---
abstract: 'We prove that there is no consistent polynomial quantization of the coordinate ring of a basic non-nilpotent coadjoint orbit of a semisimple Lie group.'
author:
- |
    <span>**Mark J. Gotay**</span>[^1]\
    Department of Mathematics\
    University of Hawai'i\
    2565 The Mall\
    Honolulu, HI 96822 USA\
    <span>E-mail: lgraham@example.com</span>
date: |
    November 30, 2001\
    (Revised )
title: |
    On quantizing semisimple basic algebras, II:\
    The general case
---

Introduction {#intro}
============

In a recent paper (Gotay \[2002\]) we showed that there do not exist polynomial quantizations of the coordinate ring $P(M)$ of a basic semisimple coadjoint orbit $M \subset {\rm sl}(2,\r)^*$. Here we extend this result to any basic non-nilpotent coadjoint orbit of a general ßLie group:

Let ${{\mathfrak b}}$ be a <span>finite-dimensional</span> ß <span>Lie algebra</span>, and $M$ a basic non-nilpotent coadjoint orbit in ${{\mathfrak b}}^*$. Then there are no polynomial quantizations of the coordinate ring $P(M)$. \[thm:nogo\]

We refer the reader to Gotay \[2000\] for definitions and discussions of basic coadjoint orbits and quantization.

In particular, a coadjoint orbit $M \subset {{\mathfrak b}}^*$ is *basic* provided ${{\mathfrak b}}$ has no subalgebras which act transitively on $M$ and which globally separate points of $M$. Unfortunately, it is difficult to determine exactly which orbits are basic. From §4 in Gotay, Grabowski, and Grundling \[2000\] and Prop. 2.1 in Gotay \[2002\] we know that $M \subset {{\mathfrak b}}^*$ will be basic for a ß <span>Lie algebra</span> ${{\mathfrak b}}$ whenever

1.  (*i*) ${{\mathfrak b}}$ is compact and $M$ is principal,

2.  (*ii*) ${{\mathfrak b}}$ is compact and simple, and

3.  (*iii*) $M$ is principal nilpotent.

Consider the symmetric algebra $S({{\mathfrak b}})$, regarded as the ring of polynomials on ${{\mathfrak b}}^*.$ The Lie bracket on ${{\mathfrak b}}$ may be extended via the Leibniz rule to a Poisson bracket on $S({{\mathfrak b}})$, so that the latter becomes a Poisson algebra. Let $I(M)$ be the associative ideal in $S({{\mathfrak b}})$ consisting of all polynomials which vanish on $M$ and set $P(M) = S({{\mathfrak b}})/I(M)$. Since $M$ is an orbit $I(M)$ is also a Lie ideal, hence a Poisson ideal, so the coordinate ring $P(M)$ of $M$ inherits the structure of a <span>Poisson algebra</span> from $S({{\mathfrak b}})$. We denote the <span>Poisson bracket</span>s on both $P(M)$ and $S({{\mathfrak b}})$ by $\{\cdot,\cdot\}$.

Here we are interested in quantizing the coordinate ring $P(M)$. By a *quantization* of $P(M)$ we mean a Lie representation ${{\cal Q}}$ thereof by symmetric operators preserving a fixed dense domain $D$ in some separable Hilbert space ${{\mathcal H}}$, such that ${{\cal Q}}\!\restriction\! {{\mathfrak b}}$ is irreducible, integrable, and faithful. Let ${{\mathcal A}}$ be the associative operator algebra generated over ${{\mathbb C}}$ by $I$ and $\{{{\cal Q}}(b)
\mid b
\in
{{\mathfrak b}}\}$. We say that a quantization ${{\cal Q}}$ of $P(M)$ is *polynomial* if ${{\cal Q}}$ is valued in ${{\mathcal A}}$.

Proof of Theorem \[thm:nogo\] {#sec:pf}
=============================

Suppose to the contrary that ${{\cal Q}}$ were a polynomial quantization of $P(M)$ in a dense invariant domain $D$ in a Hilbert space ${{\mathcal H}}$. By extending ${{\cal Q}}$ to be complex linear, we obtain a Lie representation ${{\cal Q}}_{{\mathbb C}}$ of the Poisson algebra $P(M,{{\mathbb C}})$ of complex-valued polynomials on $M$ in $D$.

By assumption the representation of ${{\mathfrak b}}$ in $D$ provided by ${{\cal Q}}$ may be integrated to a strongly continuous unitary representation $\Pi$ of the 1-connected Lie group $B$ with <span>Lie algebra</span> ${{\mathfrak b}}$ in ${{\mathcal H}}$. Let $B_{{\mathbb C}}$ be the universal complexification of $B$; since $B$ is simply connected, $B_{{\mathbb C}}$ can be identified with the 1-connected ß  complex analytic group with <span>Lie algebra</span> the complexification ${{\mathfrak b}}_{{\mathbb C}}$ of ${{\mathfrak b}}$. (See Varadarajan \[1984\], pps. 256–258 and 400–404 for background on complexifications of Lie groups.) Since $B$ is ß, $B$ is a closed subgroup of $B_{{\mathbb C}}$, and so we may use induction to obtain a strongly continuous unitary representation $\Pi_{{\mathbb C}}$ of $B_{{\mathbb C}}$ in a certain <span>infinite-dimensional</span> Hilbert space ${{\mathcal K}}$.

Now let $C$ be a compact real form of $B_{{\mathbb C}}$, and denote by $\Gamma$ the restriction of $\Pi_{{\mathbb C}}$ to $C$. As every strongly continuous unitary representation of a compact Lie group is completely reducible, we may decompose $${{\mathcal K}}= {\stackrel{\wedge}{\bigoplus_{i \in I}}}\,{{\mathcal K}}_i
\label{eq:topds}$$ for some index set $I \subset \z$, where the <span>finite-dimensional</span> invariant subspaces ${{\mathcal K}}_i$ are the carriers of the irreducible constituents $\Gamma_i$ of $\Gamma$. Let ${{\mathfrak c}}$ be the <span>Lie algebra</span> of $C$; then for each $i \in I$, we have the derived representation $d\Gamma_i$ of ${{\mathfrak c}}$ in ${{\mathcal K}}_i$. Set $d\Gamma = \oplus_{i
\in I}\, d\Gamma_i$; this gives a representation of ${{\mathfrak c}}$ in the dense subspace $$D_C = \bigoplus_{i \in I} {{\mathcal K}}_i.$$

Choose a basis $\{c_1,\ldots,c_r\}$ of ${{\mathfrak c}}$. Since ${{\mathfrak c}}_{{\mathbb C}}= {{\mathfrak b}}_{{\mathbb C}}$ and as by assumption ${{\cal Q}}$ is valued in ${{\mathcal A}}$, for every $f \in P(M,{{\mathbb C}})$ we may expand $${{\cal Q}}_{{\mathbb C}}(f) = \sum_{n_1,\ldots,n_r} a^f_{n_1 ,\ldots,n_r} {{\cal Q}}_{{\mathbb C}}(c_1)^{
n_1} \cdots {{\cal Q}}_{{\mathbb C}}(c_r)^{ n_r}
\label{eq:polyq}$$ for some coefficients $a^f_{n_1 ,\ldots,n_r}$. By means of this formula we can extend the representation $d\Gamma$ of ${{\mathfrak c}}$ to a Lie representation $\gamma$ of $P(M,{{\mathbb C}})$ in $D_C$: $$\gamma(f) = \sum_{n_1,\ldots,n_r} a^f_{n_1 ,\ldots,n_r} d\Gamma(c_1)^{
n_1} \cdots d\Gamma(c_r)^{ n_r}
\label{eq:polygamma}$$ with the *same* coefficients. As each subspace ${{\mathcal K}}_i$ is invariant, $\gamma$ restricts to a representation $\gamma_i$ of $P(M,{{\mathbb C}})$ in ${{\mathcal K}}_i$. We will show that the existence of these representations $\gamma_i$ leads to a contradiction.

To this end we recall the following algebraic fact, the proof of which is given in Gotay, Grabowski, and Grundling \[2000\].

\[lem:gra\] If $L$ is a finite-codimensional Lie ideal of an infinite-dimensional Poisson algebra $P$ with identity, then either $L$ contains the derived ideal $\{ P,P\}$ or there is a maximal finite-codimensional associative ideal $J$ of $P$ such that $\{ P,P\}\subset J$.

We apply Lemma \[lem:gra\] to each $L_i = \ker \gamma_i$ which, as ${{\mathcal K}}_i$ is finite-dimensional, has finite codimension in $P = P(M,{{\mathbb C}})$. First suppose there is an $i$ for which $\{P,P\} \not\subset L_i$. Then there must exist a maximal <span>finite-codimensional</span> associative ideal $J_i$ in $P$ with $\{P,P\} \subset J_i.$ If $\rho$ is the projection $S({{\mathfrak b}}_{{\mathbb C}}) {\rightarrow}P$, then $I_i =
\rho^{-1}(J_i)$ is a maximal <span>finite-codimensional</span> associative ideal in $S({{\mathfrak b}}_{{\mathbb C}})$ with $\{S({{\mathfrak b}}_{{\mathbb C}}),S({{\mathfrak b}}_{{\mathbb C}})\} \subset I_i.$ Since by semisimplicity $${{\mathfrak b}}_{{\mathbb C}}=
\{{{\mathfrak b}}_{{\mathbb C}},{{\mathfrak b}}_{{\mathbb C}}\} \subset \{S({{\mathfrak b}}_{{\mathbb C}}),S({{\mathfrak b}}_{{\mathbb C}})\} \subset I_i,$$ and since $1 \not \in I_i$ (as $I_i$ is proper), it follows that $I_i$ is the associative ideal generated by ${{\mathfrak b}}_{{\mathbb C}}.$ (Actually, this shows that $S({{\mathfrak b}}_{{\mathbb C}}) = {{\mathbb C}}\oplus I_i.$)

Since the orbit $M$ is not nilpotent, there is a nonzero Casimir $\Omega \in S({{\mathfrak b}}_{{\mathbb C}})$, i.e. $\rho(\Omega) = \omega$ for some constant $\omega \neq 0.$ Since ${{\mathfrak b}}_{{\mathbb C}}$ is ß it follows from the above observations that $\Omega \in
I_i.$ But then $\Omega - \omega \not \in I_i$, which is a contradiction since $\Omega - \omega \in \ker \rho \subset I_i.$

Thus for *every* $i$ it must be the case that $\{P,P\} \subset
L_i$. Again semisimplicity gives ${{\mathfrak b}}_{{\mathbb C}}= \{{{\mathfrak b}}_{{\mathbb C}},{{\mathfrak b}}_{{\mathbb C}}\} \subset L_i$, and so $\gamma\!\restriction\!{{\mathfrak b}}_{{\mathbb C}}= 0$. In particular, then, $d\Gamma = 0$. Since ${{\mathfrak c}}$ is a compact real form of ${{\mathfrak b}}_{{\mathbb C}}$, the Cartan decomposition of ${{\mathfrak b}}_{{\mathbb C}}$ implies that $d\Pi_{{\mathbb C}}= 0$. It follows from the induction construction that the original derived representation $d\Pi$ of ${{\mathfrak b}}$ in the domain $D$ must be zero as well. But then ${{\cal Q}}\! \restriction \! {{\mathfrak b}}= 0$, which contradicts the requirement that a quantization represent ${{\mathfrak b}}$ faithfully. This concludes the proof of Theorem \[thm:nogo\].

We remark that Theorem \[thm:nogo\] was already known when ${{\mathfrak b}}$ is compact (Gotay, Grabowski and Grundling \[2000\]), in which case the proof above simplifies greatly and provides an alternate means of establishing Theorem 2 *ibid.* Notice also that when ${{\mathfrak b}}$ is compact every quantization of $P(M)$ is necessarily polynomial; this follows from the observation that since ${{\cal Q}}\restriction {{\mathfrak b}}$ is irreducible the representation space ${{\mathcal H}}$ must be <span>finite-dimensional</span> together with a well known fact about enveloping algebras (Prop. 2.6.5 in Dixmier \[1976\]).

Discussion
==========

The key observation underlying Theorem \[thm:nogo\] is that as $M
\subset
{{\mathfrak b}}^*$ is non-nilpotent, its ideal $I(M)$ is nonhomogeneous. If $M$ is a basic nilpotent orbit, on the other hand, then $I(M)$ is homogeneous, and from Theorem 1.1 in Gotay \[2002\] we know that there do exist polynomial quantizations of $P(M)$. (Although it is not clear to what extent these are "nontrivial" in general.) Taken together, these two results serve to establish a conjecture of Gotay \[2000\] when ${{\mathfrak b}}$ is ß: *There exists a consistent polynomial quantization of $P(M)$ if and only if $I(M)$ is homogeneous.*

Finally, we remark that the restriction to basic coadjoint orbits is for physical reasons, cf. Gotay \[2000\]. For our purposes here, we may delete the adjective "basic" and consider arbitrary orbits. Both Theorem \[thm:nogo\] and the conjecture above will remain valid in this extended context.

References {#references .unnumbered}
==========

Dixmier, J. \[1977\] <span>*Enveloping Algebras.*</span> (North Holland, Amsterdam).

Gotay, M. J. \[2000\] Obstructions to quantization. In: <span>*Mechanics: From Theory to Computation (Essays in Honor of Juan-Carlos Simo)*</span>, J. Nonlinear Sci. Eds. (Springer, New York), 171–216.

Gotay, M. J. \[2002\] On quantizing semisimple basic algebras, I: sl$(2,\r)$. To appear in: <span>*The Jerry Marsden 60$^{th}$ Birthday Volume*</span>, P. Holmes, P. Newton, and A. Weinstein, Eds. (Springer, New York). Preprint math-ph/0012034.

Gotay, M. J., Grabowski, J., and Grundling, H. B. \[2000\] An obstruction to quantizing compact symplectic manifolds. <span>*Proc. Amer. Math. Soc.*</span> <span>**28**</span>, 237–243.

Varadarajan, V. S. \[1984\] <span>*Lie Groups, Lie Algebras, and their Representations.*</span> (Springer, New York).

[^1]: Supported in part by NSF grant DMS 00-72434.

{'source': 'arXiv_out/math-ph0208031'}