Abstract
We define so-called “elementary representations” T R,µ p , p ∈ ℤ, of the group B ℤ0 of finite upper-triangular matrices infinite in both directions by using quasi-invariant measures on certain homogeneous spaces and give a criterion for the irreducibility and equivalence of the representations constructed. We also give a criterion for the irreducibility of the tensor product of finitely many and infinitely many elementary representations.
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REFERENCES
J. Dixmier, Les C*-Algèbres et leur Representations, Gautirs-Villars, Paris (1969).
A. Weyl, L'intégration Dans les Groups Topologique et Ses Application, Hermann, Paris (1951).
Xia-Dao-Xing, Measures and Integration in Infinite-Dimensional Spaces, Academic Press, New York-London (1978).
S. Albeverio and R. Høegh-Krohn, The Energy Representation of Sobolev-Lie Group, Univ. Bielefeld (1976).
R. S. Ismagilov, “Representations of the group of smooth mappings of a segment in a compact Lie group,” Funkts. Anal. Prilozhen., 15, No. 2, 73–74 (1981).
S. Albeverio, R. Høegh-Krohn, and D. Testard, “Irreducibility and reducibility for the energy representation of a group of mappings of a Riemannian manifold into a compact Lie group,” J. Funct. Anal., 41, 378–396 (1981).
S. Albeverio, R. Høegh-Krohn, D. Testard, and A. Vershik, “Factorial representations of path groups,” J. Funct. Anal., 51, 115–131 (1983).
A. V. Kosyak, “Irreducibility criterion for regular Gaussian representations of group of finite upper-triangular matrices,” Funkts. Anal. Prilozhen., 24, No. 3, 82–83 (1990).
A. V. Kosyak, “Criteria for irreducibility and equivalence of regular Gaussian representations of group of finite upper-triangular matrices of infinite order,” Selecta. Math. Soviet., 11, 241–291 (1992).
A. V. Kosyak, “Irreducible regular Gaussian representations of the group of the interval and circle diffeomorphisms,” J. Funct. Anal., 125, 493–547 (1994).
A. V. Kosyak, “Criteria for irreducibility of regular representations corresponding to product measures of group of finite uppertriangular matrices,” Methods Funct. Anal. Topology, 6, No. 4, 43–56 (2000).
A. V. Kosyak, “Irreducibility of the regular Gaussian representations of the group B ℤ0 265-1,” Methods Funct. Anal. Topology, 7, No. 2, 42–51 (2001).
A. V. Kosyak and R. Zekri, “Regular representations of infinite-dimensional groups and factors. I,” Methods Funct. Anal. Topology, 6, No. 2, 50–59 (2000).
A. V. Kosyak, “Elementary representations of the group B ℤ0 . I,” Methods Funct. Anal. Topology, 7, No. 1, 33–44 (2001).
J. Dixmier, Les Algèbres d'Operateurs Dans L'espace Hilbertien, Gautirs-Villars, Paris (1969).
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Kosyak, O.V. Elementary Representations of the Group B ℤ0 of Upper-Triangular Matrices Infinite in Both Directions. I. Ukrainian Mathematical Journal 54, 253–265 (2002). https://doi.org/10.1023/A:1020138613311
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DOI: https://doi.org/10.1023/A:1020138613311