irreducible unitary representations of su 2 2 we give an explicit classification of the irreducible unitary representations of the simple lie group su 2 2

d split sequences and derived equivalences abstract in this paper we introduce the so called d split sequences and show that each d split sequence gives rise to a derived equivalence via a tilting module in particular we obtain derived equivalences from auslander reiten sequences via bb tilting modules or from n almost split sequences via n bb tilting modules and auslander reiten triangles further we recover n almost split sequences from n bb tilting modules over n auslander algebras

magic identities for the conformal four point integrals the minkowski metric case abstract the original magic identities are due to j m drummond j henn v a smirnov and e sokatchev they assert that all n loop box integrals for four scalar massless particles are equal to each other 3 the authors give a proof of the magic identities for the euclidean metric case only and claim that the result is also true in the minkowski metric however the minkowski case is much more subtle and requires specification of the relative positions of cycles of integration to make these identities correct in this article we prove the magic identities in the minkowski metric case and in particular specify the cycles of integration our proof of magic identities relies on previous results from 7 8 where we give a mathematical interpretation of the n loop box integrals in the context of representations of a lie group u 2 2 and quaternionic analysis the main result of 7 8 is a weaker operator version of the magic identities no prior knowledge of physics or feynman diagrams is assumed from the reader we provide a summary of all relevant results from quaternionic analysis to make the article self contained

representation dimension and quasi hereditary algebras we study auslander s representation dimension of artin algebras which is by definition the minimal projective dimension of coherent functors on modules which are both generators and cogenerators we show the following statements 1 if an artin algebra a is stably hereditary then the representation dimension of a is at most 3 2 if two artin algebras are stably equivalent of morita type then they have the same representation dimension particularly if two self injective algebras are derived equivalent then they have the same representation dimension 3 any incidence algebra of a finite partially ordered set over a field has finite representation dimension moreover we use results on quasi hereditary algebras to show that 4 the auslander algebra of a nakayama algebra has finite representation dimension

representation dimension via tilting theory abstract let be an artin algebra over a commutative artinian ring r and let t be a separating splitting tilting module with endomorphism end t the aim of this paper is to use tilting theory to study the representation dimension of the main result asserts that for an integer n 1 if is n gorenstein of finite cohen macaulay type then r e p d i m n 2 we conclude that if is a n gorenstein artin algebra of finite cohen macaulay type then r e p d i m n 2 this in particular shows that for any artin algebra r e p d i m g l d i m 2

finitistic dimension and igusa todorov algebras abstract the notion of igusa todorov algebras is introduced in connection with the little finitistic dimension conjecture and the conjecture is proved for those algebras such algebras contain many known classes of algebras over which the finitistic dimension conjecture holds e g algebras with the representation dimension at most 3 algebras with radical cube zero monomial algebras and left serial algebras etc it is an open question whether all artin algebras are igusa todorov we provide some methods to construct many new classes of 2 igusa todorov algebras and thus obtain many algebras such that the finitistic dimension conjecture holds in particular we show that the class of 2 igusa todorov algebras is closed under taking endomorphism algebras of projective modules hence if all quasi hereditary algebras are 2 igusa todorov then all artin algebras are 2 igusa todorov by v dlab c m ringel every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring proc amer math soc 107 1 1989 1 5 and have finite finitistic dimension

on the representation dimension of tilted and laura algebras abstract we prove that the representation dimension of a tilted or of a strict laura algebra is at most three

a geometrical angle on feynman integrals a direct link between a one loop n point feynman diagram and a geometrical representation based on the n dimensional simplex is established by relating the feynman parametric representations to the integrals over contents of n 1 dimensional simplices in non euclidean geometry of constant curvature in particular the four point function in four dimensions is proportional to the volume of a three dimensional spherical or hyperbolic tetrahedron which can be calculated by splitting into birectangular ones it is also shown that the known formula of reduction of the n point function in n 1 dimensions corresponds to splitting the related n dimensional simplex into n rectangular ones

realization of quaternionic discrete series on the unit ball in h d abstract we give a realization for the quaternionic discrete series for the group sp 1 d as the kernel of certain elementary cauchy riemann fueter operator and we compute the corresponding reproducing kernels

radical embeddings and representation dimension given a representation finite algebra b and a subalgebra a of b such that the jacobson radicals of a and b coincide we prove that the representation dimension of a is at most three by a result of igusa and todorov this implies that the finitistic dimension of a is finite

tilting modules and representation dimensions in this paper we compare the representation dimensions of two algebras linked by certain tilting modules our main results can be stated as follows suppose t is a tilting module over a and b enda t then 1 if t is separating and splitting then rep dim a rep dim b 2 if t p 1s is an apr tilting module and the injective dimension of s is at most 2 then rep dim b rep dim a 1

on the finitistic dimension conjecture iii related to the pair eae a let a be an artin algebra and e an idempotent element in a in this paper we use co homological conditions on a to control the finitistic dimension of eae such a consideration is of particular interest for understanding the finitistic dimension conjecture let us denote the finitistic dimension and global dimension of a by fin dim a and gl dim a respectively suppose gl dim a 4 then fin dim eae if one of the following conditions holds 1 a aea has representation dimension at most 3 2 a 3 a is an a aea module 3 proj dim sa 3 for all simple a aea modules s this result can be considered as a first step to the question of whether gl dim a 4 implies fin dim eae moreover we show the following let a be an arbitrary artin algebra and e an idempotent element of a such that the syzygy dimension or the gorenstein dimension of the eae module ae is finite if fin dim a then fin dim eae

on the representation dimension of finite dimensional algebras the importance of the representation finite artin algebras for the whole representation theory of artin algebras is well understood and there is much literature on the subject see 1 6 12 these are artin algebras such that every indecomposable module is finitely generated and every module is a direct sum of indecomposable modules moreover there is a bijective correspondence between the class of representation finite artin algebras and that of artin algebras with global dimension of at most 2 and with dominant dimension of at least 2 this result was established by auslander in 1 motivated by this correspondence auslander introduced the concept of representation dimension for artin algebras to study the connection of arbitrary artin algebras with representation finite artin algebras it is hoped that this notion gives a reasonable way of measuring how far an artin algebra is from being representation finite type 1 p 134 representation dimension is a morita invariant of artin algebras and its definition involves homological dimensions and modules which are both generators and cogenerators unfortunately little seems to be known about representation dimension so there are a lot of essential questions on this invariant dealing with the computation and estimation of the representation dimension of an artin algebra there are a few cases known auslander proved that the representation dimension of an artin algebra is two if and

global dimension of the endomorphism ring and n modules abstract we show that if t is a selfsmall and selforthogonal module over a noetherian ring r of finite global dimension with the endomorphism ring a then fd t a gd a id r t fd t a applying the result we give answers to two questions left in j wei et al j algebra 168 2 2003 404 418 concerning basic properties of n modules by showing that the flat dimension of a n module with n 3 over its endomorphism ring can even be arbitrarily far from the integer n while the flat dimension of a 2 module over its endomorphism ring is always bounded by the integer 2 and showing that n modules are not finitely generated in general even in case n 2

cramer rule for the unique solution of restricted matrix equations over the quaternion skew field in this paper we establish the determinantal representations of the generalized inverses a r t 1 s 1 2 a l t 2 s 2 2 and a t 1 t 2 s 1 s 2 2 over the quaternion skew field by the theory of the column and row determinants in addition we derive some generalized cramer rules for the unique solution of some restricted quaternion matrix equations the findings of this paper extend some known results in the literature

global µ stability criteria for quaternion valued neural networks with unbounded time varying delays in this paper we first propose quaternion valued neural networks qvnns with unbounded time varying delays some sufficient conditions on the global stability in the form of both complex valued and real valued linear matrix inequalities lmis are provided by solving two difficulties one is decomposing the qvnn into two complex valued systems with the plural decomposition method of quaternion which can reduce the complexity of calculations by avoiding the non commutativity of quaternion multiplication the other is choosing the appropriate lyapunov krasovskii functional in the form of hermitian matrices which is a big challenge finally two numerical examples are provided to verify the effectiveness of the obtained results

analysis on the minimal representation of o p q abstract for the group o p q we give a new construction of its minimal unitary representation via euclidean fourier analysis this is an extension of the q 2 case where the representation is the mass zero spin zero representation realized in a hilbert space of solutions to the wave equation the group o p q acts as the mobius group of conformal transformations on r p 1 q 1 and preserves a space of solutions of the ultrahyperbolic laplace equation on r p 1 q 1 we construct in an intrinsic and natural way a hilbert space of solutions so that o p q becomes a continuous irreducible unitary representation in this hilbert space we also prove that this representation is unitarily equivalent to the representation on l2 c where c is the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the lie algebra of o p q

on the representation dimension of some classes of algebras abstract we show that the representation dimension of the following classes of algebras is at most 3 a artin algebras a such that the functor hom a d a has finite length or dually hom a a has finite length these algebras coincide with the right left glued algebras as introduced in i assem f u coelho j pure appl algebra 96 3 1994 225 and b trivial extensions of iterated tilted algebras

effective tracking through tree search a new contour tracking algorithm is presented tracking is posed as a matching problem between curves constructed out of edges in the image and some shape space describing the class of objects of interest the main contributions of the paper are to present an algorithm which solves this problem accurately and efficiently in a provable manner in particular the algorithm s efficiency derives from a novel tree search algorithm through the shape space which allows for much of the shape space to be explored with very little effort this latter property makes the algorithm effective in highly cluttered scenes as is demonstrated in an experimental comparison with a condensation tracker

constructions of stable equivalences of morita type for finite dimensional algebras ii suppose k is a field let a and b be two finite dimensional k algebras such that there is a stable equivalence of morita type between a and b in this paper we prove that 1 if a and b are representation finite then their auslander algebras are stably equivalent of morita type 2 the n th hochschild homology groups of a and b are isomorphic for all n 1 a new proof is also provided for hochschild cohomology groups of self injective algebras under a stable equivalence of morita type

the representation dimension of k x y x2 yn the representation dimension of an artin algebra was defined by m auslander in 1970 the precise value is not known in general and is very hard to compute even for small examples for group algebras it is known in the case of cyclic sylow subgroups for some group algebras in characteristic 2 of rank at least 3 the precise value of the representation dimension follows from recent work of r rouquier there is a gap for group algebras of rank 2 in this paper we show that for all n 0 and any field k the commutative algebras k x y x2 y2 n have representation dimension 3 for the proof we give an explicit inductive construction of a suitable generator cogenerator as a consequence we obtain that the group algebras in characteristic 2 of the groups c2 c2m have representation dimension 3 note that for m 3 these group algebras have wild representation type

invariant properties of representations under excellent extensions abstract in this paper we introduce the notion of weak excellent extensions of rings as a generalization of that of excellent extensions of rings let be a weak excellent extension of an artinian algebra we prove that if is of finite representation type resp cm finite cm free then so is furthermore if is an excellent extension of then the converse also holds true we also study when the representation dimension of an artinian algebra is invariant under excellent extensions

split quaternionic analysis and separation of the series for sl 2 r and sl 2 c sl 2 r abstract we extend our previous study of quaternionic analysis based on representation theory to the case of split quaternions h r the special role of the unit sphere in the classical quaternions h identified with the group su 2 is now played by the group sl 2 r realized by the unit quaternions in h r as in the previous work we use an analogue of the cayley transform to relate the analysis on sl 2 r to the analysis on the imaginary lobachevski space sl 2 c sl 2 r identified with the one sheeted hyperboloid in the minkowski space m we study the counterparts of cauchy fueter and poisson formulas on h r and m and show that they solve the problem of separation of the discrete and continuous series the continuous series component on h r gives rise to the minimal representation of the conformal group sl 4 r while the discrete series on m provides its k types realized in a natural polynomial basis we also obtain a surprising formula for the plancherel measure of sl 2 r in terms of the poisson type integral on the split quaternions h r finally we show that the massless singular functions of four dimensional quantum field theory are nothing but the kernels of projectors onto the discrete and continuous series on the imaginary lobachevski space sl 2 c sl 2 r our results once again reveal the central role of the minkowski space in quaternionic and split quaternionic analysis as well as a deep connection between split quaternionic analysis and the four dimensional quantum field theory

on the finitistic dimension conjecture ii related to finite global dimension abstract in this paper we study the finitistic dimensions of artin algebras by establishing a relationship between the global dimensions of the given algebras on the one hand and the finitistic dimensions of their subalgebras on the other hand this is a continuation of the project in j pure appl algebra 193 2004 287 305 for an artin algebra a we denote by gl dim a fin dim a and rep dim a the global dimension finitistic dimension and representation dimension of a respectively the jacobson radical of a is denoted by rad a the main results in the paper are as follows let b be a subalgebra of an artin algebra a such that rad b is a left ideal in a then 1 if gl dim a 4 and rad a rad b a then fin dim b 2 if rep dim a 3 then fin dim b the results are applied to pullbacks of algebras over semi simple algebras moreover we have also the following dual statement 3 let b a be a surjective homomorphism between two algebras b and a suppose that the kernel of is contained in the socle of the right b module b b if gl dim a 4 or rep dim a 3 then fin dim b finally we provide a class of algebras with representation dimension at most three 4 if a is stably hereditary and rad b is an ideal in a then rep dim b 3

construction of stable equivalences of morita type for finite dimensional algebras i in the representation theory of finite groups the stable equivalence of morita type plays an important role for general finite dimensional algebras this notion is still of particular interest however except for the class of self injective algebras one does not know much on the existence of such equivalences between two finite dimensional algebras in fact even a non trivial example is not known in this paper we provide two methods to produce new stable equivalences of morita type from given ones the main results are corollary 1 2 and theorem 1 3 here the algebras considered are not necessarily self injective as a consequence of our constructions we give an example of a stable equivalence of morita type between two algebras of global dimension 4 such that one of them is quasi hereditary and the other is not this shows that stable equivalences of morita type do not preserve the quasi heredity of algebras as another by product we construct a morita equivalence inside each given stable equivalence of morita type between algebras a and b this leads not only to a general formulation of a result by linckelmann 1996 but also to a nice correspondence of some torsion pairs in a mod with those in b mod if both a and b are symmetric algebras moreover under the assumption of symmetric algebras we can get a new stable equivalence of morita type finally we point out that stable equivalences of morita type are preserved under separable extensions of ground fields

quaternionic analysis representation theory and physics we develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group we first review the cauchy fueter and poisson formulas and explain their representation theoretic meaning the requirement of unitarity of representations leads us to the extensions of these formulas in the minkowski space which can be viewed as another real form of quaternions representation theory also suggests a quaternionic version of the cauchy formula for the second order pole remarkably the derivative appearing in the complex case is replaced by the maxwell equations in the quaternionic counterpart we also uncover the connection between quaternionic analysis and various structures in quantum mechanics and quantum field theory such as the spectrum of the hydrogen atom polarization of vacuum one loop feynman integrals we also make some further conjectures the main goal of this and our subsequent paper is to revive quaternionic analysis and to show profound relations between quaternionic analysis representation theory and four dimensional physics

analysis on the minimal representation of o p q i realization via conformal geometry abstract this is the first in a series of papers devoted to an analogue of the metaplectic representation namely the minimal unitary representation of an indefinite orthogonal group this representation corresponds to the minimal nilpotent coadjoint orbit in the philosophy of kirillov kostant we begin by applying methods from conformal geometry of pseudo riemannian manifolds to a general construction of an infinite dimensional representation of the conformal group on the solution space of the yamabe equation by functoriality of the constructions we obtain different models of the unitary representation as well as giving new proofs of unitarity and irreducibility the results in this paper play a basic role in the subsequent papers where we give explicit branching formulae and prove unitarization in the various models

on the quasi heredity of birman wenzl algebras abstract in this paper we consider the birman wenzl algebras over an arbitrary field and prove that they are cellular in the sense of graham and lehrer furthermore we determine for which parameters the birman wenzl algebras are quasi hereditary so the general theory of cellular algebras and quasi hereditary algebras applies to birman wenzl algebras as a consequence we can determine all irreducible representations of the birman wenzl algebras by linear algebra methods we prove also that the new hecke algebras induced from birman wenzl algebras are frobenius over a field but not always cellular

the conformal four point integrals magic identities and representations of u 2 2 abstract in 3 5 we found mathematical interpretations of the one loop conformal four point feynman integral as well as the vacuum polarization feynman integral in the context of representations of a lie group u 2 2 and quaternionic analysis then we raised a natural question of finding mathematical interpretation of other feynman diagrams in the same setting in this article we describe this interpretation for all conformal four point integrals using this interpretation we give a representation theoretic proof of an operator version of the magic identities for the conformal four point integrals described by the box diagrams the original magic identities are due to j m drummond j henn v a smirnov and e sokatchev they assert that all n loop box integrals for four scalar massless particles are equal to each other 2 the authors give a proof of the magic identities for the euclidean metric case only and claim that the result is also true for the minkowski metric case however the minkowski case is much more subtle in this article we prove an operator version of the magic identities in the minkowski metric case and in particular specify the relative positions of cycles of integration that make these identities correct no prior knowledge of physics or feynman diagrams is assumed from the reader we provide a summary of all relevant results from quaternionic analysis to make the article self contained

analysis on the minimal representation of o p q ii branching laws this is a second paper in a series devoted to the minimal unitary representation of o p q by explicit methods from conformal geometry of pseudo riemannian manifolds we find the branching law corresponding to restricting the minimal unitary representation to natural symmetric subgroups in the case of purely discrete spectrum we obtain the full spectrum and give an explicit parseval plancherel formula and in the general case we construct an infinite discrete spectrum r 2003 published by elsevier science usa msc 22e45 22e46 53a30 43a85

representation dimension of exterior algebras we determine the representation dimension of exterior algebras this provides the first known examples of representation dimension 3 we deduce that the loewy length of the group algebra over f2 of a finite group is strictly bounded below by the 2 rank of the group a conjecture of benson a key tool is the use of the concept of dimension of a triangulated category

learning to track the visual motion of contours abstract a development of a method for tracking visual contours is described given an untrained tracker a training motion of an object can be observed over some extended time and stored as an image sequence the image sequence is used to learn parameters in a stochastic differential equation model these are used in turn to build a tracker whose predictor imitates the motion in the training set tests show that the resulting trackers can be markedly tuned to desired curve shapes and classes of motions

on the finitistic dimension conjecture of artin algebras abstract let a be an artin algebra and e be an idempotent element of a we prove that if a has representation dimension at most three then the finitistic dimension of eae is finite and deduce that if quasi hereditary algebras have representation dimensions at most three then the finitistic dimension conjecture holds

on subcategories closed under predecessors and the representation dimension abstract let be an artin algebra a and c be full subcategories of the category of finitely generated modules consisting of indecomposable modules and closed under predecessors and successors respectively in this paper we relate under various hypotheses the representation dimension of to those of the left support algebra of a and the right support algebra of c our results are then applied to the classes of laura algebras ada algebras and nakayama oriented pullbacks