Odd family algebras

  • A. A. Kirillov
  • L. G. Rybnikov
Chapter
Part of the Progress in Mathematics book series (PM, volume 243)

Summary

A new class of associative algebras related to simple complex Lie algebras (or root systems) was introduced and studied in tikya[K1] and tikya[K2]. They were named classical and quantum family algebras. The aim of this paper is to introduce the odd analogue of these algebras and expose some results about their structure. In particular, we describe the structure of \( \mathfrak{g} \) -module Λ\( \mathfrak{g} \) and compute the odd exponents for some cases.

Key words

Semisimple Lie algebra exterior algebra Clifford algebra universal enveloping algebra irreducible representation classical and quantum family algebras 
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References

  1. [AM]
    A. Alekseev and E. Meinrenken, The non-commutative Weil algebra, Invent. Math., 139 (2000), No 1, 135–172.MATHMathSciNetCrossRefGoogle Scholar
  2. [F]
    D. B. Fuchs, Cohomology of infinite-dimensional Lie algebras, Translated from the Russian by A. B. Sosinskii. Contemporary Soviet Mathematics. Consultants Bureau, New York, 1986, xii+328 pp.Google Scholar
  3. [K1]
    A. A. Kirillov, Family algebras, Electronic Research Announcements AMS, 6 (2000), No 1.Google Scholar
  4. [K2]
    A. A. Kirillov, Introduction to family algebras, Moscow Math. J., 1 (2001), No 1, 49–63.MATHMathSciNetGoogle Scholar
  5. [Ko1]
    B. Kostant, Lie group representations on polynomial ring, Amer. J. Math., 85 (1963), 327–404.MATHCrossRefMathSciNetGoogle Scholar
  6. [Ko2]
    B. Kostant, Clifford algebra analogue of the Hopf-Kozul-Samelson theorem, the ρ-decomposition C(\( \mathfrak{g} \) ) = End Vρ ⊗ C(P), and the \( \mathfrak{g} \) -module structure of Λ\( \mathfrak{g} \) ), Adv. Math., 125 (1997), 275–350.MATHCrossRefMathSciNetGoogle Scholar
  7. [OV]
    A. Onishchik and E. Vinberg, A seminar on Lie groups and algebraic groups, Nauka, Moscow, 1988, 1995 (second edition) 44 pp, In Russian. English translation: Series in Soviet Mathematics, Springer-Verlag, 1990, xx+328 pp.Google Scholar

Copyright information

© Birkhäuser Boston 2006

Authors and Affiliations

  • A. A. Kirillov
    • 1
    • 3
  • L. G. Rybnikov
    • 2
  1. 1.Department of MathematicsThe University of PennsylvaniaPhiladelphiaUSA
  2. 2.Moscow State UniversityMoscow
  3. 3.Institute for Problems of Information Transmission of Russian Academy of SciencesMoscowRussia

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