Polarization (Lie algebra)

Let $G$ be a Lie group, $\backslash mathfrak\{g\}$ the corresponding Lie algebra and $\backslash mathfrak\{g\}^*$ its dual. Let $\backslash langle\; f,\backslash ,X\backslash rangle$ denote the value of the linear form (covector) $f\backslash in\backslash mathfrak\{g\}^*$ on a vector $X\backslash in\backslash mathfrak\{g\}$. The subalgebra $\backslash mathfrak\{h\}$ of the algebra $\backslash mathfrak\; g$ is called subordinate of $f\backslash in\backslash mathfrak\{g\}^*$ if the condition

:$\backslash forall\; X,\; Y\backslash in\backslash mathfrak\{h\}\backslash \; \backslash langle\; f,\backslash ,[X,\backslash ,Y]\backslash rangle\; =\; 0$,

or, alternatively,

:$\backslash langle\; f,\backslash ,[\backslash mathfrak\{h\},\backslash ,\backslash mathfrak\{h\}]\backslash rangle\; =\; 0$

is satisfied. Further, let the group $G$ act on the space $\backslash mathfrak\{g\}^*$ via coadjoint representation $\backslash mathrm\{Ad\}^*$. Let $\backslash mathcal\{O\}\_f$ be the orbit of such action which passes through the point $f$ and let $\backslash mathfrak\{g\}^f$ be the Lie algebra of the stabilizer $\backslash mathrm\{Stab\}(f)$ of the point $f$. A subalgebra $\backslash mathfrak\{h\}\backslash subset\backslash mathfrak\{g\}$ subordinate of $f$ is called a polarization of the algebra $\backslash mathfrak\{g\}$ with respect to $f$, or, more concisely, polarization of the covector $f$, if it has maximal possible dimensionality, namely

:$\backslash dim\backslash mathfrak\{h\}\; =\; \backslash frac\{1\}\{2\}\backslash left(\backslash dim\backslash ,\backslash mathfrak\{g\}\; +\; \backslash dim\backslash ,\backslash mathfrak\{g\}^f\backslash right)\; =\; \backslash dim\backslash ,\backslash mathfrak\{g\}\; -\; \backslash frac\{1\}\{2\}\backslash dim\backslash ,\backslash mathcal\{O\}\_f$.

The following condition was obtained by L. Pukanszky:{{cite journal

| last1 = Dixmier| first1 = Jacques

| last2 = Duflo| first2 = Michel

| last3 = Hajnal| first3 = Andras

| last4 = Kadison| first4 = Richard

| last5 = Korányi| first5 = Adam

| last6 = Rosenberg| first6 = Jonathan

| last7 = Vergne| first7 = Michele

| title = Lajos Pukánszky (1928 – 1996)

| journal = Notices of the American Mathematical Society

| volume = 45

| issue = 4

| pages = 492–499

| publisher = American Mathematical Society

| location =

| date = April 1998

| url = https://www.ams.org/journals/notices/199804/199804FullIssue.pdf

| issn = 1088-9477

| doi =

| mr =

| zbl = }}

Let $\backslash mathfrak\{h\}$ be the polarization of algebra $\backslash mathfrak\{g\}$ with respect to covector $f$ and $\backslash mathfrak\{h\}^\backslash perp$ be its annihilator: $\backslash mathfrak\{h\}^\backslash perp\; :=\; \backslash \{\backslash lambda\backslash in\backslash mathfrak\{g\}^*|\backslash langle\backslash lambda,\backslash ,\backslash mathfrak\{h\}\backslash rangle\; =\; 0\backslash \}$. The polarization $\backslash mathfrak\{h\}$ is said to satisfy the Pukanszky condition if

:$f\; +\; \backslash mathfrak\{h\}^\backslash perp\backslash in\backslash mathcal\{O\}\_f.$

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.{{cite journal

| last1 = Pukanszky | first = Lajos

| title = On the theory of exponential groups

| journal = Transactions of the American Mathematical Society

| volume = 126

| issue =

| pages = 487–507

| publisher = American Mathematical Society

| location =

| date = March 1967

| url = https://www.ams.org/journals/tran/1967-126-03/S0002-9947-1967-0209403-7/S0002-9947-1967-0209403-7.pdf

| issn = 1088-6850

| doi = 10.1090/S0002-9947-1967-0209403-7

| mr = 0209403

| zbl = 0207.33605 | doi-access = free

}}

• Polarization is the maximal totally isotropic subspace of the bilinear form $\backslash langle\; f,\backslash ,[\backslash cdot,\backslash ,\backslash cdot]\backslash rangle$ on the Lie algebra $\backslash mathfrak\{g\}$.{{Citation | last1=Kirillov | first1=A. A. | title=Elements of the theory of representations | orig-year=1972 | publisher=Springer-Verlag | location=Berlin, New York | series= Grundlehren der Mathematischen Wissenschaften | isbn=978-0-387-07476-4 |mr=0412321 | year=1976 | volume=220}}
• For some pairs $(\backslash mathfrak\{g\},\backslash ,f)$ polarization may not exist.
• If the polarization does exist for the covector $f$, then it exists for every point of the orbit $\backslash mathcal\{O\}\_f$ as well, and if $\backslash mathfrak\{h\}$ is the polarization for $f$, then $\backslash mathrm\{Ad\}\_g\backslash mathfrak\{h\}$ is the polarization for $\backslash mathrm\{Ad\}^*\_g\; f$. Thus, the existence of the polarization is the property of the orbit as a whole.
• If the Lie algebra $\backslash mathfrak\{g\}$ is completely solvable, it admits the polarization for any point $f\backslash in\backslash mathfrak\{g\}^*$.{{citation | last1=Dixmier | first1=Jacques | author-link=Jacques Dixmier | title=Enveloping algebras | orig-year=1974 | url=https://books.google.com/books?isbn=0821805606 | publisher=American Mathematical Society | location=Providence, R.I. | series=Graduate Studies in Mathematics | isbn=978-0-8218-0560-2 | mr=0498740 | year=1996 | volume=11}}
• If $\backslash mathcal\{O\}$ is the orbit of general position (i. e. has maximal dimensionality), for every point $f\backslash in\backslash mathcal\{O\}$ there exists solvable polarization.

{{reflist}}

Category:Bilinear forms

Category:Representation theory of Lie algebras