Skein construction of idempotents in Birman-Murakami-Wenzl algebras

We give skein theoretic formulas for minimal idempotents in the Birman-Murakami-Wenzl algebras. These formulas are then applied to derive various known results needed in the construction of quantum invariants and modular categories. In particular, an elementary proof of the Wenzl formula for quantum dimensions is given. This proof does not use the representation theory of quantum groups and the character formulas.Comment: 26 pages, LaTeX with figures; Section 8 and details to the proof of Theorem 3.1 are adde

Formal Desingularization of Surfaces - The Jung Method Revisited -

In this paper we propose the concept of formal desingularizations as a substitute for the resolution of algebraic varieties. Though a usual resolution of algebraic varieties provides more information on the structure of singularities there is evidence that the weaker concept is enough for many computational purposes. We give a detailed study of the Jung method and show how it facilitates an efficient computation of formal desingularizations for projective surfaces over a field of characteristic zero, not necessarily algebraically closed. The paper includes a generalization of Duval's Theorem on rational Puiseux parametrizations to the multivariate case and a detailed description of a system for multivariate algebraic power series computations.Comment: 33 pages, 2 figure

An Intuitionistic Formula Hierarchy Based on High-School Identities

We revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal) propositional sequent calculi are formula (i.e. sequent) isomorphisms corresponding to the high-school identities, we show that one can obtain a more compact variant of a proof system, consisting of non-invertible proof rules only, and where the invertible proof rules have been replaced by a formula normalisation procedure. Moreover, for certain proof systems such as the G4ip sequent calculus of Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the non-invertible proof rules as strict inequalities between exponential polynomials; a careful combinatorial treatment is given in order to establish this fact. Finally, we extend the exponential polynomial analogy to the first-order quantifiers, showing that it gives rise to an intuitionistic hierarchy of formulas, resembling the classical arithmetical hierarchy, and the first one that classifies formulas while preserving isomorphism

Invariant subalgebras of affine vertex algebras

Given a finite-dimensional complex Lie algebra g equipped with a nondegenerate, symmetric, invariant bilinear form B, let V_k(g,B) denote the universal affine vertex algebra associated to g and B at level k. For any reductive group G of automorphisms of V_k(g,B), we show that the invariant subalgebra V_k(g,B)^G is strongly finitely generated for generic values of k. This implies the existence of a new family of deformable W-algebras W(g,B,G)_k which exist for all but finitely many values of k.Comment: Final version, proof of main result simplified. arXiv admin note: substantial text overlap with arXiv:1006.562

Search and witness problems in group theory

Decision problems are problems of the following nature: given a property P and an object O, find out whether or not the object O has the property P. On the other hand, witness problems are: given a property P and an object O with the property P, find a proof of the fact that O indeed has the property P. On the third hand(?!), search problems are of the following nature: given a property P and an object O with the property P, find something "material" establishing the property P; for example, given two conjugate elements of a group, find a conjugator. In this survey our focus is on various search problems in group theory, including the word search problem, the subgroup membership search problem, the conjugacy search problem, and others

Dependent Inductive and Coinductive Types are Fibrational Dialgebras

In this paper, I establish the categorical structure necessary to interpret dependent inductive and coinductive types. It is well-known that dependent type theories \`a la Martin-L\"of can be interpreted using fibrations. Modern theorem provers, however, are based on more sophisticated type systems that allow the definition of powerful inductive dependent types (known as inductive families) and, somewhat limited, coinductive dependent types. I define a class of functors on fibrations and show how data type definitions correspond to initial and final dialgebras for these functors. This description is also a proposal of how coinductive types should be treated in type theories, as they appear here simply as dual of inductive types. Finally, I show how dependent data types correspond to algebras and coalgebras, and give the correspondence to dependent polynomial functors.Comment: In Proceedings FICS 2015, arXiv:1509.0282