Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p : independence of p

The method disclosed herein may be practiced by passing the vapors to be sampled along a path with halogen vapor, preferably chlorine vapor, heating the mixed vapors to halogenate those of the sampled vapors subject to halogenation, removing unreacted halogen vapor, and then sensing the vapors for organic halogenated compounds. The apparatus disclosed herein comprises means for flowing the vapors, both sample and halogen vapors, into a common path, means for heating the mixed vapors to effect the halogenation reaction, means for removing unreacted halogen vapor, and a sensing device for sensing halogenated compounds. By such a method and means, the vapors of low molecular weight hydrocarbons, ketones and alcohols, when present, such as methane, ethane, acetone, ethanol, and the like are converted, at least in part, to halogenated compounds, then the excess halogen removed or trapped, and the resultant vapors of the halogenated compounds sensed or detected. The system is highly sensitive. For example, acetone in a concentration of 30 parts per billion (volume) is readily detected.

[1]  G. Lusztig.  Quantum deformations of certain simple modules over enveloping algebras , 1988 .

[2]  W. Soergel.  n-Cohomology of simple highest weight modules on walls and purity , 1989 .

[3]  I. Gel'fand,et al.  SCHUBERT CELLS AND COHOMOLOGY OF THE SPACES G/P , 1973 .

[4]  G. Lusztig.  Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra , 1990 .

[5]  H. H. Andersen,et al.  Quantum groups at roots of ±1 , 1996 .

[6]  Jurgen Fuchs,et al.  Affine Lie Algebras and Quantum Groups , 1995 .

[7]  Wolfgang Soergel,et al.  The combinatorics of Harish-Chandra bimodules. , 1992 .

[8]  O. Gabber,et al.  Towards the Kazhdan-Lusztig conjecture , 1981 .

[9]  Michel Demazure,et al.  Invariants symétriques entiers des groupes de Weyl et torsion , 1973 .

[10]  Brian Parshall,et al.  Quantum Linear Groups , 1991 .

[11]  S. Donkin.  MODULN MIT EINEM HÖCHSTEN GEWICHT (Lecture Notes in Mathematics, 750.) , 1981 .

[12]  G. Lusztig.  Monodromic systems on affine flag manifolds , 1994, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[13]  V. Kac,et al.  Representations of quantum groups at roots of 1 , 1992 .

[14]  V. Kats,et al.  Irreducible representations of Lie p-algebras , 1971 .

[15]  S. Kato.  On the Kazhdan-Lusztig polynomials for affine Weyl groups , 1985 .

[16]  Shrawan Kumar,et al.  Cohomology of quantum groups at roots of unity , 1993 .

[17]  J. Humphreys.  Algebraic groups and modular lie algebras , 1967 .

[18]  J. Jantzen.  Support Varieties of Weyl Modules , 1987 .

[19]  J. Jantzen.  Über Darstellungen höherer Frobenius-Kerne halbeinfacher algebraischer Gruppen , 1979 .

[20]  H. H. Andersen.  Filtrations of cohomology modules for Chevalley groups , 1983 .

[21]  C. Curtis,et al.  Methods of representation theory--with applications to finite groups and orders , 1981 .

[22]  J. Humphreys.  Modular representations of classical Lie algebras and semisimple groups , 1971 .

[23]  George Lusztig,et al.  Canonical bases arising from quantized enveloping algebras , 1990 .

[24]  E. Friedlander,et al.  Modular representation theory of Lie algebras , 1988 .

[25]  J. Jantzen.  Über das Dekompositionsverhalten gewisser modularer Darstellungen halbeinfacher Gruppen und ihrer Lie-Algebren , 1977 .

[26]  J. Jantzen.  Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne. , 1980 .

[27]  W. Soergel.  Kategorie , perverse Garben und Moduln über den Koinvarianten zur Weylgruppe , 1990 .

[28]  M. Kashiwara,et al.  Characters of the Negative Level Highest-Weight Modules for Affine Lie Algebras , 1995 .

[29]  E. Friedlander,et al.  Deformations of Lie Algebra Representations , 1990 .

[30]  Jian-pan Wang,et al.  Cohomology of infinitesimal quantum groups, I , 1992 .

[31]  G. Lusztig.  Some problems in the representation theory of nite Cheval-ley groups , 1979 .

[32]  G. Lusztig.  Hecke algebras and Jantzen's generic decomposition patterns , 1980 .

[33]  B Kostant,et al.  The nil Hecke ring and cohomology of G/P for a Kac-Moody group G. , 1986, Proceedings of the National Academy of Sciences of the United States of America.

[34]  D. Kazhdan,et al.  Tensor structures arising from affine Lie algebras. III , 1993 .

[35]  Nicolai Reshetikhin,et al.  Quantum Groups , 1993 .

[36]  R. Carter.  REFLECTION GROUPS AND COXETER GROUPS (Cambridge Studies in Advanced Mathematics 29) , 1991 .

[37]  N. Xi.  Irreducible modules of quantized enveloping algebras at roots of 1 , 1996 .