Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-27T11:39:49.328Z Has data issue: false hasContentIssue false
  • English
  • Français

Infinite dimensional representations of

Published online by Cambridge University Press:  18 May 2009

A. Dean  and
F. Zorzitto
A. Dean
Affiliation:
Department of MathematicsBishop's University Lennoxville, QuebecCanadaJim 1Z7
F. Zorzitto
Affiliation:
Department of Pure MathematicsUniversity of WaterlooWaterloo, OntarioCanadaN2L 3G1
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

By a representation of the extended Dynkin diagram we shall mean a list of 5 vector spaces P, E1, E2, E3, E4 over an algebraically closed field K, and 4 linear maps a1, a2, a3, a4 as shown.

The spaces need not be of finite dimension.

In their solution of the 4-subspace problem [6], Gelfand and Ponomarev have classified such representations when the spaces are finite dimensional. A representation like (1) can also be viewed as a module over the K-algebra R4 consisting of all 5 × 5 matrices having zeros off the first row and off the main diagonal.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 32 , Issue 1 , January 1990 , pp. 25 - 33
Copyright
Copyright © Glasgow Mathematical Journal Trust 1990

References

REFERENCES

1.Aronszajn, N. and Fixman, U., Algebraic spectral problems, Studia Math. 30 (1968), 273–338.CrossRefGoogle Scholar
2.Cohn, P. M., On the free product of associative rings, Math. Z. 71 (1959), 380–398.CrossRefGoogle Scholar
3.Dean, A. and Zorzitto, F., A criterion for pure simplicity, to appear in J. Algebra.Google Scholar
4.Dlab, V. and Ringel, C. M., Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 173 (1976).Google Scholar
5.Fixman, U., On algebraic equivalence between pairs of linear transformations, Trans. Amer. Math. Soc. 113 (1964), 424–453.CrossRefGoogle Scholar
6.Gel'fand, I. M. and Ponomarev, V. A., Quadruples of subspaces of a finite dimensional space, Soviet Math. Dokl. 12 (1971), 535–539.Google Scholar
7.Lawrence, J., Okoh, F. and Zorzitto, F., Rational functions and Kronecker modules, Comm. Algebra 14 (1986), 1947–1965.CrossRefGoogle Scholar
8.Okoh, F., A bound on the rank of purely simple systems, Trans. Amer. Math. Soc. 232 (1977), 169–186.CrossRefGoogle Scholar
9.Okoh, F., Some properties of purely simple Kronecker modules I, J. Pure Appl. Algebra 27 (1983), 39–48.CrossRefGoogle Scholar
10.Okoh, F., Applications of linear functional to Kronecker modules I, Linear Algebra Appl. 76 (1986), 165–204.CrossRefGoogle Scholar
11.Ringel, C. M., Infinite-dimensional representations of finite-dimensional hereditary algebras, Symp. Math. 23 (1979), 321–412.Google Scholar
12.Ringel, C. M., The spectrum of a finite-dimensional algebra, Ring theory (Proc. Antwerp Conf., 1978), Lecture Notes in Pure and Appl. Math. 51 (Dekker, 1979), 535–597.Google Scholar