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Polarization (Lie algebra)

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In representation theory, polarization is the maximal totally isotropic subspace of a certain skew-symmetric bilinear form on a Lie algebra. The notion of polarization plays an important role in construction of irreducible unitary representations of some classes of Lie groups by means of the orbit method[1] as well as in harmonic analysis on Lie groups and mathematical physics.

Definition

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Let be a Lie group, the corresponding Lie algebra and its dual. Let denote the value of the linear form (covector) on a vector . The subalgebra of the algebra is called subordinate of if the condition

,

or, alternatively,

is satisfied. Further, let the group act on the space via coadjoint representation . Let be the orbit of such action which passes through the point and let be the Lie algebra of the stabilizer of the point . A subalgebra subordinate of is called a polarization of the algebra with respect to , or, more concisely, polarization of the covector , if it has maximal possible dimensionality, namely

.

Pukanszky condition

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The following condition was obtained by L. Pukanszky:[2]

Let be the polarization of algebra with respect to covector and be its annihilator: . The polarization is said to satisfy the Pukanszky condition if

L. Pukanszky has shown that this condition guaranties applicability of the Kirillov's orbit method initially constructed for nilpotent groups to more general case of solvable groups as well.[3]

Properties

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  • Polarization is the maximal totally isotropic subspace of the bilinear form on the Lie algebra .[4]
  • For some pairs polarization may not exist.[4]
  • If the polarization does exist for the covector , then it exists for every point of the orbit as well, and if is the polarization for , then is the polarization for . Thus, the existence of the polarization is the property of the orbit as a whole.[4]
  • If the Lie algebra is completely solvable, it admits the polarization for any point .[5]
  • If is the orbit of general position (i. e. has maximal dimensionality), for every point there exists solvable polarization.[5]

References

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  1. ^ Corwin, Lawrence; GreenLeaf, Frederick P. (25 January 1981). "Rationally varying polarizing subalgebras in nilpotent Lie algebras". Proceedings of the American Mathematical Society. 81 (1). Berlin: American Mathematical Society: 27–32. doi:10.2307/2043981. ISSN 1088-6826. Zbl 0477.17001.
  2. ^ Dixmier, Jacques; Duflo, Michel; Hajnal, Andras; Kadison, Richard; Korányi, Adam; Rosenberg, Jonathan; Vergne, Michele (April 1998). "Lajos Pukánszky (1928 – 1996)" (PDF). Notices of the American Mathematical Society. 45 (4). American Mathematical Society: 492–499. ISSN 1088-9477.
  3. ^ Pukanszky, Lajos (March 1967). "On the theory of exponential groups" (PDF). Transactions of the American Mathematical Society. 126. American Mathematical Society: 487–507. doi:10.1090/S0002-9947-1967-0209403-7. ISSN 1088-6850. MR 0209403. Zbl 0207.33605.
  4. ^ a b c Kirillov, A. A. (1976) [1972], Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, vol. 220, Berlin, New York: Springer-Verlag, ISBN 978-0-387-07476-4, MR 0412321
  5. ^ a b Dixmier, Jacques (1996) [1974], Enveloping algebras, Graduate Studies in Mathematics, vol. 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, MR 0498740