Journal of Lie Theory Volume 12 (2002) 69–79 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

Integral Structures on H-type Lie Algebras

Gordon Crandall and Józef Dodziuk^{∗}

Communicated by M. Moskowitz

Abstract. In this paper we prove that every H-type Lie algebra possesses a basis with respect to which the structure constants are integers. Existence of such an integral basis implies via the Mal’cev criterion that all simply connected H-type Lie groups contain co-compact lattices. Since the Campbell-Hausdorff formula is very simple for two-step nilpotent Lie groups we can actually avoid invoking the Mal’cev criterion and exhibit our lattices in an explicit way. As an application, we calculate the isoperimetric dimensions of H-type groups.

1. Introduction

In this paper we prove that every H-type Lie algebra [7, 8, 9] possesses a basis with respect to which the structure constants are integers. We are going to call such a basis an integral basis. Existence of an integral basis implies via the Mal’cev criterion that all simply connected H-type Lie groups contain cocompact lattices. Since the Campbell-Hausdorff formula is very simple for twostep nilpotent Lie groups we can actually avoid invoking the Mal’cev criterion and exhibit our lattices in an explicit way.

The theory of H-type Lie algebras is related very closely to the theory of Clifford algebras and Clifford modules (cf. [4, 10]) and we are going to use the classification of Clifford modules in our construction.

We briefly recall the definition of H-type Lie algebras and establish notation and conventions for the sequel. Let U and V be two finite-dimensional inner product spaces over ℝ of dimensions m and n respectively. Let J : U ⟶ End(V), z ↦ J_{z}, be a linear mapping satisfying

|J_{z}(v)| = |z| |v|  (1)
J_{z} ^{}J_{z}(v) = −|z|^{2} v  (2)

for all z ∈ U and v ∈ V. Such a mapping J is called an orthogonal multiplication. Because of (2), by the universal property of Clifford algebras [10, Proposition 1.1, Chapter 1], J extends to an algebra homomorphism ϕ of the Clifford algebra

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag C(U) into End(V) so that V acquires the structure of a module over C(U). We will often write ϕ(α)v = αv and call αv the Clifford product of α and v.

∗ The second author was supported in part by NSF Grant #DMS-9704743 and the PSC-CUNY Research Award Program

After squaring, polarization of (1) yields

(J_{z_{1}}(v), J_{z_{2}}(v)) = (z_{1}, z_{2}) |v|^{2}  (3)
(J_{z}(v_{1}), J_{z}(v_{2})) = |z|^{2}(v_{1}, v_{2})  (4)

holding for all z, z_{1}, z_{2} ∈ U and v, v_{1}, v_{2} ∈ V. In order not to complicate notation, we use the same notation for norms and inner products in U and V. It follows immediately that

(J_{z}(v_{1}), v_{2}) = −(v_{1}, J_{z}(v_{2})),  (5)

i.e. J_{z}(v) is a skew-adjoint endomorphism of V, J_{z}^{∗} = −J_{z}. Now the Lie algebra structure is defined on N = U ⊕ V by requiring that U be contained in the center and that the bracket of two elements of V belong to U and satisfy

(z, [v_{1}, v_{2}]) = (J_{z}(v_{1}), v_{2}).

In this way, N becomes a two-step nilpotent Lie algebra which is referred to as a Heisenberg-type or H-type Lie algebra. Orthonormal bases z_{1}, …, z_{m} and v_{1}, …, v_{n} of U and V respectively give rise to the basis z_{1}, …, z_{m}, v_{1} …, v_{n} of N which is orthonormal. The only nonzero structure constants for N with respect to this basis occur among numbers A_{i,j}^{k} defined by

[v_{i}, v_{j}] = A_{i,j}^{k}z_{k}.  (6)

We are now ready to state our main result.

Theorem 1.1. For every H-type Lie algebra N = U ⊕ V as above there exist orthonormal bases z_{1}, …, z_{m} and v_{1}, …, v_{n} of U and V respectively so that the structure constants A_{i,j}^{k} of the Lie algebra N with respect to the basis z_{1}, …, z_{m}, v_{1} …, v_{n} are integers and in fact take values 0, 1, −1.

The numbers A_{i,j}^{k} are clearly equal to (J_{z_{k}}(v_{i}), v_{j}) = (z_{k}v_{i}, v_{j}), i.e. depend only on the Clifford module structure and the inner product of V. According to [10, Proposition 5.16], every Clifford module, i.e. a finite dimensional module over C(U), admits an inner product such that the Clifford multiplication by elements of U ⊂ C(U) is an orthogonal multiplication. We remark that the Clifford multiplication is orthogonal if and only if elements of U act by skewadjoint transformations, i.e. if and only if (5) holds.

Thus every Clifford module gives rise to an H-type Lie algebra and Theorem 1.1 can be reformulated as a statement about Clifford modules as follows.

Theorem 1.2. Given an inner product space U with an orthonormal basis z_{1}, …, z_{m} and a module V over the Clifford algebra C(U) there exists an inner product (·, ·) on V and an orthonormal basis v_{1} …, v_{n} of V = (V, (·, ·)) so that the Clifford multiplication J_{z}(v) = zv by elements of U satisfies (1) and (2) and (z_{i}v_{p}, v_{q}) is equal to 0, 1 or −1 for all i, p, q.

We will say that the choice of an inner product and a basis for V as above is an integral structure and that V with this additional structure is an integral Clifford module. Observe that the fact that the values of (e_{i}v_{p}, v_{q}) are 0, 1, −1 is equivalent to the assertion that each of the generators e_{i} acts on the basis v_{1}, v_{2}, …, v_{n} by a permutation and, possibly, some changes of sign.

¿From now on we will abandon the notational distinction between an inner product space U and the underlying vector space U.

2. Proof of Theorem 1.2

Since Clifford modules are completely reducible [10, p. 31], it suffices to prove Theorem 1.2 for irreducible Clifford modules. Suppose dim U = k and let e_{1}, …, e_{k} be an orthonormal basis of U. We are going to identify U with ℝ^{k} and C(U) with the algebra C_{k} generated over ℝ by e_{1}, …, e_{k} subject to relations

e_{i}e_{j} = −e_{j}e_{i},  e_{i}^{2} = −1.  (7)

According to the classification of irreducible Clifford modules, for every k ≠ 3 (mod 4) there exists only one isomorphism class of irreducible modules over C_{k}. If k = 3 (mod 4) there are two such classes, but the dimensions as vector spaces over ℝ of non-isomorphic modules are equal and the H-type groups associated to them are isomorphic. We will denote an irreducible module over C_{k} by V_{k}.

Classification of irreducible Clifford modules proceeds by induction on k, cf. [10, p. 33], and we shall retrace this induction proving at every stage existence of an integral basis. It will be convenient to first classify ℤ_{2}-graded Clifford modules. We briefly recall their definition. Denote by C_{k}^{0} the subspace generated by products of even numbers of generators e_{1}, …, e_{k} and by C_{k}^{1} the subspace generated by products of an odd number of generators. Then C_{k}^{0} is a subalgebra, C_{k}^{0}⊕C_{k}^{1} = C_{k}, and

C_{k}^{i} · C_{k}^{j} ⊂ C_{k}^{i+j}

for i, j ∈ ℤ_{2}. A finite dimensional space W over ℝ is a ℤ_{2}-graded Clifford module if W = W^{0} ⊕ W^{1} and

C_{k}^{i} · W^{j} ⊂ W^{i+j}

with i, j ∈ ℤ_{2}.

We need to define an analog of integral structure on ℤ_{2}-graded Clifford modules (we use definitions and notation of [4, Chapter 11, Sections 4,6] regarding ℤ_{2}-graded tensor products of Clifford algebras and Clifford modules).

Definition 2.1. Suppose W^{0} is a module over C_{k}^{0}. A choice of inner product ( · , ·) on W^{0} and an orthonormal basis w_{1}, … w_{m} with respect to this inner product is called integral if the basis elements are permuted with a possible change in sign by Clifford multiplication by all double products e_{i}e_{j} and

(ze_{k}w, ze_{k}w) = |z|^{2}|w|^{2}

for all w ∈ W^{0} and z ∈ ℝ^{k}. We will call W^{0} with the inner product and a basis satisfying the conditions above integral.

If W^{0} is the 0-component of a ℤ_{2}-graded Clifford module W then multiplication by e_{k} is an isomorphism of W^{0} onto W^{1}. Therefore we can transfer the inner product on W^{0} to W^{1} and define the inner product on W^{0} ⊕ W^{1} = W by requiring the two summands to be orthogonal. A simple calculation then shows that the Clifford multiplication by elements of ℝ^{k} is an orthogonal multiplication and the elements of the basis w_{1}, …, w_{m}, e_{k}w_{1}, … e_{k}w_{m} are permuted by Clifford multiplication by every e_{i} with a possible change in sign. In particular, Theorem 1.2 holds for W. In the sequel we are only going to use inner products such that W^{0} ⊥ W^{1} and the multiplication by e_{k} maps W^{0} onto W^{1} isometrically.

Our proof of Theorem 1.2 will be carried out by showing that for every k there exists an irreducible ℤ_{2}-graded Clifford module W_{k} with an integral structure. This will imply that every (ungraded) irreducible Clifford module is integral. It will follow a posteriori that all ℤ_{2}-graded Clifford modules are integral. The main fact in the classification of Clifford modules is that if W_{k} and W_{8} are irreducible ℤ_{2}-graded modules over C_{k} and C_{8} respectively, then W_{k} ˆ⊗W_{8} is an irreducible ℤ_{2}-graded module over C_{k} ˆ⊗C_{8} ≃ C_{k+8}. Our proof of Theorem 1.2 will consist of exhibiting an integral irreducible ℤ_{2}-graded Clifford module W_{k} for k = 1, …, 8 and then showing that W_{k} ˆ⊗W_{8} is integral if W_{k} carried an integral structure. To handle the low dimensional cases we need the following general lemma.

Lemma 2.2. Suppose V is an integral module over the algebra C_{k}^{0}. Then W = C_{k} ⊗_{C_{k}^{0}} V is a ℤ_{2}-graded module over C_{k} with the ℤ_{2}-grading given by W^{0} = 1⊗V and W^{1} = e_{k} ⊗V . Thus W^{0} has integral structure transfered from V via the isomorphism v ↦ 1 ⊗ v. In addition, if V is irreducible as a C_{k}^{0}-module then W is irreducible as a ℤ_{2}-graded module over C_{k}.

Proof. This is obvious (see [4, Chapter 11, Proposition 6.3]).

Our next task is to describe integral structures on Clifford modules over C_{k} for k ≤ 8. For k ≤ 7, we refer to the description of (ungraded) Clifford modules given in [1].

Case k = 1. ℝ ≃ iℝ is acting on ℂ by complex multiplication. The real and imaginary parts are 0- and 1-components of ℤ_{2}-grading respectively and the standard inner product together with 1 as the basis of the 0-component give the integral structure.

Case k = 2. Consider the space of quaternions ℍ equipped with the standard inner product. Identify ℝ^{2} with the span of i, j and let it act on ℍ by quaternion multiplication. This makes ℍ into an irreducible module over C_{2}. In addition, the decomposition ℍ = span {1, k} ⊕ span {i, j} is a ℤ_{2}-grading and an integral structure is given by the basis 1, k, i, j.

Case k = 3. In this case the space of quaternions becomes a module over the Clifford algebra C_{3} represented in the algebra of endomorphisms of ℍ as the subalgebra generated by quaternion multiplications by i, j, and k. Clearly, the standard basis 1, i, j, k is permuted with possibly a change in sign by these endomorphisms so that ℍ as a module over C_{3} carries an integral structure. However, this module does not have a natural ℤ_{2}-graded structure. We regard ℍ as a module over C_{3}^{0} ⊂ C_{3} and apply Lemma 2.2 to create an integral ℤ_{2}- graded irreducible C_{3}-module structure on C_{3} ⊗_{C_{3}^{0}} ℍ.

To cover the remaining low-dimensional cases we will use the algebra of octonions 𝕆 with its standard generators 1, i_{1}, … i_{7} and multiplication table given in [5, Page 448] with c = −1.

Case k = 4. This is analogous to the case k = 2. ℝ^{4} is identified with span{i_{1}, i_{2}, i_{3}, i_{4}} which acts on 𝕆 by octonion multiplication. The resulting Clifford module is ℤ_{2}-graded with the 0-component equal to span{1, i_{5}, i_{6}, i_{7}} and the 1-component span{i_{1}, i_{2}, i_{3}, i_{4}}. These bases are orthonormal with respect to the standard inner product and give rise to an integral structure. This follows from inspection of the multiplication table.

Case k = 5, 6, 7. We treat these three cases simultaneously. The Clifford algebra C_{k} can be represented as an algebra of endomorphisms of 𝕆 generated by transformations of octonion multiplications by i_{j}, 1 ≤ j ≤ k. The resulting Clifford module V_{k} is irreducible but not ℤ_{2}-graded. As above, we regard it as a module over C_{k}^{0} which allows us to create an irreducible, ℤ_{2}-graded, integral Clifford module W_{k} = C_{k} ⊗_{C_{k}^{0}} 𝕆 using Lemma 2.2.

Case k = 8. This uses the isomorphism, for every k ≥ 2, ϕ : C_{k−1} ⟶ C_{k}^{0} defined as follows [4, Chapter 11, Section 6]. Let x = x_{0} +x_{1} be the decomposition of x ∈ C_{k} into its 0 and 1 components. Then ϕ(x) = x_{0} + e_{k}x_{1} where, given standard generators e_{1}, …, e_{k} of C_{k}, we regard C_{k−1} as the subalgebra generated by e_{1}, …, e_{k−1}. The octonions are a module over C_{7} as above (we relabel i_{j} as e_{j}) which allows us to regard 𝕆 as a module over C_{8}^{0} by defining the multiplication xo as ϕ^{−1}(x)o. Since ϕ^{−1}(e_{j}) = e_{j} and ϕ^{−1}(e_{8}e_{j}) = e_{j} for j ≤ 7 and 𝕆 was integral as a module over C_{7}, the resulting module over C_{8}^{0} is integral. Applying Lemma 2.2 we obtain an integral, irreducible, ℤ_{2}-graded module W_{8} = C_{8} ⊗_{C_{8}^{0}} 𝕆. We remark that W_{8} is irreducible as an ungraded Clifford module as well. This is because W_{8}^{0} is irreducible as a module over C_{8}^{0} (since the pair (W_{8}^{0}, C_{8}^{0}) is isomorphic to (V_{7}, C_{7}) and V_{7} is irreducible as a module over C_{7}).

We are now ready for the inductive step in the argument. Suppose W_{k} and W_{l} are integral, ℤ_{2}-graded Clifford modules over C_{k} and C_{l} respectively. Let e_{1}, …, e_{k} and f_{1}, …, f_{l} be the standard generators of C_{k} and C_{l}. Suppose W_{k}^{0} and W_{l}^{0} are equipped with inner products and integral bases v_{1}, …, v_{m} and w_{1}, …, w_{n} respectively. Recall that as a vector space W_{k} ˆ⊗W_{l} is isomorphic to W_{k} ⊗ W_{l} so that we can equip W_{k} ˆ⊗W_{l} with an inner product by requiring that

(x_{1} ⊗ y_{1}, x_{2} ⊗ y_{2}) = (x_{1}, x_{2}) · (y_{1}, y_{2}).

In particular, different components W_{k}^{i} ⊗ W_{l}^{j} for (i, j) ∈ ℤ_{2} × ℤ_{2} are orthogonal. Under the isomorphism C_{k} ˆ⊗C_{l} ≃ C_{k+l}, cf. [4, Corollary 4.8] (note that our notation differs from the notation in [4] since we use the symbol ˆ⊗ instead of ⊗ to denote the tensor product of ℤ_{2}-graded algebras), the standard generators of C_{k+l} correspond to e_{1} ⊗ 1, …, e_{k} ⊗ 1, 1 ⊗ f_{1}, … 1 ⊗ f_{l}. To exhibit an integral structure we need a basis of (W_{k} ˆ⊗W_{l})^{0} = (W_{k}^{0} ⊗ W_{l}^{0}) ⊕ (W_{k}^{1} ⊗ W_{l}^{1}) in addition. The basis that we are going to use is

{v_{i} ⊗ w_{j}, e_{k}v_{i} ⊗ f_{l}w_{j} | 1 ≤ i ≤ m, 1 ≤ j ≤ n}.  (8)

Lemma 2.3. The bases described above give rise to an integral structure on (W_{k} ˆ⊗W_{l})^{0} and, consequently, on W_{k} ˆ⊗W_{l}.

Proof. We first verify that the Clifford multiplication by elements of the space U spanned by the generators of C_{k} ˆ⊗C_{l} is orthogonal. Recall that it suffices to verify that each element z ∈ U acts on W_{k} ˆ⊗W_{l} as a skew-symmetric endomorphism. We abuse the notation and write z for an endomorphism associated with z. Clearly,

(e_{i} ⊗ 1)^{∗} = e_{i}^{∗} ⊗ 1 = −e_{i} ⊗ 1
(1 ⊗ f_{j})^{∗} = 1 ⊗ f_{j}^{∗} = −1 ⊗ f_{j}

since Clifford multiplications on W_{k} and W_{l} are orthogonal. Since every element of U is a linear combination of such products, our assertion follows.

To verify properties of the integral structure, it suffices to show that elements of the basis (8) are permuted up to sign by the double products of generators. There are three cases to consider (e_{i} ⊗ 1) · (e_{j} ⊗ 1) = e_{i}e_{j} ⊗ 1, (1 ⊗ f_{i}) · (1 ⊗ f_{j}) = 1 ⊗ f_{i}f_{j}, and (e_{i} ⊗ 1) · (1 ⊗ f_{j}) = (e_{i} ⊗ f_{j}). In the first two cases the action is as desired since the multiplication by e_{p}’s permutes v_{1}, …, v_{m} up to sign and the multiplication by f_{q} ’s acts the same way on w_{1}, …, w_{n}. To treat the third case, note that e_{p} = ±e_{k}e_{p}e_{k} and f_{q} = ±f_{l}f_{q}f_{l}. Thus, up to signs,

(e_{p} ⊗ f_{q}) · (v_{i} ⊗ w_{j}) = e_{k}e_{p}e_{k}v_{i} ⊗ f_{l}f_{q}f_{l}w_{j} = e_{k}v_{i′} ⊗ f_{l}w_{j′}

since double products of generators of C_{k} and C_{l} permute up to sign the distinguished bases of W_{k}^{0} and W_{l}^{0} respectively. Similarly,

(e_{p} ⊗ f_{q}) · (e_{k}v_{i} ⊗ f_{l}w_{j}) = e_{p}e_{k}v_{i} ⊗ f_{q}f_{l}w_{j} = e_{k}v_{i′} ⊗ f_{l}w_{j′}.

This proves the lemma.

Corollary 2.4. Suppose that W_{k} is an irreducible, integral, ℤ_{2}-graded module over C_{k}. Then W_{k} ˆ⊗W_{8} is an irreducible, integral, ℤ_{2}-graded module over C_{k} ˆ⊗C_{8} ≃ C_{k+8} with the integral structure described above.

Proof. All assertions except irreducibility are contained in the lemma above. By [4, Chapter 11, 6.5] dim_{ℝ} W_{k} ˆ⊗W_{8} = dim_{ℝ} W_{k} · dim_{ℝ} W_{8} = 16 dim_{ℝ} W_{k} is equal to the dimension of an irreducible, ℤ_{2}-graded Clifford module over C_{k+8}. It follows immediately that W_{k} ˆ⊗W_{8} is irreducible.

\begin{matrix}k & a_{k} & b_{k} \\ 1 & 2 & 2 \\ 2 & 4 & 4 \\ 3 & 4 & 8 \\ 4 & 8 & 8 \\ 5 & 8 & 16 \\ 6 & 8 & 16 \\ 7 & 8 & 16 \\ 8 & 16 & 16 \end{matrix}

Table 1: Dimensions of irreducible Clifford modules

An easy induction using the explicit description of irreducible, ℤ_{2}-graded Clifford modules over C_{k}, 1 ≤ k ≤ 8 and the lemma above yields existence of an irreducible, ℤ_{2}-graded Clifford module W_{k} over C_{k} for every k.

Some additional work is required to show that every (ungraded) Clifford module has an integral structure. Let a_{k} be the dimension of an irreducible Clifford module over C_{k}, and let b_{k} denote the dimension of an irreducible, ℤ_{2}- graded module over C_{k}. It follows from [4, Propositions 6.2 and 6.3, Chapter 11] that a_{k} = b_{k+1}/2. From basic periodicity [4, 6.5, Chapter 11], a_{k+8} = 16a_{k} and b_{k+8} = 16b_{k}. All values of a_{k} and b_{k} can now be computed from Table 1, which summarizes some of the information about Clifford modules of low dimensions.

Recall now (cf. [4, Chapter 6]) that if k = 1, 2, 4, 8 (mod 8) then the dimensions over ℝ of an irreducible Clifford module and an irreducible ℤ_{2}-graded Clifford module are equal. Thus ℤ_{2}-graded Clifford modules W_{k} constructed above are irreducible as (ungraded) Clifford modules. The classification also says that for these values of k there exists exactly one isomorphism class of Clifford modules over C_{k} which proves Theorem 1.2 for k = 1, 2, 4, 8 (mod 8).

Next consider the case k = 3 (mod 4). According to the classification there are two non-isomorphic Clifford modules over C_{k}. We are going to show that their direct sum is isomorphic as an ungraded Clifford module to W_{k} and that the integral structure of W_{k} induces integral structures on the two modules. The key role in the proof of this fact is played by the “volume element” ω = e_{1}e_{2} … e_{k}. ω belongs to the center of C_{k} and satisfies ω^{2} = 1, cf. [10, Proposition 3.3, Chapter 1]. In addition, the multiplication by ω is a symmetric operator on every orthogonal Clifford module. This can be seen as follows. If k = 4l + 3, then

ω^{∗} = e_{k}^{∗}e_{k−1}^{∗} … e_{1}^{∗} = (−1)^{4l+3}e_{k}e_{k−1} … e_{1} = − (−1)^{\frac{(4l+4)(4l+3)}{2}} e_{1}e_{2} … e_{k} = ω,

since multiplications by e_{p}’s are skew-symmetric and e_{p}’s anti-commute. Now define ϕ_{+} and ϕ_{−} to be multiplications by central elements (1+ω)/2 and (1−ω)/2 respectively. ϕ_{±} are self-adjoint and satisfy ϕ_{±}^{2} = ϕ_{±}, i.e. they are orthogonal projections onto their images. Since (1 + ω)(1 − ω) = 0 and ϕ_{+} + ϕ_{−} = 1 the ranges of these projections are perpendicular. Since 1 ± ω are central, they are in fact (ungraded) Clifford submodules V_{+} and V_{−} of W_{k}. Since ωW_{k}^{0} = W_{k}^{1}, ϕ_{±} is injective on W_{k}^{0}. A count of dimensions shows that W_{k} = ϕ_{+}W_{k}^{0} ⊕ ϕ_{−}W_{k}^{0} is an orthogonal direct decomposition of W_{k} into two Clifford submodules. ω acts on V_{±} by multiplication by ±1 so that the two modules are non-isomorphic. To conclude our analysis we exhibit integral bases of the two modules. We do the argument for V_{+} since the argument for V_{−} is identical after 1 + ω is replaced by 1 − ω. Let w_{1}, …, w_{m} be an integral basis of W_{0}, i.e. an orthonormal basis which is permuted up to sign by multiplications by double products e_{i}e_{j}. Let

v_{p} = \frac{1}{\sqrt{2}}(1 + ω)w_{p} ∈ V_{+}

for p = 1, …, m. Since ωW_{k}^{0} = W_{k}^{1} is perpendicular to W_{k}^{0}, the Pythagorean theorem insures that this is an orthonormal basis. For a generator e_{i} of C_{k} we calculate

(e_{i}v_{p}, v_{q}) = (e_{i}w_{p} + e_{i}ωw_{p}, w_{q} + ωw_{q})/2.

Observe that e_{i}ωw_{p} and w_{q} belong to W_{k}^{0} while e_{i}w_{p} and ωw_{q} are in W_{k}^{1}. Therefore the inner product above simplifies to

((e_{i}w_{p}, ωw_{q}) + (e_{i}ωw_{p}, w_{q}))/2 = (e_{i}w_{p}, ωw_{q}),

since ω is central and self-adjoint. Now ωw_{q} can be expressed as a product of e_{i} times a product of an even number of generators of C_{k}. It therefore follows from the definition of an integral basis that e_{i}ωw_{q} is up to sign equal to e_{i}w_{q′} so that, finally

(e_{i}v_{p}, e_{i}v_{q′}) = (v_{p}, v_{q′}) = δ_{p,q′}

which proves that the basis v_{1}, …, v_{m} is integral.

The remaining two cases of Theorem 1.2 are k = 5, 6 (mod 8) say k = 8l + r, with r equal to 5 or 6. By the classification, in either case there is only one isomorphism class of irreducible modules over C_{k} and the dimension over ℝ of an irreducible module in this class is equal to the dimension of the Clifford module V_{+} over C_{4l+7} constructed above. Clearly, C_{k} can be regarded as the subalgebra of C_{4l+7} generated by the first k generators so that V_{+} = V_{k} becomes a module over C_{k}. It is irreducible since its dimension is that of an irreducible module and, trivially, integral. Theorem 1.2 is proved.

We conclude this section with a very simple proof, available now, of the fact that two non-isomorphic Clifford modules over C_{k}, k = 3 (mod 4) give rise to isomorphic H-type algebras. This is very well known, cf. [11], but we give the proof for completeness. The structure constants of the H-type Lie algebra U ⊕V_{+} associated to the Clifford module V_{+} have been computed above and are equal to (e_{i}v_{p}, v_{q}) = (e_{i}w_{p}, ωw_{q}) (here U = span{e_{1}, …, e_{k}}). In the analogous calculation for V_{−} ω is replaced by −ω so that the corresponding structure constants for the Lie algebra U ⊕ V_{−} (with respect to the basis e_{1}, …, e_{k}, v′_{1}, …, v′_{m} where v′_{p} = (1/\sqrt{2})(1 − ω)w_{p} are negatives of the structure constants for U ⊕ V_{+} if we choose the obvious correspondence between bases of V_{+} and V_{−}. However, if we deploy the set −e_{1}, …, −e_{k}, v′_{1}, …, v′_{1}, …, v′_{m} as the basis for U ⊕ V_{−} instead, the structure constants for the two Lie algebras are equal so that the algebras are isomorphic.

3. Examples and applications

In this section we show that every simply connected Lie group of Heisenberg type contains a co-compact lattice. We are also going to calculate the isoperimetric dimension of such groups.

We choose and fix an integral basis as in Theorem 1.1. Since H-type Lie algebras are two-step nilpotent (in particular nilpotent) the exponential mapping is a global diffeomorphism of the Lie algebra and the group. We will therefore use it to identify the two. In addition the Campbell-Hausdorff formula that expresses the group multiplication in terms of the Lie bracket takes a particularly simple form

X · Y = X + Y + \frac{1}{2}[X, Y]  (9)

where X, Y ∈ N = U ⊕ V as in the Introduction. Let U_{ℤ} be the abelian subgroup of U consisting of linear combination with integer coefficients of the generators e_{1}, …, e_{k} of C(U) ≃ C_{k}. Let V_{ℤ} be the lattice in V generated by elements of an integral basis. Since [V_{ℤ}, V_{ℤ}] ⊂ U_{ℤ} our candidate for the lattice L in the group (N, ·) is the subset

L = \frac{1}{2}U_{ℤ} ⊕ V_{ℤ}.

Verification of this claim is straightforward after first checking that −X is the inverse of X with respect to the group operation. Thus L is a subgroup of N and it is obviously discreet. The product of an arbitrary element u + v of N with w ∈ V is equal to u + (1/2)[v, w] + v + w and has u + (1/2)[v, w] as its U component and v + w as the component in V . Given v we can choose w ∈ V_{ℤ} so that v + w has all coefficients in the interval [0, 1] when expanded with respect to the distinguished basis of V . We can then act on u + (1/2)[v, w] by elements u_{1} of (1/2)U_{ℤ}. Since this action is by ordinary translation (U_{ℤ} is contained in the center of N) we can easily achieve that u + v + (1/2)[v, w] + u_{1} has its coefficients with respect to the basis e_{1}, …, e_{k} in the interval [0, 1/2]. Hence all coefficients in the expansion of

(u + v) · (u_{1} + w) = u + u_{1} + \frac{1}{2}[v, w] + v + w

with respect to the integral basis chosen lie in [0, 1], which proves that L is cocompact.

To calculate the isoperimetric dimension of N we will need to calculate the ranks of groups in the lower central series of L. Using our explicit description of the product in N one verifies very easily that the bracket operation of the Lie algebra coincides with the commutator operation in the group. We therefore have [L, L] = [V_{ℤ}, V_{ℤ}] = U_{ℤ}. The second equality requires a proof. Observe first that

|(z, [v, w])| = |(zv, w)| ≤ |z| · |v| · |w|

if z ∈ U, v, w ∈ V . It follows that

|[v, w]| ≤ |v| · |w|.

Now let e_{i} be one of the generators of C_{k}. Then e_{i} induces a permutation of the basis vectors of V with a possible change in sign. Thus given v_{p}, e_{k}v_{p} = ±v_{q} for some q. It follows that |(e_{i}, [v_{p}, v_{q}])| = |(e_{1}v_{p}, v_{q})| = 1 so that e_{i} = ±[v_{p}, v_{q}]. Thus each of the generators e_{i} is a commutator which proves the claim. We see therefore that

L/[L, L] ≃ (\frac{1}{2}U_{ℤ}/U_{ℤ}) ⊕ V_{ℤ} ≃ ℤ_{2} ⊕ V_{ℤ}.

We remark that there is no ambiguity in the interpretation of the quotient above since [L, L] ⊂ U and the group multiplication by u ∈ U amounts to the the translation by u because of (9).

A theorem of Bass [2] gives the estimate of growth of L in the word metric with respect to a finite set of generators. In our case we can take for example the generating set S given by the vectors of an integral basis and their negatives. The lower central series of L reduces to L_{0} = L ⊃ L_{1} = [L, L] ⊃ L_{2} = 0. The ranks of successive quotients are therefore equal to dim V and dim U. If d = dim V + 2 dim U then by Bass’ theorem L has polynomial growth of degree d, i.e. the number g(R) of distinct elements of L in the metric ball of radius R in L satisfies

g(R) ≥ cR^{d}.

where the constant c depends on the choice of the generating set. A result of Coulhon and Saloff-Coste [3] asserts that the Cayley graph of a group of polynomial growth of degree d satisfies a d-dimensional isoperimetric inequality. Since every Lie group has bounded geometry we can invoke Kanai’s theorem [6] to conclude that the H-type group under consideration satisfies an isoperimetric inequality of the same kind. More precisely we have the following.

Theorem 3.1. Suppose N = U ⊕ V is an H-type group equipped with a leftinvariant metric. There exists a positive constant c such that for every relatively compact subset F of N with smooth boundary ∂F we have

\frac{A(∂F)}{V (F)^{1−1/d}} ≥ c,

where A(∂F) and v(F) denote the n − 1-dimensional volume of the boundary of F and V (F) stands for the volume of F.

References

[1] Barbano, P., Automorphisms and quasi-conformal mappings of Heisenbergtype groups, J. Lie Theory 8 (1998), 255–277.

[2] Bass, H., The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. 3 #25 (1972), 603–614.

[3] Coulhon, T., and L. Saloff-Coste, Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamericana 9 (1993), 293–314.

[4]  Husemoller, D., “Fibre Bundles”, New York: McGraw Hill, 1966.

[5] Jacobson, N., “Basic Algebra I”, New York: W. H. Freeman and Company, 1985.

[6] Kanai, M., Rough isometries and combinatorial approximations of geometries of non-compact Riemannian manifolds, J. Math. Soc. Japan 37 #3 (1985), 391–413.

[7] Kaplan, A., Fundamental solutions of a class of hypoelliptic PDE, Transactions Amer. Math. Soc. 258 (1980), 147–153.

[8] —, Riemannian nil-manifolds attached to Clifford modules, Geometriae Dedicata 11 (1981), 127–136.

Crandall & Dodziuk 79

[9] —, On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983), 35–42.

[10] Lawson, B., and L. Michelson, “Spin Geometry”, Princeton, NJ: Princeton University Press, 1989.

[11] Ricci. F., Harmonic analysis on groups of H-type, preprint.

Gordon Crandall Department of Mathematics LaGuardia Community College The City University of New York 31-10 Thomson Avenue Long Island City, NY 11101 crandallgo@lagcc.cuny.edu Józef Dodziuk Ph.D. Program in Mathematics Graduate Center The City University of New York 365 Fifth Avenue New York, NY 10016 jdodziuk@gc.cuny.edu

Received September 12, 2000 and in final form February 7, 2001