Stephen Sawin

• Fairfield University

We prove a version of Gabriel's theorem for locally finite-dimensional representations of infinite quivers. Specifically, we show that if $\Omega$ is any connected quiver, the category of locally finite-dimensional representations of $\Omega$ has unique representation type (meaning no two indecomposable representations have the same dimension vecto...

Gabriel’s Theorem states that the quivers which have finitely many isomorphism classes of indecomposable representations are exactly those with underlying graph one of the ADE Dynkin diagrams and that the indecomposables are in bijection with the positive roots of this graph. When the underlying graph is $$\varvec{A_n}$$ A n , these indecomposable...

The Hennings invariant for the small quantum group associated to an arbitrary simple Lie algebra at a root of unity is shown to agree with the Witten–Reshetikhin–Turaev (WRT) three-manifold invariant arising from Chern–Simons field theory for the same Lie algebra and the same root of unity on all integer homology three-spheres, at roots of unity wh...

An alternative construction of the invariant of homology three spheres valued in a completion of an integral polynomial ring associated to each quantized complex simple Lie algebra by Habiro and Lê [Unified quantum invariants for integral homology spheres associated with simple Lie algebras, Geom. Topol. 20 (2016) 2687–2835, doi:10.2140/gt.2016.20....

Gabriel's Theorem states that the quivers which have finitely many isomorphism classes of indecomposable representations are exactly those with underlying graph one of the ADE Dynkin diagrams and that the indecomposables are in bijection with the positive roots of this graph. When the underlying graph is An, these indecomposable representations are...

Using linear algebraic methods we show that every (possibly infinite-dimensional) representation of a quiver with underlying graph $A_{\infty, \infty}$ is Krull-Schmidt, as long as the arrows in the quiver eventually point outward.

The purpose of this paper is to discuss the categorical structure for a method of defining quantum invariants of knots, links and three-manifolds. These invariants can be defined in terms of right integrals on certain Hopf algebras. We call such an invariant of 3-manifolds a Hennings invariant. The work reported in this paper has its background in...

Many introductory courses in quantum mechanics include Feynman's time-slicing definition of the path integral, with a complete derivation of the propagator in the simplest of cases. However, attempts to generalize this, for instance to non-quadratic potentials, encounter formidable analytic issues in showing the successive approximations in fact co...

Feynman's time-slicing construction approximates the path integral by a product, determined by a partition of a finite time interval, of approximate propagators. This paper formulates general conditions to impose on a short-time approximation to the propagator in a general class of imaginary-time quantum mechanics on a Riemannian manifold which ens...

The Hennings invariant for the small quantum group associated to an arbitrary simple Lie algebra at a root of unity is shown to agree with the Chern-Simons (aka Jones-Witten or Reshetikhin-Turaev) invariant for the same Lie algebra and the same root of unity on all integer homology three- spheres, at roots of unity where both are defined. This part...

Feynman's time-slicing construction approximates the path integral by a product, determined by a partition of a finite time interval, of approximate propagators. This paper formulates general conditions to impose on a short-time approximation to the propagator in a general class of imaginary-time quantum mechanics on a Riemannian manifold which ens...

We introduce the south-pointing chariot, an intriguing mechanical device from ancient China. We use its ability to keep track of a global direction as it travels on an arbitrary path as a tool to explore the geometry of curved surfaces. This takes us as far as a famous result of Gauss on the impossibility of a faithful map of the globe, which start...

Following Feynman's prescription for constructing a path integral representation of the propagator of a quantum theory, a short-time approximation to the propagator for imaginary-time, N = 1 supersymmetric quantum mechanics on a compact, even-dimensional Riemannian manifold is constructed. The path integral is interpreted as the limit of products,...

This paper bounds the computational cost of computing the Kauffman bracket of a link in terms of the crossing number of that link. Specifically, it is shown that the image of a tangle with $g$ boundary points and $n$ crossings in the Kauffman bracket skein module is a linear combination of $O(2^g)$ basis elements, with each coefficient a polynomial...

This paper bounds the computational cost of computing the Kauffman bracket of a link in terms of the crossing number of that link. Specifically, it is shown that the image of a tangle with g boundary points and n crossings in the Kauffman bracket skein module is a linear combination of O(2g) basis elements, with each coefficient a polynomial with a...

Following Feynman's prescription for constructing a path integral representation of the propagator of a quantum theory, a short-time approximation to the propagator for imaginary time, N=1 supersymmetric quantum mechanics on a compact, even-dimensional Riemannian manifold is constructed. The path integral is interpreted as the limit of products of...

Following Feynman's prescription for constructing a path integral representation of the propagator of a quantum theory, a short-time approximation to the propagator for imaginary time, N=1 supersymmetric quantum mechanics on a compact, even-dimensional Riemannian manifold is constructed. The path integral is interpreted as the limit of products, de...

In a rigorous construction of the path integral for supersymmetric quantum mechanics on a Riemann manifold, based on Bär and Pfäffle’s use of piecewise geodesic paths, the kernel of the time evolution operator is the heat kernel for the Laplacian on forms. The path integral is approximated by the integral of a form on the space of piecewise geodesi...

For a finite-dimensional (but possibly noncompact) symplectic manifold with a compact group acting with a proper moment map, we show that the square of the moment map is an equivariantly perfect Morse function in the sense of Kirwan, show that certain integrals of equivariant cohomology classes localize as a sum of contributions from these compact...

For an arbitrary simple Lie algebra $\g$ and an arbitrary root of unity $q,$ the closed subsets of the Weyl alcove of the quantum group $U_q(\g)$ are classified. Here a closed subset is a set such that if any two weights in the Weyl alcove are in the set, so is any weight in the Weyl alcove which corresponds to an irreducible summand of the tensor...

For a finite-dimensional (but possibly noncompact) symplectic manifold with a compact group acting with a proper moment map, we show that the square of the moment map is an equivariantly perfect Morse function in the sense of Kirwan, and that the set of critical points of the square of the moment map is a countable discrete union of compact sets. W...

We develop the basic representation theory of all quantum groups at all roots of unity (that is, for q any root of unity, where q is defined as in [18]), including Harish–Chandra's theorem, which allows us to show that an appropriate quotient of a subcategory gives a semisimple ribbon category. This work generalizes previous work on the foundations...

The quotient process of Müger and Bruguières 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 TQF...

A version of Kirby calculus for spin and framed three-manifolds is given and is used to construct invariants of spin and framed three-manifolds in two situations. The first is ribbon *-categories which possess odd degenerate objects. This case includes the quantum group situations corresponding to the half-integer level Chern–Simons theories conjec...

The notion of 2-framed three-manifolds is defined. The category of 2-framed cobordisms is described, and used to define a 2-framed three-dimensional TQFT. Using skeletonization and special features of this category, a small set of data and relations is given that suffice to construct a 2-framed three-dimensional TQFT. These data and relations are e...

A version of Kirby calculus for spin and framed three-manifolds is given and is used to construct invariants of spin and framed three-manifolds in two situations. The first is ribbon *-categories which possess odd degenerate objects. This case includes the quantum group situations corresponding to the half-integer level Chern-Simons theories conjec...

The quotient process of M\"uger and Brugui\`eres 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...

We extend the theory of diffeomorphism-invariant spin network states from the real-analytic category to the smooth category. Suppose thatGis a compact connected semisimple Lie group andP→Mis a smooth principalG-bundle. A “cylinder function” on the space of smooth connections onPis a continuous complex function of the holonomies along finitely many...

Suppose thatGis a compact connected Lie group andP→Mis a smooth principalG-bundle. We define a “cylinder function” on the space of smooth connections onPto be a continuous complex function of the holonomies along finitely many piecewise smoothly immersed curves inM. Completing the algebra of cylinder functions in the sup norm, we obtain a commutati...

We extend the theory of diffeomorphism-invariant spin network states from the real-analytic category to the smooth category. Suppose that G is a compact connected semisimple Lie group and P -> M is a smooth principal G-bundle. A `cylinder function' on the space of smooth connections on P is a continuous complex function of the holonomies along fini...

The Jones polynomial and the Kauffman bracket are constructed, and their relation with knot and link theory is described. The quantum groups and tangle functor formalisms for understanding these invariants and their descendents are given. The quantum group $U_q(sl_2)$, which gives rise to the Jones polynomial, is constructed explicitly. The $3$-man...

. A positive, diffeomorphism-invariant generalized measure on the space of metrics of a two-dimensional smooth manifold is constructed. We use the term generalized measure analogously with the generalized measures of Ashtekar and Lewandowski and of Baez. A family of actions is presented which, when integrated against this measure give the two-dimen...

For each unitary representation (hence for each irreducible representation) of any complex semisimple quantum group U q (g) for q a positive real different from 1, a locally trivial subfactor pair is constructed analogous to the Jones subfactor pair constructed in [Jon83]. The index is computed, and the factor is given a description as a fixed poin...

The best known examples of Vassiliev invariants are the coefficients of a Jones-type polynomial expanded after exponential substitution. We show that for a given knot, the first $N$ Vassiliev invariants in this family determine the rest for some integer $N$.

Let $G$ be a compact connected Lie group and $P \to M$ a smooth principal $G$-bundle. Let a `cylinder function' on the space $\A$ of smooth connections on $P$ be a continuous function of the holonomies of $A$ along finitely many piecewise smoothly immersed curves in $M$, and let a generalized measure on $\A$ be a bounded linear functional on cylind...

The decomposition of an arbitrary axiomatic topological quantum field theory or TQFT into indecomposable theories is given. In particular, unitary TQFT's in arbitrary dimensions are shown to decompose into a sum of theories in which the Hilbert space of the sphere is one-dimensional, and indecomposable two-dimensional theories are classified. Comme...

A representation of the Hecke algebra with real positive parameter is given within the hyperfinite type III λ factor obtained from the Powers construction with r by r matrices. The Markov trace for the representation is obtained from the Powers state in the larger algebra. By constructing a sequence in the representation which converges strongly to...

Thesis (Ph. D. in Mathematics)--University of California, Berkeley, Apr. 1991. Includes bibliographical references (leaves 35-37).