A resolution (minimal model) of the PROP for bialgebras

This paper is concerned with a minimal resolution of the PROP for bialgebras (Hopf algebras without unit, counit and antipode). We prove a theorem about the form of this resolution (Theorem 15) and give, in Section 5, a lot of explicit formulas for the differential.     

Radford's biproduct for quasi-Hopf algebras and bosonization

Let B be a braided Hopf algebra (with bijective antipode) in the category of left Yetter-Drinfeld modules over a quasi-Hopf algebra H. As in the case of Hopf algebras (J. Algebra 92 (1985) 322), the smash product B#H defined in (Comm. Algebra 28(2) (2000) 631) and a kind of smash coproduct afford a quasi-Hopf algebra structure on B⊗H. Using this, we obtain the structure of quasi-Hopf algebras with a projection. Further we will use this biproduct to describe the Majid bosonization (J. Algebra 163 (1994) 165) for quasi-Hopf algebras.     

Periodic polynomial of trace maps

Let σ be an endomorphism of the free group on two generators and Φσ the trace map associated with σ. A polynomial P is said to be periodic for σ if, for some positive integer n, it is invariant under , i.e., . In this note we study the structure of the ring of periodic polynomials for σ.     

A property that characterizes Euler characteristic among invariants of combinatorial manifolds

If a real valued invariant of compact combinatorial manifolds (with or without boundary) depends only on the number of simplices in each dimension in the manifold, then the invariant is completely determined by the Euler characteristic of the manifold and its boundary. So essentially, the Euler characteristic is the unique invariant of this type.     

Comparison of admissibility conditions for cyclotomic Birman-Wenzl-Murakami algebras

We show the equivalence of admissibility conditions proposed by Wilcox and Yu (in press) [11] and by Rui and Xu (2009) [9] for the parameters of cyclotomic BMW algebras.     

Projection operators and states in the tensor product of quaternion hilbert modules

Following the construction of tensor product spaces of quaternion Hilbert modules in our previous paper, we define the analogue of a ray (in a complex quantum mechanics) and the corresponding projection operator, and through these the notion of a state and density operators. We find that there is a one-to-one correspondence between a state and an equivalence class of vectors from the tensor product space, which gives us another method to define the gauge transformations.On sabbatical leave from the School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. Work supported in part by a fellowship from the Ambrose Monell Foundation.     

Tensor product of quaternion hilbert modules

One of the main problems in the theory of quaternion quantum mechanics has been the construction of a tensor product of quaternion Hilbert modules. A solution to this problem is given by studying the tensor product of quaternion algebras (over the reals) and some of its quotient modules. Real, complex, and (covariant) quaternion scalar products are found in the tensor product spaces. Annihilationcreation operators are constructed, corresponding to the second quantization of the quaternion quantum theory with Bose-Einstein or Fermi-Dirac statistics. The gauge transformations of a tensor product vector and the gauge fields are studied.On Sabbatical leave from the School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. Work supported in part by a fellowship from the Ambrose Monell Foundation.     

Filtered ends of infinite covers and groups

Let f:A→B be a covering map. We say that A has e filtered ends with respect to f (or B) if, for some filtration {K_n} of B by compact subsets, A−f⁻¹(K_n) “eventually” has e components. The main theorem states that if Y is a (suitable) free H-space, if K<H has infinite index, and if Y has a positive finite number of filtered ends with respect to H?Y, then Y has one filtered end with respect to K?Y. This implies that if G is a finitely generated group and K<H<G are subgroups each having infinite index in the next, then implies that , where is the number of filtered ends of a pair of groups in the sense of Kropholler and Roller.     

Submersions on open symplectic manifolds

Let M be an open manifold with a symplectic form Ω, and N a manifold with dimN<dimM. We prove that submersions with symplectic fibres satisfy the h-principle. Such submersions define Dirac manifold structures on the given manifold. As an application to this result we show that CPn?CPk−1 admits a submersion into R2(2k−n) with symplectic fibres for n/2<k?n.     

Polynomial identities for the Jordan algebra of a degenerate symmetric bilinear form

Let _Jn$J_n$ be the Jordan algebra of a degenerate symmetric bilinear form. In the first section we classify all possible G  -gradings on _Jn$J_n$ where G   is any group, while in the second part we restrict our attention to a degenerate symmetric bilinear form of rank n−1$n-1$, where n is the dimension of the vector space V   defining _Jn$J_n$. We prove that in this case the algebra _Jn$J_n$ is PI-equivalent to the Jordan algebra of a nondegenerate bilinear form.     

Combinatorial Lie bialgebras of curves on surfaces

Goldman (Invent. Math. 85(2) (1986) 263) and Turaev (Ann. Sci. Ecole Norm. Sup. (4) 24 (6) (1991) 635) found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in the generators of the fundamental group and their inverses. We give a combinatorial algorithm to compute this Lie bialgebra on this vector space of cyclic words. Using this presentation, we prove a variant of Goldman's result relating the bracket to disjointness of curve representatives when one of the classes is simple. We exhibit some examples we found by programming the algorithm which answer negatively Turaev's question about the characterization of simple curves in terms of the cobracket. Further computations suggest an alternative characterization of simple curves in terms of the bracket of a curve and its inverse. Turaev's question is still open in genus zero.     

Semisimple Hopf actions on commutative domains

Let H be a semisimple (so, finite dimensional) Hopf algebra over an algebraically closed field k of characteristic zero and let A be a commutative domain over k. We show that if A arises as an H-module algebra via an inner faithful H-action, then H must be a group algebra. This answers a question of E. Kirkman and J. Kuzmanovich and partially answers a question of M. Cohen.     

Discrete Morse theory for totally non-negative flag varieties

In a seminal 1994 paper Lusztig (1994) [26], Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P)_?0 of an arbitrary (generalized, partial) flag variety G/P. He referred to this space as a “remarkable polyhedral subspace”, and conjectured a decomposition into cells, which was subsequently proven by the first author Rietsch (1998) [33]. In Williams (2007) [40] the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron, by conjecturing it to be a regular CW complex that is homeomorphic to a closed ball. In this article we use discrete Morse theory to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that the boundaries of the cells are homotopic to spheres, and the closures of cells are contractible. The latter part generalizes a result of Lusztig's (1998) [28], that (G/P)_?0 - the closure of the top-dimensional cell - is contractible. Concerning our result on the boundaries of cells, even the special case that the boundary of the top-dimensional cell (G/P)_>0 is homotopic to a sphere, is new for all G/P other than projective space.     

Discrete and continuous Yang-Mills measure for non-trivial bundles over compact surfaces

In this paper, we construct one Yang-Mills measure on an orientable compact surface for each isomorphism class of principal bundles with compact connected structure group over this surface. For this, we refine the discretization procedure used in a previous construction [9] and define a discrete theory on a new configuration space which is essentially a covering of the usual one. We prove that the measures corresponding to different isomorphism classes of bundles or to different total areas of the base space are mutually singular. We give also a combinatorial computation of the partition functions which relies on the formalism of fat graphs.     

Right distributive quasigroups on algebraic varieties

In this paper we develop a structure theory of algebraic right distributive quasigroups which correspond to closed and connected conjugacy classes generating algebraic Fischer groups (in the sense of [6]) such that the mappingx x ^–1 ax, fora , is an automorphism of (as variety). We also give examples of algebraic Fischer groups where this does not happen. It becomes clear that the class of algebraic right distributive quasigroups has nice properties concerning subquasigroups, normal subquasigroups and direct product.We give a complete classification of one- and two-dimensional as well as of minimal algebraic right distributive quasigroups.