# 1998 seminars

## 23/01/2003, Thursday, 11:00–12:00

José Velhinho, Univ. Beira Interior and CENTRA
Representations of holonomy algebras and shadow states

It has been argued that the Ashtekar-Lewandowski representation of the Ashtekar-Isham holonomy algebra is fundamental, in the sense that any other representation can be obtained by a suitable limit procedure. We propose to clarify that statement, providing, in particular, a canonical way of mapping GNS states to a family of vectors of the Ashtekar-Lewandowski Hilbert space. The so-called family of shadow states thus obtained converges, as states of the algebra, to the original GNS state.

#### References

1. M. Varadarajan, Phys. Rev D. 64 , 104003 (2001); gr-qc/0104051
2. J.M. Velhinho, Commun. Math. Phys. 227, 541 (2002); math-ph/0107002
3. A. Ashtekar and J. Lewandowski, Class. Quant. Grav. 18, L117 (2001); gr-qc/0107043
4. T. Thiemann, gr-qc/0206037
5. H. Sahlmann, gr-qc/0207112
6. A. Ashtekar, J. Lewandowski and H. Sahlmann, gr-qc/0211012

## 23/01/2003, Thursday, 13:30–14:30

Lina Oliveira, Instituto Superior Técnico
Ideals of nest algebras

Complex Banach spaces are naturally endowed with an algebraic structure, other than that of a vector space. The holomorphic structure of the open unit ball in a complex Banach space A leads to the existence of a closed subspace ${A}_{s}$ of $A$, known as the symmetric part of $A$, and of a partial triple product $\left(a,b,c\right)\to abc$ mapping $A×{A}_{s}×A$ to $A$. The existence of a Jordan triple identity satisfyied by this algebraic structure relates any complex Banach space to the Banach Jordan triple systems important in infinite-dimensional holomorphy.
A nest algebra, which is a primary example of a non-self-adjoint algebra of operators, is also an interesting case of a complex Banach space whose symmetric part is a proper subspace.
The ideals of nest algebras related to its associative multiplication have been extensively investigated, and whilst it is clear that ideals in the associative sense provide examples of ideals in the partial triple sense, the converse assertion remains in general an open problem. It is the aim of this talk to show that, in a large class of nest algebras, the weak*-closed ideals in the partial triple sense are also weak*-closed ideals in the associative algebra sense.
A brief overview of how the partial triple produtct arises from the holomorphic structure of the open unit ball of the nest algebra will also be given.

#### References

1. Jonathan Arazy, An application of infinite dimensional holomorphy to the geometry of Banach space, Geometrical aspects of functional analysis, Lecture Notes in Mathematics, Vol. 1267, Springer-Verlag, Berlin/Heidelberg/New York, 1987.
2. Lina Oliveira, Weak*-closed Jordan ideals of nest algebras, Math. Nach., to appear.

## 15/10/2003, Wednesday, 15:00–16:00

Igor Kanatchikov, Institute of Theoretical Physics, Free University of Berlin
Precanonical quantization of the Yang-Mills fields and the mass gap problem

We overview the ideas of the precanonical quantization approach and apply it to the Yang-Mills fields. We show how the approach reduces the mass gap problem to a quantum mechanical spectral problem similar to that for the magnetic Schroedinger operator with a Clifford-valued magnetic field. Reference: hep-th/0301001.

## 30/10/2003, Thursday, 16:30–17:30

Pedro Lopes, Instituto Superior Técnico
Nós e os Quandles I

Apresentação dos resultados da tese de doutoramento.

References:

1. P. Lopes, Quandles at finite temperatures I, J. Knot Theory Ramifications, 12(2):159-186 (2003), math.QA/0105099
2. F. M. Dionisio and P. Lopes, Quandles at finite temperatures II, J. Knot Theory Ramifications to appear, math.GT/0205053
3. J. Bojarczuk and P. Lopes, Quandles at finite temperatures III, submitted to J. Knot Theory Ramifications

## 06/11/2003, Thursday, 17:00–18:00

Pedro Lopes, Instituto Superior Técnico
Nós e os Quandles II

Apresentação dos resultados da tese de doutoramento.

References:

1. P. Lopes, Quandles at finite temperatures I, J. Knot Theory Ramifications, 12(2):159-186 (2003), math.QA/0105099
2. F. M. Dionisio and P. Lopes, Quandles at finite temperatures II, J. Knot Theory Ramifications to appear, math.GT/0205053
3. J. Bojarczuk and P. Lopes, Quandles at finite temperatures III, submitted to J. Knot Theory Ramifications

## 17/12/2003, Wednesday, 14:00–15:00

J. Scott Carter, Univ. South Alabama
Quandle homology theories and cocycle invariants of knots

Cohomology theories have been developed for certain self-distributive groupoids called quandles. Variations of invariants of knots and knotted surfaces have been defined using quandle cocycles and the state-sum form. We review these developments, and also discuss quandle modules and their relation to generalizations of Alexander modules, and topological applications of these invariants.