On complex H-type Lie algebras
Nathaniel Eldredge \footnote{ School of Mathematical Sciences
University of Northern Colorado
501 20th St.
Campus Box 122
Greeley
\textsc{CO
H-type Lie algebras were introduced by Kaplan as a large class of real Lie algebras which, when equipped with a suitable inner product, have certain rigidity properties that naturally generalize the familiar Heisenberg Lie algebra $\mathfrak{h^3$. Among them are the complex Heisenberg Lie algebras $\mathfrak{h^{2n+1_{\mathbb{C$ equipped with the (Hermitian) Euclidean inner product. One might ask whether there are other finite-dimensional complex Lie algebras which, when equipped with some Hermitian inner product, are H-type. In this short note, we show that there are not.
complex-htype.tex
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\begin{document}

\title{On complex H-type Lie algebras}
\author{Nathaniel Eldredge
  \footnote{
    School of Mathematical Sciences,
    University of Northern Colorado,
    501 20th St., Campus Box 122,
    Greeley, \textsc{CO}, 80639,
    \textsc{USA}.
    Email: \texttt{neldredge@unco.edu}
 }
}
\maketitle

\begin{abstract}
H-type Lie algebras were introduced by Kaplan as a large class of real
Lie algebras which, when equipped with a suitable inner product, have
certain rigidity properties that naturally generalize the familiar
Heisenberg Lie algebra $\mathfrak{h}^3$.  Among them are the complex
Heisenberg Lie algebras $\mathfrak{h}^{2n+1}_{\mathbb{C}}$ equipped
with the (Hermitian) Euclidean inner product.  One might ask whether
there are other finite-dimensional complex Lie algebras which, when
equipped with some Hermitian inner product, are H-type.  In this short
note,
we show that there are not.
\end{abstract}

Since their introduction by Kaplan \cite{kaplan80}, H-type Lie
algebras, and their corresponding nilpotent Lie groups, have attracted
interest as a natural generalization of the classical real Heisenberg
Lie algebra $\mathfrak{h}^3$ of dimension 3 and the corresponding real
Heisenberg group $\mathbb{H}^3$.  The Heisenberg group is a motivating
example in many areas of mathematics, and in many cases, facts about
the Heisenberg group carry over into the H-type setting.  For
instance, H-type groups carry a natural structure as sub-Riemannian
manifolds, and the analysis of their sub-Laplacians has attracted
considerable interest.  As a sampling, we mention
\cite{inglis-kontis-zegarlinski,eldredge-gradient,liu-wang-restriction,li-gradient-h-type,balogh-tyson}.
The H-type Lie algebras are a large class; in particular, it is shown
in \cite{kaplan80} that their centers may have any finite dimension.

One notable example of an H-type Lie algebra is the complex Heisenberg
Lie algebra $\mathfrak{h}_{\mathbb{C}}^3$ with its usual Euclidean
inner product, which has the additional
property of being a complex Lie algebra.  As such, analysis on the
complex Heisenberg group can take advantage of all the tools of
complex geometry, together with the many results for H-type groups
mentioned above.  The same is true for the higher-dimensional complex
Heisenberg (or Heisenberg--Weyl) Lie algebras
$\mathfrak{h}_{\mathbb{C}}^{2n+1}$ defined below.  However, the
purpose of this note is to record the elementary fact, which does not
seem to have appeared previously, that there are no other examples of
complex Lie algebras which are H-type under a Hermitian inner product.

We begin by recalling the definition of an H-type Lie algebra, as
formulated in \cite[Definition 18.1.1]{blu-book}. (The original
definition in \cite{kaplan80} is equivalent but slightly less
convenient for our purposes.)  Let $\mathfrak{g}$ be a real
finite-dimensional Lie algebra equipped with an inner product
$\inner{\cdot}{\cdot} : \mathfrak{g} \times \mathfrak{g} \to
\mathbb{R}$.  Let $\mathfrak{z}$ be the center of $\mathfrak{g}$, and
let $\mathfrak{v} = \mathfrak{z}^\perp$.  For $z \in \mathfrak{z}$ and
$u \in \mathfrak{v}$, define $J_z v$ as the unique element of
$\mathfrak{v}$ satisfying
\begin{equation}\label{Jdef}
  \inner{J_z u}{v} = \inner{z}{[u,v]} \quad \text{for all $v \in \mathfrak{v}$}.
\end{equation}
It is clear that $J_z : \mathfrak{v} \to \mathfrak{v}$ is a linear
map, and moreover is linear in $z$.

\begin{definition*}\label{Htype-def}
We say that $(\mathfrak{g}, \inner{\cdot}{\cdot})$ is \textbf{H-type}
if the following two conditions hold:
\begin{enumerate}
\item $[\mathfrak{v}, \mathfrak{v}] \subset \mathfrak{z}$
\item For each $z \in \mathfrak{z}$ with $\|z\|=1$, $J_z :
  \mathfrak{v} \to \mathfrak{v}$ is an isometry with respect to
  $\inner{\cdot}{\cdot}$.
\end{enumerate}
\end{definition*}
We observe that an H-type Lie algebra is necessarily nilpotent of step 2.

Now suppose that $\mathfrak{g}$ is a complex Lie algebra, whose
complex structure we denote by $i$.  If we wish to equip
$\mathfrak{g}$ with a real inner product, it is natural to demand some
compatibility with the complex structure.  Specifically, we woud like the inner
product to be \textbf{Hermitian}, i.e., for $x,y \in \mathfrak{g}$ we have
$\inner{ix}{iy} = \inner{x}{y}$.  We may then define $J$ as in
(\ref{Jdef}).  We observe for later use that, as a consequence of the
Hermitian property of the inner product, we have for $\alpha, \beta
\in \mathbb{C}$ and $u,z \in \mathfrak{g}$,
\begin{equation}\label{complex-J}
  J_{\alpha z}(\beta u) = \alpha \bar{\beta} J_z u.
\end{equation}
That is, $J_z u$ is complex linear in $z$ and conjugate linear in
$u$.

We may then speak of a \textbf{complex H-type Lie algebra} as a
complex Lie algebra $\mathfrak{g}$ equipped with a Hermitian inner
product $\inner{\cdot}{\cdot}$ such that $(\mathfrak{g},
\inner{\cdot}{\cdot})$ is H-type in the sense of the above definition.

\begin{example*}
  The \textbf{complex Heisenberg Lie algebra} of complex
  dimension $2n+1$ is the complex Lie algebra
  $\mathfrak{h}^{2n+1}_{\mathbb{C}}$ generated (over $\mathbb{C}$) by
  vectors $\{x_1, y_1, \dots, x_n, y_n, z\}$ with the bracket defined
  by $[x_k, y_k]=z$, and for $j \ne k$, $[x_j, y_k] = [x_j, z] = [y_j,
    z] = 0$.  We may equip $\mathfrak{h}^{2n+1}_{\mathbb{C}}$ with the
  real inner product $\inner{\cdot}{\cdot}$ that makes all of $x_k,
  ix_k, y_k, iy_k, z, iz$ orthonormal; it is clear that this inner
  product is Hermitian.  The center $\mathfrak{z}$ of
  $\mathfrak{h}^{2n+1}_{\mathbb{C}}$ is spanned (over $\mathbb{C}$) by
  $z$, so we clearly have $[\mathfrak{v}, \mathfrak{v}] =
  \mathfrak{z}$.  Defining $J$ as above, it is easy to compute
  \begin{equation*}
    J_z x_k = y_k \quad J_z y_k = -x_k \quad J_z ix_k = -iy_k
    \quad J_z iy_k = ix_k
  \end{equation*}
  so that $J_z$ is an isometry.  Moreover, every element $w \in
  \mathfrak{z}$ is of the form $w = \alpha z$ for some $\alpha \in \mathbb{C}$, and $\|w\| = |\alpha|$,
  so using (\ref{complex-J}) we see that $J_w$ is an isometry whenever
  $\|w\| = 1$.  Thus $(\mathfrak{h}^{2n+1}_{\mathbb{C}},
  \inner{\cdot}{\cdot})$ is a  complex H-type Lie algebra.
\end{example*}

We shall now prove that these are, up to isometric isomorphism, the
only complex H-type Lie algebras.

We first recall the well-known Clifford algebra identity for H-type
Lie algebras: 
\begin{equation}\label{clifford}
  J_z J_w + J_w J_z = -2 \inner{z}{w} I, \quad z,w \in \mathfrak{z}.
\end{equation}
To prove this, first consider the case when $w=z$ and $\|z\|=1$.  Then
for any $u,v \in \mathfrak{v}$, we have
\begin{align*}
  \inner{J_z^2 u}{v} = \inner{z}{[J_z u, v]} = -\inner{z}{[v, J_z u]}
  = -\inner{J_z v}{J_z u} = -\inner{v}{u}
\end{align*}
since $J_z$ is an isometry.  So $J_z^2 = -I$.  The general case
follows by scaling and polarization.

\begin{theorem}\label{mainthm}
  Let $(\mathfrak{g}, \inner{\cdot}{\cdot})$ be a complex H-type Lie
  algebra as defined above.  Then for some $n$, $(\mathfrak{g},
  \inner{\cdot}{\cdot})$ is isometrically isomorphic to
  $\mathfrak{h}^{2n+1}_{\mathbb{C}}$ with its standard Hermitian inner
  product.
\end{theorem}

In particular, complex H-type Lie algebras are completely classified
by their dimension, and all have centers of complex dimension 1.  Of
course, the analogous statement holds for their corresponding
nilpotent complex Lie groups, for which the inner product corresponds
to a left-invariant Riemannian metric which is Hermitian.

\begin{proof}
  Suppose $(\mathfrak{g}, \inner{\cdot}{\cdot})$ is complex H-type,
  and let $\mathfrak{v}$, $\mathfrak{z}$ and $J$ be defined as above.

  We first note that $\mathfrak{z}$ must have complex dimension 1.  If
  not, then we can find $z,w \in \mathfrak{z}$ with $\|z\| = \|w\| =
  1$ and $\inner{z}{w} = \inner{iz}{w} = 0$.  Then by (\ref{clifford})
  and (\ref{complex-J}) we have
  \begin{align*}
    0 &= -2 \inner{z}{w} I = J_z J_w + J_w J_z \\
    0 &= -2 \inner{iz}{w} I = J_{iz} J_w + J_w J_{iz} = i J_z J_w +
    J_w i J_z = i(J_z J_w - J_w J_z).
  \end{align*}
  Thus $J_w J_z = J_z J_w = 0$, contradicting the requirement that
  $J_z, J_w$ be isometries.

  Therefore, $\mathfrak{z}$ is the complex span of a single unit
  vector $z$.  We recursively construct an orthonormal basis for
  $\mathfrak{v}$ of the form $\{x_k, ix_k, y_k, iy_k : k=1,\dots,n\}$.
  Suppose $\{x_k, ix_k, y_k, iy_k : k = 1, \dots, m-1\}$ have been
  constructed and do not span $\mathfrak{v}$.  Let $x_m$ be any unit
  vector orthogonal to all of $x_k, ix_k, y_k, iy_k$ for
  $k=1,\dots,m$.  Then set $y_m = J_z x_m$.  We have $\|y_m\| = 1$,
  and a few straightforward computations verify that $\{x_k, ix_k,
  y_k, iy_k : k = 1,\dots, m\}$ are now orthogonal.  When the process
  terminates, we have our orthonormal basis.

  To compute brackets, for $j \ne k$ we have
  \begin{align*}
    \inner{z}{[x_k, y_k]} &= \inner{J_z x_k}{y_k} = \inner{y_k}{y_k}
    = 1 \\
    \inner{z}{[x_k, x_j]} &= \inner{J_z x_k}{x_j} = \inner{y_k}{x_j}
    = 0 \\
    \inner{z}{[y_k,y_j]} &= \inner{J_z y_k}{y_j} = \inner{J_z y_k}{J_z
      x_j} = \inner{y_k}{x_j} = 0 \\
    \inner{z}{[x_k, y_j]} &= \inner{J_z x_k}{y_j} =
    \inner{y_k}{y_j} = 0.
  \end{align*}
  Similar computations show that if $z$ is replaced by $iz$, all of
  the above expressions vanish.  Each bracket is in $\mathfrak{z}$ and
  hence a complex scalar multiple of $z$, so we have
  \begin{equation*}
    [x_k, y_k] = z, \quad [x_k, x_j] = [y_k, y_j] = [x_k, y_j] = 0.
  \end{equation*}
  The corresponding brackets for $ix_k, iy_k$, etc, follow from the
  complex bilinearity of the bracket.  These are precisely the
  same relations as for the complex Heisenberg Lie algebra
  $\mathfrak{h}^{2n+1}_{\mathbb{C}}$, and the basis is orthonormal,
  just as for the standard inner product on
  $\mathfrak{h}^{2n+1}_{\mathbb{C}}$.  Therefore, $(\mathfrak{g},
  \inner{\cdot}{\cdot})$ is isometrically isomorphic to
  $\mathfrak{h}^{2n+1}_{\mathbb{C}}$ with its standard inner product.
\end{proof}

\begin{remark*}
  It is well known that, given an H-type Lie algebra $(\mathfrak{g},
  \inner{\cdot}{\cdot})$, the map $\mathfrak{z} \ni z \mapsto J_z
  \in \operatorname{End}(\mathfrak{v})$ extends to a representation on
  $\mathfrak{v}$ of the Clifford algebra $C\ell(\mathfrak{z},
  \inner{\cdot}{\cdot})$.  This gives a one-to-one correspondence
  between H-type Lie algebras and Clifford algebra representations.
  (For further details, see \cite[Section 2.2]{eldredge-thesis}.)
  As such, Theorem \ref{mainthm} could be restated in terms of a real
  Clifford algebra $\mathcal{A}$ with a representation on a real
  vector space $V$, each of which is equipped with additional
  structure corresponding to the complex structure on $\mathfrak{g}$.
  However, this does not appear to be terribly enlightening.  In
  particular, we cannot simply state Theorem \ref{mainthm} in terms of
  a complex Clifford algebra with a complex representation, because as
  noted in (\ref{complex-J}), the map $J_z$ is not complex linear on
  $\mathfrak{v}$ but instead conjugate linear, so it doesn't
  correspond naturally to an endomorphism of the complex vector space
  $\mathfrak{v}$.
\end{remark*}

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\end{document}

complex-htype.bbl
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\bibitem[Balogh and Tyson(2002)]{balogh-tyson}
Zolt{\'a}n~M. Balogh and Jeremy~T. Tyson.
\newblock Polar coordinates in {C}arnot groups.
\newblock \emph{Math. Z.}, 241\penalty0 (4):\penalty0 697--730, 2002.
\newblock ISSN 0025-5874.
\newblock \doi{10.1007/s00209-002-0441-7}.
\newblock URL \url{http://dx.doi.org/10.1007/s00209-002-0441-7}.

\bibitem[Bonfiglioli et~al.(2007)Bonfiglioli, Lanconelli, and
  Uguzzoni]{blu-book}
A.~Bonfiglioli, E.~Lanconelli, and F.~Uguzzoni.
\newblock \emph{Stratified {L}ie groups and potential theory for their
  sub-{L}aplacians}.
\newblock Springer Monographs in Mathematics. Springer, Berlin, 2007.
\newblock ISBN 978-3-540-71896-3; 3-540-71896-6.

\bibitem[Eldredge(2009)]{eldredge-thesis}
Nathaniel Eldredge.
\newblock \emph{Hypoelliptic heat kernel inequalities on {H}-type groups}.
\newblock PhD thesis, {U}niversity of {C}alifornia, {S}an {D}iego, 2009.
\newblock URL \url{http://arxiv.org/abs/1406.1840}.
\newblock arXiv:1406.1840.

\bibitem[Eldredge(2010)]{eldredge-gradient}
Nathaniel Eldredge.
\newblock Gradient estimates for the subelliptic heat kernel on {$H$}-type
  groups.
\newblock \emph{J. Funct. Anal.}, 258\penalty0 (2):\penalty0 504--533, 2010.
\newblock ISSN 0022-1236.
\newblock \doi{10.1016/j.jfa.2009.08.012}.
\newblock URL \url{http://dx.doi.org/10.1016/j.jfa.2009.08.012}.
\newblock arXiv:0904.1781.

\bibitem[Hu and Li(2010)]{li-gradient-h-type}
Jun-Qi Hu and Hong-Quan Li.
\newblock Gradient estimates for the heat semigroup on {H}-type groups.
\newblock \emph{Potential Anal.}, 33\penalty0 (4):\penalty0 355--386, 2010.
\newblock ISSN 0926-2601.
\newblock \doi{10.1007/s11118-010-9173-1}.
\newblock URL \url{http://dx.doi.org/10.1007/s11118-010-9173-1}.

\bibitem[Inglis et~al.(2011)Inglis, Kontis, and
  Zegarli{\'n}ski]{inglis-kontis-zegarlinski}
J.~Inglis, V.~Kontis, and B.~Zegarli{\'n}ski.
\newblock From {$U$}-bounds to isoperimetry with applications to {H}-type
  groups.
\newblock \emph{J. Funct. Anal.}, 260\penalty0 (1):\penalty0 76--116, 2011.
\newblock ISSN 0022-1236.
\newblock \doi{10.1016/j.jfa.2010.08.003}.
\newblock URL \url{http://dx.doi.org/10.1016/j.jfa.2010.08.003}.

\bibitem[Kaplan(1980)]{kaplan80}
Aroldo Kaplan.
\newblock Fundamental solutions for a class of hypoelliptic {PDE} generated by
  composition of quadratic forms.
\newblock \emph{Trans. Amer. Math. Soc.}, 258\penalty0 (1):\penalty0 147--153,
  1980.
\newblock ISSN 0002-9947.

\bibitem[Liu and Wang(2011)]{liu-wang-restriction}
Heping Liu and Yingzhan Wang.
\newblock A restriction theorem for the {H}-type groups.
\newblock \emph{Proc. Amer. Math. Soc.}, 139\penalty0 (8):\penalty0 2713--2720,
  2011.
\newblock ISSN 0002-9939.
\newblock \doi{10.1090/S0002-9939-2011-10907-9}.
\newblock URL \url{http://dx.doi.org/10.1090/S0002-9939-2011-10907-9}.

\end{thebibliography}
