Summary: Starting from the physical point of view on the Miura transformation as reflectionless potential and its connection with supersymmetry we define a scaling -deformation of this to obtain -deformed supersymmetric quantum mechanics. An application to an inverse scattering problem is given.
This paper is devoted to new algebraic structures, called qualgebras and squandles. Topologically, they emerge as an algebraic counterpart of knotted 3-valent graphs, just like quandles can be seen as an "algebraization" of knots. Algebraically, they are modeled after groups with conjugation and multiplication/squaring operations. We discuss basic properties of these structures, and introduce and study the notions of qualgebra/squandle 2-cocycles and 2-coboundaries. Knotted 3-valent graph invariants...
This article establishes the algebraic covering theory of quandles. For every connected quandle Q with base point q ∈ Q, we explicitly construct a universal covering p: (Q̃,q̃̃) → (Q,q). This in turn leads us to define the algebraic fundamental group , where Adj(Q) is the adjoint group of Q. We then establish the Galois correspondence between connected coverings of (Q,q) and subgroups of π₁(Q,q). Quandle coverings are thus formally analogous to coverings of topological spaces, and resemble Kervaire’s...
A symmetric quandle is a quandle with a good involution. For a knot in ℝ³, a knotted surface in ℝ⁴ or an n-manifold knot in , the knot symmetric quandle is defined. We introduce the notion of a symmetric quandle presentation, and show how to get a presentation of a knot symmetric quandle from a diagram.
In this paper we show that the multiplicities of holomorphic discrete series representations relative to reductive subgroups satisfy the credo “quantization commutes with reduction”.
A construction of the noncommutative-geometric counterparts of classical classifying spaces is presented, for general compact matrix quantum structure groups. A quantum analogue of the classical concept of the classifying map is introduced and analyzed. Interrelations with the abstract algebraic theory of quantum characteristic classes are discussed. Various non-equivalent approaches to defining universal characteristic classes are outlined.
From the text: The author reviews recent research on quantum deformations of the Poincaré supergroup and superalgebra. It is based on a series of papers (coauthored by P. Kosiński, J. Lukierski, P. Maślanka and A. Nowicki) and is motivated by both mathematics and physics. On the mathematical side, some new examples of noncommutative and noncocommutative Hopf superalgebras have been discovered. Moreover, it turns out that they have an interesting internal structure of graded bicrossproduct. As far...
We give criteria for framed links and 3-manifolds to be periodic of prime order. As applications we show that the Poincaré sphere is of periodicity 2, 3, 5 only and the Brieskorn sphere Σ(2,3,7) is of periodicity 2, 3, 7 only.
We propose a direction of study of nonabelian theta functions by establishing an analogy between the Weyl quantization of a one-dimensional particle and the metaplectic representation on the one hand, and the quantization of the moduli space of flat connections on a surface and the representation of the mapping class group on the space of nonabelian theta functions on the other. We exemplify this with the cases of classical theta functions and of the nonabelian theta functions for the gauge group...
A general theory of characteristic classes of quantum principal bundles is presented, incorporating basic ideas of classical Weil theory into the conceptual framework of noncommutative differential geometry. A purely cohomological interpretation of the Weil homomorphism is given, together with a geometrical interpretation via quantum invariant polynomials. A natural spectral sequence is described. Some interesting quantum phenomena appearing in the formalism are discussed.