polynomial identity over a field of characteristic zero is pi equivalent to the grassmann envelope g a of a finite dimensional superalgebra a in this paper by exploiting the basic properties of the exponent of a pi algebra proved by giambruno and zaicev 1999 we define and classify the minimal superalgebras of a given exponent over a field of characteristic zero in particular we prove that these algebras can be realized as block triangular matrix algebras over the base field the importance of such algebras is readily proved a is a minimal superalgebra if and only if the ideal of identities of g a is a product of verbally prime t ideals also such superalgebras allow us to classify all minimal varieties of a given exponent i e varieties v such that exp v d 2 and exp u d for all proper subvarieties u of v this proves in the positive a conjecture of drensky 1988 as a corollary we obtain that there is only a finite number of minimal varieties for any given exponent a classification of minimal varieties of finite basic rank was proved by the authors 2003 as an application we give an effective way for computing the exponent of a t ideal given by generators and we discuss the problem of what functions can appear as growth functions of varieties of algebras

multialternating jordan polynomials and codimension growth of matrix algebras abstract let r be the jordan algebra of k k matrices over a field of characteristic zero we exhibit a noncommutative jordan polynomial f multialternating on disjoint sets of variables of order k 2 and we prove that f is not a polynomial identity of r we then study the growth of the polynomial identities of the jordan algebra r through an analysis of its sequence of jordan codimensions by exploiting the basic properties of the polynomial f we are able to prove that the exponential rate of growth of the sequence of jordan codimensions of r in precisely k 2

function order of positive operators based on the mond pecaric method we shall show function order preserving operator inequalities under general setting based on kantorovich type inequalities for convex functions due to mond pecaric let a and b be positive operators on a hilbert space h satisfying mi b mi 0 let f t be a continuous convex function on m m if g t is a continuous increasing convex function on m m sp a then for a given 0 a b 0implies g a i f b where maxm t m f m f m f m t m m m g t as applications we shall extend kantorovich type operator inequalities by furuta yamazaki and yanagida and present operator inequalities on the usual order and the chaotic order via ky fan furuta constant among others we show the following inequality if a b 0 and mi b mi 0 then mp 1mq 1aq q 1 q 1qq mp mp q m m mmp mmp q 1aq bp holds for all p 1 and q 1 such that qmp 1 mp mpm m qmp 1

proper identities lie identities and exponential codimension growth abstract the exponent exp a of a pi algebra a in characteristic zero is an integer and measures the exponential rate of growth of the sequence of codimensions of a a giambruno m zaicev on codimension growth of finitely generated associative algebras adv math 140 1998 145 155 a giambruno m zaicev exponential codimension growth of p i algebras an exact estimate adv math 142 1999 221 243 in this paper we study the exponential rate of growth of the sequences of proper codimensions and lie codimensions of an associative pi algebra we prove that the corresponding proper exponent exists for all pi algebras except for some algebras of exponent two strictly related to the grassmann algebra we also prove that the lie exponent exists for any finitely generated pi algebra the value of both exponents is always equal to exp a or exp a 1

some functions reversing the order of positive operators abstract as a complement to our previous results about the function preserving the operator order we shall show the following reversing version let a and b be positive operators on a hilbert space h satisfying mi b mi 0 let f t be a continuous convex function on m m if g t is a continuous decreasing convex function on m m sp a then for a given 0 a b 0 implies g b i f a where max m t m f m f m f m t m m m g t our main result is to classify complementary inequalities on power means of positive operators as a matter of fact we determine real constants 1 and 1 such that 2 m k s a m k r a 1 m k s a if r s where m k r a j 1 k j a j r 1 r r r 0 is weighted power mean of positive operators a j sp a j m m for some scalars 0 m m and j r such that j 1 k j 1 j 1 k

graded polynomial identities and codimensions computing the exponential growth abstract let g be a finite abelian group and a a g graded algebra over a field of characteristic zero this paper is devoted to a quantitative study of the graded polynomial identities satisfied by a we study the asymptotic behavior of c n g a n 1 2 the sequence of graded codimensions of a and we prove that if a satisfies an ordinary polynomial identity lim n c n g a n exists and is an integer we give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple g z 2 graded algebra related to a

asymptotics of h identities for associative algebras with an h invariant radical abstract we prove the existence of the hopf pi exponent for finite dimensional associative algebras a with a generalized hopf action of an associative algebra h with 1 over an algebraically closed field of characteristic 0 assuming only the invariance of the jacobson radical j a under the h action and the existence of the decomposition of a j a into the sum of h simple algebras as a consequence we show that the analog of amitsur s conjecture holds for g codimensions of finite dimensional associative algebras over a field of characteristic 0 with an action of an arbitrary group g by automorphisms and anti automorphisms and for differential codimensions of finite dimensional associative algebras with an action of an arbitrary lie algebra by derivations

on the codimension growth of finite dimensional lie algebras abstract we study the exponential growth of the codimensions c n l of a finite dimensional lie algebra l over a field of characteristic zero we show that if the solvable radical of l is nilpotent then lim n c n l exists and is an integer

homological characteristics of nakayama algebras in a previous paper 3 1 we described a homological characterization of pro uniserial rings the purpose of this paper is to give a similar charac terization of injec uniserial rings and of nakayama algebras throughout this paper all rings are assumed to be finite dimensional algebras over some fixed algebraically closed field f and all modules are left modules if r is a ring an r module is called uniserial iff it has a unique composition series r is called

graded polynomial identities group actions and exponential growth of lie algebras abstract consider a finite dimensional lie algebra l with an action of a finite group g over a field of characteristic 0 we prove the analog of amitsur s conjecture on asymptotic behavior for codimensions of polynomial g identities of l as a consequence we prove the analog of amitsur s conjecture for graded codimensions of any finite dimensional lie algebra graded by a finite abelian group

group graded algebras and almost polynomial growth let f be a field of characteristic 0 g a finite abelian group and a a g graded algebra we prove that a generates a variety of g graded algebras of almost polynomial growth if and only if a has the same graded identities as one of the following algebras 1 fcp the group algebra of a cyclic group of order p where p is a prime number and p g 2 ut2g f the algebra of 2 2 upper triangular matrices over f endowed with an elementary g grading 3 e the infinite dimensional grassmann algebra with trivial g grading 4 in case 2 g ez2 the grassmann algebra with canonical z2 grading

identities of graded algebras and codimension growth let a g g a g be a g graded associative algebra over a field of characteristic zero in this paper we develop a conjecture that relates the exponent of the growth of polynomial identities of the identity component a e to that of the whole of a in the case where the support of the grading is finite we prove the conjecture in several natural cases one of them being the case where a is finite dimensional and a e has polynomial growth

on weak equivalences of gradings abstract when one studies the structure e g graded ideals graded subspaces radicals or graded polynomial identities of graded algebras the grading group itself does not play an important role but can be replaced by any other group that realizes the same grading here we come to the notion of weak equivalence of gradings two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component the following question arises naturally when a group grading on a finite dimensional algebra is weakly equivalent to a grading by a finite group it turns out that this question can be reformulated purely group theoretically in terms of the universal group of the grading namely a grading is weakly equivalent to a grading by a finite group if and only if the universal group of the grading is residually finite with respect to a special subset of the grading group the same is true for all the coarsenings of the grading if and only if the universal group of the grading is hereditarily residually finite with respect to the same subset we show that if n 349 then on the full matrix algebra m n f there exists an elementary group grading that is not weakly equivalent to any grading by a finite semi group and if n 3 then any elementary grading on m n f is weakly equivalent to an elementary grading by a finite group

asymptotic values for degrees associated with strips of young diagrams abstract the sums s l n occur in the representations of the symmetric and the general linear groups in combinatorics and in pi algebras we give asymptotic values for these sums

on codimension growth of finitely generated associative algebras abstract letabe a pi algebra over a fieldf we study the asymptotic behavior of the sequence of codimensionscn a ofa we show that ifais finitely generated overftheninv a limn c n a always exists and is an integer we also obtain the following characterization of simple algebras ais finite dimensional central simple overfif and only ifinv a dim a

on the homological structures associated with certain artin algebras if r is a ring there are several graded algebras which are associated with r if m is any r module then exti m m c o ext m m is a graded algebra where the product is the yoneda product if jz c rmod is any collection of r modules then 0 cm6 cnt a ext m n is also a graded ring where the product is the yoneda product where it is defined and zero elsewhere in this paper we shall give a description of these structures for certain classes of modules over rings satisfying certain conditions in particular in this paper all rings are assumed to be finite dimensional algebras over some fixed algebraically closed field f and all modules are left modules if r is such a ring yr or just 9 if the context ensures no ambiguity e rmod will be a collection containing one representative of each isomorphism class of simple r modules we shall compute the structure of exts s s for sey and of ext r csg y ctttfextr s t when r is pro uniserial injec uniserial or nakayama

a complementary result of kantorovich type order preserving inequalities by micic pecaric seo abstract we shall discuss operator inequalities which are obtained by elementary lemma lemma 3 1 associated with holder mccarthy and kantorovich inequalities firstly we shall give the following complementary result to micic et al linear algebra appl 360 2003 15 let a and b be two strictly positive operators on a hilbert space h such that m1i a m1i 0 and m2i b m2i 0 where m1 m1 0 m2 m2 0 and a b a if p 1 and q 1 then the following inequality holds q 1 q 1 q q m 2 p m 2 p q m 2 m 2 m 2 m 2 p m 2 m 2 p q 1 a q b p for m 2 p 1 q m 2 p m 2 p m 2 m 2 m 2 p 1 q b if p m 1 m 1 p m 1 m 1 p q 1 m 1 m 1 q 1 m 1 p m 1 p q m 1 m 1 p m 1 m 1 p q b q a p for m 1 p 1 q m 1 p m 1 p m 1 m 1 m 1 p 1 q we remark that a is shown in linear algebra appl 360 2003 15 as an extension of two variable version of our previous one variable one j inequal appl 2 1998 137 secondly we shall show the following extension of two parameters type of an extension of fujii et al sci math 1 1998 307 on the determinant of an operator let t be strictly positive operators on a hilbert space h such that mi t mi 0 then the following inequality holds s h p q x t q t p x x x t p for p 0 and q 0 wheresh p q is defined by s h p q m p q h q h p 1 e log h q h p 1 if q h p 1 log h qh p andthe determinant x t for strictly positive operator t at a unit vector x in hilbert space h is defined by x t exp log t x x as an application of this result we shall give an alternative proof of two variable version of characterization of the chaotic order in linear algebra appl ibid theorem 4 4