witten s invariants of rational homology spheres at prime values of k and trivial connection contribution we establish a relation between the coefficients of asymptotic expansion of the trivial connection contribution to witten s invariant of rational homology spheres and the invariants that t ohtsuki extracted from witten s invariant at prime values ofk we also rederive the properties of primek invariants discovered by h murakami and t ohtsuki we do this by using the bounds on taylor series expansion of the jones polynomial of algebraically split links studied in our previous paper these bounds are enough to prove that ohtsuki s invariants are of finite type the relation between ohtsuki s invariants and trivial connection contribution is verified explicitly for lens spaces and seifert manifolds

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

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

deformations of quasi homogeneous surface singularities let a a be a finitely generated normal graded domain over e so that i 0 v spec a has an isolated singularity we say v is a quasi homogeneous singularity or admits a good ie action or is defined by weighted homogeneous polynomials we are concerned with two questions about the deformations of v there are two natural notions of equisingular deformation of v first one has deformations of weight 0 that is one perturbs the defining equations of vby terms of weight at least as big to make a good notion out of this dock sang rim observed one is really deforming a and the weight filtration ij where ij ai

analysis in space time bundles iii higher spin bundles the universal cosmos m is the unique four dimensional globally causal space time manifold to which the dirac and maxwell equations among others maximally and covariantly extend a systematic treatment is presented of general fields over m of arbitrary spin considered are fields induced from all irreducible representation of the isotropy group scale extended poincare group to g the connected causal group of m restricted to any species of such fields the k invariant canonical dirac operator k maximal essentially compact subgroup of g is shown g covariant for a unique conformal weight a normalized k finite basis for such fields is constructed the basis actions thereon of the dirac operator infinitesimal generators of g discrete symmetries second order casimir and the essentially unique third order noncentral quantum number enveloping algebra element invariant under k are derived composition series under g of a class of these field spaces namely the extension to m of the relativistic fields considered by bargman and wigner or arbitrary spin and conformal weight are determined distinguishing by invariance and causality features alone the essentially conventional positive energy mass 0 subspaces and massive invariant sub quotient spaces whose unitarity under g is given a new proof the completely positive subclass cf below of representations is determined a more detailed treatment of spin one bundles vector and two form of arbitrary conformal weight is included the exterior derivative transformations are diagonalized and the conformally invariant massive spin one scalar product is identified with a mathematical version of the conventional electromagnetic field lagrangian

on the classification of clifford algebras and their relation to spinors in n dimensions a classification of all the clifford algebras is given in terms of kronecker products of the quaternion and dihedral groups the relationship to spinors in n dimensions is explicitly determined we show that the real clifford algebra in minkowski spacetime is distinct from both the algebra of dirac matrices and the algebra of majorana matrices and cannot be realized by the spinor framework the matrix representations of clifford algebras are discussed and are utilized to give a classification of the real forms of lie algebras we are thus able to relate clifford lie and spinor algebras in an intrinsic geometrical setting

reduced weighted complete intersection and derivations abstract let a f x 1 x n f 1 f n be a zero dimensional weighted complete intersection char f 0 we prove a general result on the homogeneous derivations of a in particular we deduce that in the generic case when f x 1 x n f 1 f n 1 is reduced deg f i deg f n i 1 n 1 there are no nontrivial derivations of strictly negative degree if moreover all the weights of the variables x i are even it follows that the serre spectral sequence of any orientable fibration f e b with h f a as graded algebras collapses at the e 2 term thus verifying a conjecture of halperin we also discuss several examples

conformally invariant differential equations and projective geometry abstract we combine harmonic analysis on certain pseudo riemannian symmetric spaces with results on conformally invariant linear and non linear differential equations this gives in many cases part of the decomposition of certain representations of the conformal group of a manifold when restricted to the isometry group

on the new k th yau algebras of isolated hypersurface singularities let v be a hypersurface with an isolated singularity at the origin defined by the holomorphic function f mathbb c n 0 rightarrow mathbb c 0 the yau algebra l v is defined to be the lie algebra of derivations of the moduli algebra a v mathcal o n f frac partial f partial x 1 cdots frac partial f partial x n i e l v text der a v a v and plays an important role in singularity theory it is known that l v is a finite dimensional lie algebra and its dimension lambda v is called yau number in this article we generalize the yau algebra and introduce a new series of k th yau algebras l k v which are defined to be the lie algebras of derivations of the moduli algebras a k v mathcal o n f m k j f k ge 0 i e l k v text der a k v a k v and where m is the maximal ideal of mathcal o n in particular it is yau algebra when k 0 the dimension of l k v is denoted by lambda k v these numbers i e k th yau numbers lambda k v are new numerical analytic invariants of an isolated singularity in this paper we studied these new series of lie algebras l k v and also compute the lie algebras l 1 v for fewnomial isolated singularities we also formulate a sharp upper estimate conjecture for the lambda k v of weighted homogeneous isolated hypersurface singularities and we prove this conjecture in case of k 1 for large class of singularities

real representations of finite clifford algebras i classification a classification of real matrix irreducible representations of finite dimensional real clifford algebras has been made in contrast to the case of complex representation three distinct types of representations can be obtained which we call normal almost complex and quaternionic the dimension of the latter two cases is twice as large as that of the normal representation a criteria for a given clifford algebra to possess a particular type of the representations is also given with some applications

analysis in space time bundles iv natural bundles deforming into and composed of the same invariant factors as the spin and form bundles abstract we study two interesting new bundles over the universal cosmos m or maximal isotropic space time which may be physically applicable the treatment is from a homogeneous vector bundle point of view and uses the notation and some of the results of the treatment in papers i iii s m paneitz and i e segal j funct anal 47 1982 78 142 49 1982 335 414 54 1983 18 22 of conventional bundles over m the spannor bundle deforms into essentially the usual spinor bundle as a conformally invariant parameter that may be interpreted as the space curvature becomes arbitrarily small from a minkowski space standpoint however the spannors involve a nontrivial action of space time translations that deforms into a trivial action in the spinor limit and also have more complex transformation properties under discrete symmetries also studied are the plyors consisting of the dual to the bundle product of the spannors with themselves composition series for the spannor and plyor section spaces are treated relative to the conformal group and irreducible subquotients are identified with certain that occur in conventional bundles in particular factors corresponding to the maxwell and massless dirac equations and which may represent certain of the observed elementary particles are determined a gauge and conformally invariant nonlinear coupling between spannors and plyors constituting essentially a generalization of that used in quantum electrodynamics is developed and an associated invariant nonlinear partial differential equation is derived covariant and causal quantization for spannors as fermions and plyors as bosons is formulated algebraically the present treatment is basically mathematical but physical motivations and possible interpretations are briefly noted

the trivial connection contribution to witten s invariant and finite type invariants of rational homology spheres we derive an analog of the melvin morton bound on the power series expansion of the colored jones polynomial of algebraically split links and boundary links this allows us to produce a simple formula for the trivial connection contribution to witten s invariant of rational homology spheres we show that then th term in the 1 k expansion of the logarithm of this contribution is a finite type invariant of ohtsuki order 3n and of at most garoufalidis ordern

realization extension and classification of certain physically important groups and algebras an associative algebra of differential forms with division has been constructed the algebra of forms in each different space provides a practical realization of the universal clifford algebra of that space a classification of all such algebras is given in terms of two distinct types of algebras nk and sk the former include the dihedral quaternion and majorana algebras the latter include the complex spinor and dirac algebras the associative product expresses hodge duality as multiplication by a basis element this makes possible the realization of higher order algebras in a calculationally useful algebraic setting the fact that the associative algebras as well as the enveloped lie algebras are precisely those arising in physics suggests that this formalism may be a convenient setting for the formulation of basic physical laws

computer calculation of witten s 3 manifold invariant witten s 2 1 dimensional chern simons theory is exactly solvable we compute the partition function a topological invariant of 3 manifolds on generalized seifert spaces thus we test the path integral using the theory of 3 manifolds in particular we compare the exact solution with the asymptotic formula predicted by perturbation theory we conclude that this path integral works as advertised and gives an effective topological invariant

unitary representations of the 4 1 de sitter group on irreducible representation spaces of the poincare group the construction of the principal continuous series of unitary representations of the simply connected covering group of the 4 1 de sitter group on unitary irreducible representation spaces of the poincare group is presented a unitary irreducible representation space of this covering group of the de sitter group is realized as the direct sum of two irreducible representation spaces of the poincare group possible physical implications are indicated in particular an interpretation of the instantaneous velocity operator in the dirac theory as the spin part of the de sitter boosts is given we obtain a simple method of computing the matrix elements of the generators of the de sitter group in an so 4 basis using the matrix elements of the generators of the four dimensional euclidean group also we obtain explicit expressions for certain matrix elements between the spinor and so 4 basis of the representation space as functions on the coset space so 4 so 3

electromagnetism and the holomorphic properties of spacetime the cauchy riemann equations of holomorphy are extended to fields in higher dimensional spaces in a framework of clifford algebras the equations of holomorphy in minkowski spacetime turn out to be the maxwell equations in vacuum the lorentz gauge condition is a result of the holomorphy sources can be included in an extension of the residue theorem where charges correspond to the residues

the relationship between finite groups and clifford algebras clifford algebras are traditionally realized in terms of a specific set of representation matrices this paper provides a more effective alternative by giving the finite group associated with each clifford algebra all the representation independent algebraic results which are really direct consequences of the underlying group structure can thus be derived in an easier and more general manner there are five related but distinct classes of finite groups associated with the clifford algebras these groups are constructed from the complex cyclic quaternion and dihedral groups in a way which is discussed here in detail of particular utility is a table which lists the order structure of each group this permits the immediate identification of any clifford algebra in any dimension

canonical q deformations of noncompact lie super algebras the author proposes a procedure for q deformations of the real forms g of complex lie super algebras associated with generalized cartan matrices the procedure gives different q deformations for the nonconjugate cartan subalgebras of g the author gives several illustrations e g q deformed lorentz and conformal super algebras the q deformed conformal algebra contains as a subalgebra a q deformed poincare algebra and as hopf subalgebras two conjugate 11 generator q deformed weyl algebras the q deformed lorentz algebra is a hopf subalgebra of both weyl algebras

chern simons witten invariants of lens spaces and torus bundles and the semiclassical approximation we derive explicit formulas for the chern simons witten invariants of lens spaces and torus bundles overs1 for arbitrary values of the levelk most of our results are for the groupg su 2 though some are for more general compact groups we explicitly exhibit agreement of the limiting values of these formulas ask with the semiclassical approximation predicted by the chern simons path integral

a formula for casson s invariant suppose h is a homology 3 sphere obtained by dehn surgery on a link l in a homology 3 sphere m if every pair of components of l has zero linking number in m then we give a formula for the casson invariant x h in terms of a m the surgery coefficients of l and a certain coefficient from each of the conway polynomials of l and all its sublinks a few consequences of this formula are given introduction andrew casson has discovered an integral invariant of oriented homology 3 spheres which reduces mod 2 to the rochlin or n invariant 1 in this paper we give a formula for casson s invariant in the case where the homology sphere h is obtained by dehn surgery on a link l in a homology sphere m and furthermore the linking number between every pair of components of l is zero note that every homology 3 sphere can be obtained in this way with m s3 in this case the casson invariant x h can be expressed in terms of x m the surgery coefficients of l and a certain coefficient from each of the conway polynomials of l and all its sublinks to illustrate the utility of this formula we compute the casson invariant of homology spheres that arise as cyclic branched covers branched along untwisted double knots we also describe how the x invariant changes when a crossing of the framed link l between two strands of the same component is changed this provides at least in this setting an effective method for computing the invariant an interesting question is whether it is possible to show directly that the formula we give is preserved by the kirby rolfsen calculus and thus give an alternative proof of the invariance of x if l1 and l2 are two rationally framed links representing the same manifold then l1 can be transformed into l2 by a sequence of three types of moves homeomorphism twisting and trivial insertion or deletion 8 if the linking number between every pair of components of l1 is zero then the first two types of moves will preserve this property while the third move may not however if l1 can be transformed into l2 through a sequence of links all of whose linking numbers are all zero then we show that the formula for x is preserved we do not know if such a transformation is in general possible in order to state casson s theorem and the formula for x h we first establish some notation suppose l k1 kn is a framed link in a 3 manifold m with received by the editors october 22 1985 1980 mathematics subject classification 1985 revision primary 57m25 supported in part by the nsf under grant no dms 8502940 x 1986 american mathematical society 0002 9947 86 1 00 25 per page

self maps of flag manifolds rationally a map between flag manifolds is seen to be determined up to homotopy by the homomorphism it induces on cohomology two algebraic results for cohomology endomorphisms then serve a to determine those flag manifolds which have nontrivial self maps that factor through a complex projective space and b for a special class of flag manifolds to classify the self maps of their rationalizations up to homotopy 0 introduction the homotopy classes of self maps of a complex projective space are classified by an integral degree and we show that for rational homotopy there is an analogous result for a much larger class of homogeneous spaces these spaces are quotients of a unitary group by a closed connected subgroup of maximal rank and just as for projective spaces the degree of a self map is determined by its effect on a two dimensional cohomology class the first step is to observe that homotopy classes of endomorphisms of the minimal model for the rational homotopy type of such a space correspond bijectively with endomorphisms of the rational cohomology algebra see theorem 1 1 the second step is to classify the cohomology endomorphisms such a classification was given in gh1 for the grassmann manifold of complex p planes in complex n space for n greater than 2p2 and it follows that the rational homotopy classes of self maps are indeed classified by a rational degree here in theorems 1 3 and 1 4 we give the analogous result for the flag manifolds m 1 p q u 1 p q u 1 x u p x u q the cohomology endomorphisms fall into a finite number often one of families each of which is classified by a rational degree some of these families are represented by the endomorphisms induced by conjugation in u 1 p q by elements of the normalizer of u 1 x u p x u q but if 1 p and q are distinct we get only the identity in this way the rest are represented by idempotents which we call projective endomorphisms because they factor through the cohomology of complex projective space such projective endomorphisms for any flag manifold are in turn classified in theorem 1 3 by factorizations over z of the polynomial

unitary representations of the 4 1 de sitter group on unitary irreducible representation spaces of the poincare group equivalence with their realizations as induced representations in a previous work we have constructed realizations of the principal continuous series of unitary irreducible representations of the simply connected covering group of the 4 1 de sitter group on unitary irreducible representation spaces of the simply connected covering group of the poincare group in this work we demonstrate the equivalence of the representations constructed in the previous work with their realizations as induced representations

n dimensional spinors their properties in terms of finite groups a classification scheme is presented for the finite multiplicative group generated by the gamma matrices associated with a given clifford algebra this group reflects the periodicities observed in n dimensional spinors and its representations and other properties are studied thus highlighting the dependence on the number of spacelike and timelike vectors the reality of the representations is examined and tabulated application is made to the imposition of majorana and weyl conditions

harmonic analysis on spannors the spannor representation of the universal covering group of the conformal group of minkowski space time is studied herein it is an example of an indecomposable representation various parallelizations for spannors are described normalized k finite basis fields for the spannors are introduced and a computation of the actions of the scale generator and other generators of the conformal group on these basis fields is given the irreducibility of all irreducible composition factors is established and it is decided which ones are infinitesimally unitary two sets of generators for maximal abelian subalgebras of the enveloping algebra of the conformal group are introduced the action of one of the sets on k finite basis fields in the spannor representation is studied this is important for understanding of the problem of degeneracy in the spannor representation since k types occur with multiplicity 2 many calculations make use of the lowest highest weight module structures of the unitarizable and positive negative energy irreducible composition factors and thus many of our results and techniques can be used to initiate a study of the quantum deformations of the associated lie algebra representations results on the poincare content of the unitarizable irreducible composition factors are stated and unitary equivalences between certain of these factors are established

a general method of generating and classifying clifford algebras a general construction program is given for the generation of higher order clifford algebras of various signatures together with their faithful representations in terms of pauli type operators the analysis is based upon several isomorphism theorems that provide a simpler understanding of the standard classification scheme of these algebras and an improved signature index notation is suggested that identifies each of the five types of clifford algebras

three dimensional quantum gravity chern simons theory and the a polynomial we study three dimensional chern simons theory with complex gauge group sl 2 which has many interesting connections with three dimensional quantum gravity and geometry of hyperbolic 3 manifolds we show that in the presence of a single knotted wilson loop in an infinite dimensional representation of the gauge group the classical and quantum properties of such theory are described by an algebraic curve called the a polynomial of a knot using this approach we find some new and rather surprising relations between the a polynomial the colored jones polynomial and other invariants of hyperbolic 3 manifolds these relations generalize the volume conjecture and the melvin morton rozansky conjecture and suggest an intriguing connection between the sl 2 partition function and the colored jones polynomial

composition of interactions in relativistic quantum theory a scheme of composition of interactions which is the generalisation of sokolov s method 1975 of packing operators is proposed this scheme can be used in all forms of relativistic quantum dynamics in contrast to the formulation of coester and polyzou 1982 the author makes no use of the properties of multichannel wave operators but uses only the properties of the poincare group unitary representations a solution to the problem of composition of interactions in the instant form is given and it is shown that this solution agrees with that obtained by coester and polyzou

su 2 su 2 shift operators and representations of so 5 su 2 su 2 shift operators analogous to the su 2 shift operators developed and used by the author for the classification and analysis of representations of lie algebras in an su 2 or so 3 basis are obtained for the su 2 su 2 lie algebra in the case where one has an additional set of operators forming an irreducible four dimensional tensor representation of su 2 su 2 the shift operators obtained are used to treat the representations of so 5 in an su 2 su 2 basis

on the melvin morton rozansky conjecture we prove a conjecture stated by melvin and morton and elucidated further by rozansky saying that the alexander conway polynomial of a knot can be read from some of the coefficients of the jones polynomials of cables of that knot i e coefficients of the colored jones polynomial we first reduce the problem to the level of weight systems using a general principle which may be of some independent interest and which sometimes allows to deduce equality of vassiliev invariants from the equality of their weight systems we then prove the conjecture combinatorially on the level of weight systems finally we prove a generalization of the melvin morton rozansky mmr conjecture to knot invariants coming from arbitrary semi simple lie algebras as side benefits we discuss a relation between the conway polynomial and immanants and a curious formula for the weight system of the colored jones polynomial

continuous family of finite dimensional representations of a solvable lie algebra arising from singularities abstract a natural mapping from the set of complex analytic isolated hypersurface singularities to the set of finite dimensional lie algebras is first defined it is proven that the image under this natural mapping is contained in the set of solvable lie algebras this approach gives rise to a continuous inequivalent family of finite dimensional representations of a solvable lie algebra

an embedding of the poincare lie algebra into an extension of the lie field of so0 1 4 the isomorphism between algebraic extensions of the lie fields of so0 1 4 and the poincare group are considered herein it is shown that the principal series of unitary ray representations of so0 1 4 are associated via the isomorphism with real mass positive and negative energy representations of the poincare lie algebra with arbitrary spin results on the most degenerate exceptional series of so0 1 4 representations are also given

micro local characterization of quasi homogeneous singularities a moduli algebra a v of hypersurface singularity v 0 is a finite dimensional c algebra in 1982 mather and yau proved that two germs of complex analytic hypersurfaces of the same dimension with isolated singularities are biholomorphically equivalent if and only if their moduli algebra are isomorphic it is a natural question to ask for a necessary and sufficient condition for a complex analytic isolated hypersurface singularity to be quasi homogeneous in terms of its moduli algebra in this paper we prove that v 0 admits a quasi homogeneous structure if and only if its moduli algebra is isomorphic to a finite dimensional nonnegatively graded algebra in 1983 yau introduced a finite dimensional lie algebra l v to an isolated hypersurface singularity v 0 l v is defined to be the algebra of derivations of the moduli algebra a v and is finite dimensional we prove that v 0 is quasi homogeneous singularity if 1 l v is isomorphic to a nonnegatively graded lie algebra without center 2 there exists e in l v of degree zero such that e d i i d i for any d i in l v of degree i and 3 for any element a m m 2 where m is the maximal ideal of a v ae is not in degree zero part of l v

dirac general covariance and tetrads i clifford and lie bundles and torsion the isomorphic map of clifford and lie bundles to arbitrary coordinate atlases using a global orthonormal tetrad field on a parallelizable space time is used to construct a fully covariant dirac spinor theory the klein gordon equation exhibits a natural spin torsion coupling of the einstein cartan form the torsion coming from the tetrad field the tetrad connection coefficients are explicitly derived in addition to their relationship to the usual levi civita coefficients various topological conditions for vanishing torsion are given the dirac and adjoint dirac equations are obtained from a simple lagrangian and the structure of the adjoint equation is discussed

algebras with three anticommuting elements i spinors and quaternions a general construction of alternative algebras with three anticommuting elements and a unit is given as an exhaustive result over the real and complex fields we obtain the clifford algebras h quaternions n1 dihedral clifford algebra which is related to real 2 spinors and s1 algebra of pauli matrices which is related to complex 2 spinors what is important is that the algebras n1 and s1 possess inverses everywhere except on a region akin to the light cone of the minkowski space while the quaternion algebra h has inverses everywhere except at the zero element we discuss the reasons why the three algebras n1 h and s1 are so difficult to distinguish in the representation space of 2 2 complex matrices

gell mann formula for simple complex lie groups and geometric quantization a lie poisson isomorphism for lie algebras g and q k k is derived in a closed form where g is complex and simple and k is the maximal compact subalgebra in g this isomorphism enables one to find a generalized gell mann formula relating unitary irreducible representations of the corresponding lie groups g and q the case g sl n c k su n is considered in detail a simple relation between characters is obtained

group representations arising from lorentz conformal geometry abstract it is shown that there exist conformally covariant differential operators d 2 l k of all even orders 2 l on differential forms of all orders k in the double cover n of the n dimensional compactified minkowski space n these act as intertwining differential operators for natural representations of o 2 n the conformal group of n for even n the resulting decompositions of differential form representations of o 2 n the orthochronous conformal group produce infinite families of unitary representations the most interesting of which are carried by positive mass squared positive frequency quotients for 2l n 2k physically these generalize unitary representations of the conformal group associated with the modified wave operator d 2 0 n 2 2 2 and the maxwell operator on vector potentials d 2 n 2 2 d all the representation spaces produced unitary and nonunitary may be viewed as infinite systems of harmonic oscillators as a by product of the spectral resolution of the d 2 l k one gets some striking wave propagative properties for all of the equations d 2 l k 0 including huygens principle in the curved spacetime n the operators d 2 l k have not been seen before except in the special cases k 0 or n and k n 2 2 l 1 the maxwell operator thus much new information is obtained even in the physical case n 4

some topological properties of kahler manifolds and homogeneous spaces on donne des calculs explicites des groupes d homotopie rationnels du monoide topologique de toutes les auto equivalences d homotopie d une variete de kahler homogene

solvability of lie algebras arising from isolated singularities and nonisolatedness of singularities defined by sl 2 c invariant polynomials

the covariant linear oscillator and generalized realization of the dynamical su 1 1 symmetry algebra an exactly solvable problem for the finite difference schrodinger equation in the relativistic configurational space is considered the appropriate finite difference generalization of the factorization method is developed the theory of new special functions the relativistic hermite polynomials in which the solutions are expressed is constructed

some group theoretical aspects of the so0 1 4 invariant theory some properties of unitary representations of the de sitter group so0 1 4 are considered in the case of spin zero and positivity of the mass operator these representations involve a decomposition which ensures the fulfilment of the cluster separability condition and allows for the conventional interpretation of the de sitter analogues of momentum and boost operators the decomposition of the tensor product of representations belonging to the principal series contains the representations of the principal series with all masses under natural assumptions concerning the interaction operators there are no bound states in the theory which is due to the universal de sitter antigravity

the problem of interactions in de sitter invariant theories the structure of representations describing systems of free particles in the theory with the invariance group is investigated the property of the particles to be free means as usual that the representation describing a many particle system is the tensor product of the corresponding single particle representations i e no interaction is introduced it is shown that the mass operator contains only continuous spectrum in the interval and such representations are unitarily equivalent to ones describing interactions gravitational electromagnetic etc this means that there are no bound states in the theory and the hilbert space of the many particle system contains a subspace of states with the following property the action of free representation operators on these states is manifested in the form of different interactions possible consequences of the results are discussed

on fibrations with formal elliptic fibers abstract we prove that for a fibration of simply connected spaces of finite type f e b with f being positively elliptic and h f q not possessing non trivial derivations of negative degree the base b is formal if and only if the total space e is formal moreover in this case the fibration map is a formal map as a geometric application we show that positive quaternion kahler manifolds are formal and so are their associated twistor fibration maps

reshetikhin s formula for the jones polynomial of a link feynman diagrams and milnor s linking numbers feynman diagrams are used to prove a formula for the jones polynomial of a link derived recently by reshetikhin this formula presents the colored jones polynomial as an integral over the coadjoint orbits corresponding to the representations assigned to the link components the large k limit of the integral can be calculated with the help of the stationary phase approximation the feynman rules allow one to express the phase in terms of integrals over the manifold and the link components its stationary points correspond to flat connections in the link complement a relation between the dominant part of the phase and milnor s linking numbers is conjectured it is checked explicitly for the triple and quartic numbers by comparing their expression through the massey product with feynman diagram integrals

derivations of the moduli algebras of weighted homogeneous hypersurface singularities abstract let r c x 1 x 2 z n f where f is a weighted homogeneous polynomial defining an isolated singularity at the origin then r and der r the lie algebra of derivations on r are graded it is well known that der r has no negatively graded component 10 j wahl conjectured that the above fact is still true in higher codimensional case provided that r c x 1 x 2 x n f 1 f 2 f m is an isolated normal and complete intersection singularity and f 1 f 2 f m are weighted homogeneous polynomials with the same weight type w 1 w 2 w n on the other hand the first author yau conjectured that the moduli algebra a v c x 1 x 2 x n f x 1 f x n has no negatively weighted derivations where f is a weighted homogeneous polynomial defining an isolated singularity at the origin assuming this conjecture has a positive answer he gave a characterization of weighted homogeneous hypersurface singularities only using the lie algebra der a v of derivations on a v the conjecture of yau can be thought as an artinian analogue of j wahl s conjecture for the low embedding dimension the yau conjecture has a positive answer in this paper we prove this conjecture for any high dimensional singularities under the condition that the lowest weight is bigger than or equal to half of the highest weight

analysis in space time bundles i general considerations and the scalar bundle abstract mathematical features involved in the systematic development of elementary particle theory on an alternative cosmos space time are presented bundles representative of physical fields are studied in the unique variant of minkowski space m 0 enjoying similar properties of causality and symmetry the universal cosmos m is the universal cover of the causal compactification of m 0 the bundles studied are induced from representations of the scale extended poincare group which forms the isotropy group in m of the universal cover g s u 2 2 of the connected component of the group of all causal transformations on m discrete symmetries and higher dimensional cases are also discussed the primary focus is on the temporal evolution especially stability involving positivity of the energy wave equations implicative of finite propagation velocity and the unitarity and or composition series of associated actions of g general spin bundles on m are treated parallelized and correlated with bundles on m 0 associated covariant wave equations and the spectral resolution of fundamental quantum numbers are studied in detail in the scalar case

some solutions of the yang mills equations in dimension greater than four the yang mills equations for gauge fields of an arbitrary semisimple lie group g in euclidean spaces rn with dimension n 4 is considered for gauge fields aa we introduce the ansatze which reduce the yang mills equations in rn to a set of nonlinear differential matrix equations rouhani ward equations this permits us to write out the explicit form of cosmological plane wave and other types of solutions

finiteness of physics and its possible consequences the modular analog of representations of the so 1 4 algebra for a system of two spinless particles is considered in the framework of approach proposed by the author earlier in which physical systems are described by the elements of a linear space over a finite field and operators of physical quantities by linear operators in this space the eigenvalues of the free mass operator and corresponding eigenvectors are found it is shown that if the finite field under consideration satisfies some properties then the particles necessarily interact with each other the corresponding interaction is universal since it is fully defined by the masses of particles and characteristics of the finite field the formula for the kernel of the interaction operator is derived but since the kernel is defined by the sum over the galois field and this sum cannot be calculated explicitly then it cannot be determined how far the above interaction manifests itself in the usual conditions

representations of the poincare lie algebra from unitary representations of so 1 4 an algebraic isomorphism of an extension of the lie field of so 1 4 and an extension of the poincare lie field was given in havlicek and moylan j math phys 34 5320 1993 here we use the isomorphism described in that paper in order to obtain matrix elements of the poincare lie algebra generators in a normalizable basis for an arbitrary unitary spherical principal series representation of so0 1 4

construction of representations of poincare group using lie fields in this paper we give an explicit construction of the unitary irreducible representations of the poincare groups in 2 3 and 4 space time dimensions on hilbert spaces associated with the schrodinger representation of the weyl algebra for n 1 2 and 3 respectively our method of constructing the representations uses extension and localization of the enveloping algebras associated with these weyl algebras and the poincare algebras

a contribution of the trivial connection to the jones polynomial and witten s invariant of 3d manifolds ii we extend the results of our previous paper 1 from knots to links by using a formula for the jones polynomial of a link derived recently by n reshetikhin we establish a relation between the parameters of this formula and the multivariable alexander polynomial this relation is illustrated by an example of a torus link we check that our expression for the alexander polynomial satisfies some of its basic properties finally we derive a link surgery formula for the loop corrections to the trivial connection contribution to witten s invariant of rational homology spheres

grassmann calculus pseudoclassical mechanics and geometric algebra a reformulation of grassmann calculus is presented in terms of geometric algebra a unified language for physics based on clifford algebra in this reformulation grassmann generators are replaced by vectors so that every product of generators has a natural geometric interpretation the calculus introduced by berezin the method of second quantization academic new york 1966 is shown to be unnecessary amounting to no more than an algebraic contraction this approach is not only conceptually clearer but it is also computationally more efficient as demonstrated by treatments of the grauss integral and the grassmann fourier transform the reformulation is applied to pseudoclassical mechanics ann phys 104 336 1977 where it is shown to lead to a new concept the multivector lagrangian to illustrate this idea the three dimensional fermi oscillator is reformulated and solved and its symmetry properties discussed as a result a new and highly compact formula for generating super lie algebras is revealed the paper ends with a discussion of quantization outlining a new approach to fermionic path integrals

chern simons perturbation theory ii we study the perturbation theory for three dimensional chern simons quantum field theory on a general compact three manifold without boundary we show that after a simple change of variables the action obtained by brs gauge fixing in the lorentz gauge has a superspace formulation the basic properties of the propagator and the feynman rules are written in a precise manner in the language of differential forms using the explicit description of the propagator singularities we prove that the theory is finite finally the anomalous metric dependence of the 2 loop partition function on the riemannian metric which was introduced to define the gauge fixing can be cancelled by a local counterterm as in the 1 loop case in fact the counterterm is equal to the chern simons action of the metric connection normalized precisely as one would expect based on the framing dependence of witten s exact solution invited talk at the xxth conference on differential geometric methods in physics new york june 1991

jones witten invariants for nonsimply connected lie groups and the geometry of the weyl alcove abstract the quotient process of muger and bruguieres is used to construct modular categories and tqfts out of closed subsets of the weyl alcove of a simple lie algebra in particular it is determined at which levels closed subsets associated to nonsimply connected groups lead to tqfts many of these tqfts are shown to decompose into a tensor product of tqfts coming from smaller subsets the prime subsets among these are classified and apart from some giving tqfts depending on homology as described by murakami ohtsuki and okada they are shown to be in one to one correspondence with the tqfts predicted by dijkgraaf and witten to be associated to chern simons theory with a nonsimply connected lie group thus in particular a rigorous construction of the dijkgraaf witten tqfts is given as a byproduct a purely quantum groups proof of the modularity of the full weyl alcove for arbitrary quantum groups at arbitrary levels is given

analysis in space time bundles ii the spinor and form bundles abstract the structures of the spin and form bundles over the universal cosmos m and their relations with corresponding bundles over the minkowski space m 0 canonically imbedded in m are treated wave equations covariant with respect to the causal group g of m are studied their solution manifolds and other stable essentially positive energy invariant subspaces of the section spaces of the bundles are determined and the indecomposability of relevant invariant subspace chains is shown explicit parallelizations of the bundles are applied to the dirac and maxwell equations on m a basis for spinor fields that diagonalizes a complete set of k covariant quantum numbers k maximal essentially compact subgroup of g is developed local multilinear invariants of bundles over m are treated and specialized to convergent mathematical versions of the fermi and yukawa interaction lagrangians that are g invariant for the appropriate conformal weights

on the integrability of certain symmetric representations of the lie algebra of so0 4 1 a proof of the existence of an essentially self adjoint extension of a symmetric so0 4 1 nelson operator which is constructed out of the generators of a positive mass arbitrary spin unitary irreducible representation of the poincare group is presented our analysis of so0 4 1 and its lie algebra provides us with an example of an observation of harish chandra there exist subspaces of the space of differentiable vectors of a representation of a noncompact group which are invariant under the lie algebra but the closures of the subspaces are not invariant under the group the chief results of this paper should hold true for so0 n 1 in particular we should have a realization of an arbitrary principal series irreducible unitary representation of so0 n 1 on the direct sum of two identical unitary irreducible representation spaces of the motion group in an n dimensional minkowski space which has one timelike dimension

linearly and nonlinearly transforming fields on homogeneous spaces of the 4 1 de sitter group scalar functions on the homogeneous spaces hr of the de sitter group g so 4 1 are studied where the spaces hr are of the form g k with k being a subgroup of the lorentz group h so 3 1 contained in so 4 1 the spaces hr can be regarded as fiber bundles er er g h h k with the base v 4 g h being a space of constant negative curvature characterized by a fundamental length parameter r 4 1 de sitter space and the fiber s h k being a homogeneous space of the lorentz group the action of g on the spaces er is a linear action on v4 and a nonlinear action on s where the latter action is defined by a generalized wigner rotation a gauge theory based on the 4 1 de sitter group is investigated with matter represented in terms of a generalized wave function x y with x u4 riemann cartan space time v 4 and y s which is defined as a map from a cross section on the bundle e e u4 f er g so 4 1 over space time u4 with fiber f er g k and structural group g so 4 1 into the complex numbers th

algebras with three anticommuting elements ii two algebras over a singular field the clifford algebra generated by the elements 1 with 2 1 is an abelian ring of dimension two with properties analogous to the complex field c the ring has a string of singular inverses and may be regarded as a singular field which circumvents both the fundamental theorem of algebra and the frobenius theorem we construct two associative algebras of dimension four over the clifford algebra 1 and the biquaternions of clifford 2 and demonstrate that both algebras possess inverses everywhere except on a singular region akin to the light cone of the minkowski space matrix representations are discussed as well as the importance of the algebras 1 and 2 in the description of physical vector fields

modular representations as a possible basis of finite physics the approach in which physical systems are described by the elements of a linear space over a finite field and operators of physical quantities by linear operators in this space is discussed the mathematical formulation of the correspondence between such a description and the conventional one is given for a case when the characteristic of the finite field is sufficiently large the above correspondence is considered in the examples of finite analogs of representations of so 2 3 so 1 4 and osp 1 4 algebras finite analogs of representations of infinite dimensional algebras are briefly discussed

on the non existence of negative weight derivations of the new moduli algebras of singularities abstract we propose a new conjecture about the non existence of negative weight derivations of the new moduli algebras of weighted homogeneous hypersurface singularities and verify this conjecture up to dimension three

variation of complex structures and variation of lie algebras given a family of complex projective hypersurfaces in cp the torelli problem studied by p griffiths and his school asks whether the period map is injective on that family i e whether the family of complex hypersurfaces can be distinguished by means of their hodge structures a complex projective hypersurface in cp can be viewed as a complex hypersurface with isolated singularity in c t let v z c 1 f z o be a complex hypersurface with isolated singularity at the origin the moduli algebra of v 0 is a v