The Grothendieck groups of the categories of finitely generated modules and finitely generated projective modules over a tower of algebras can be endowed with (co)algebra structures that, in many cases of interest, give rise to a dual pair of Hopf algebras. Moreover, given a dual pair of Hopf algebras, one can construct an algebra called the Heisenberg double, which is a generalization of the c...

We reconsider the results cocerning the extreme-quantum S = 1/2 square-lattice Heisenberg antiferromagnet with frustrating diagonal couplings (J 1 − J 2 model) drawn from a comparison with exact-diagonalization data. A combined approach using also some intrinsic features of the self-consistent spin-wave theory leads to the conclusion that the theory strongly overestimates the stabilizing role o...

This paper analyses the 3 × 3 self-adjoint matrix HN = N−1 ∑ j=1 S (j) x S (j+1) x + S (j) y S (j+1) y + S (j) z S (j+1) z , the Hamiltonian of a spin-1, one-dimensional, Heisenberg antiferromagnet. Various results are obtained, including alternative representations of HN , families of operators commuting with HN , a complete description of the spin-0 subspace (which includes all oneand two-dim...

In this problem you will consider the Heisenberg model of a one-dimensional quantum antiferromagnet. I first give you a brief summary on the Heisenberg model. You do not need to have any previous knowledge on magnetism (or the Heisenberg model) to do this problem. You will be able to solve this problem only with the methods that were discussed in class. The one-dimensional Heisenberg model is d...

We examine the Schrödinger algebra in the framework of Berezin quantization. First, the Heisenberg-Weyl and sl(2) algebras are studied. Then the Berezin representation of the Schrödinger algebra is computed. In fact, the sl(2) piece of the Schrödinger algebra can be decoupled from the Heisenberg component. This is accomplished using a special realization of the sl(2) component that is built fro...

Odometer actions of discrete, finitely generated and residually finite groups G have been defined by Cortez and Petite. In this paper we focus on the case where G is the discrete Heisenberg group. We prove a structure theorem for finite index subgroups of the Heisenberg group based on their geometry when they are considered as subsets of Z3. We use this structure theorem to provide a classifica...

Is every locally compact abelian group which admits a Heisenberg central extension isomorphic to the product of a locally compact abelian group and its Pontryagin dual? An affirmative answer is obtained for all the commonly occurring types of abelian groups having Heisenberg central extensions, including Lie groups and certain finite Cartesian products of local fields and adèles. Furthermore, f...

Heisenberg, in constructing quantum mechanics, explicitly followed certain principles exemplified, as he believed, in Einstein's construction of the special theory of relativity which for him was the paradigm for radical theoretic change in physics. These were the principles of (i) scientific realism, (ii) stability of background knowledge, (iii) E-observability, (iv) contextual re-interpretati...

We compute the braided groups and braided matrices B(R) for the solution R of the Yang-Baxter equation associated to the quantum Heisenberg group. We also show that a particular extension of the quantum Heisenberg group is dual to the Heisenberg universal enveloping algebra U q (h), and use this result to derive an action of U q (h) on the braided groups. We then demonstrate the various covaria...