COMPLETE CLASSIFICATION OF PSEUDO H-TYPE LIE ALGEBRAS: I KENRO FURUTANI, IRINA MARKINA Abstract.

KENRO FURUTANI, IRINA MARKINA

Abstract. LetN be a 2-step nilpotent Lie algebra endowed with a non-degenerate scalar product h. , .i, and let N = V ⊕⊥Z, whereZ is the centre of the Lie algebra and V its orthogonal complement. We study classification of the Lie algebras for which the space V arises as a representation space of the Clifford algebra Cl(R^r,s), and the representation map J: Cl(R^r,s)→End(V) is related to the Lie algebra structure byhJzv, wi=hz,[v, w]ifor all z∈R^r,sandv, w∈V. The classification depends on parametersrandsand is completed for the Clifford modulesV having minimal possible dimension, that are not necessary irreducible.

We find necessary conditions for the existence of a Lie algebra isomorphism according to the range of the integer parameters 0 ≤ r, s < ∞. We present a constructive proof for the isomorphism maps for isomorphic Lie algebras and determine the class of non-isomorphic Lie algebras.

1. Introduction

M´etivier studied in [31] 2-step real nilpotent Lie algebras n=V⊕Z with the centreZ such that the adjoint map ad_x: n→ Z is surjective for any x ∈ V, where V is a complement to the centre. Thus, the bracket defines a vector valued anti-symmetric form [. , .] : V ×V →Z, such that the anti-symmetric real valued bilinear formB(x, y) =ω([x, y]) is non-degenerate on V for all ω ∈ Z^∗\0. In particular, this immediately implies that the space V is even dimensional and [V, V] = Z since Z = adx(n) ⊆ [n,n] for x ∈ V. These Lie algebras were introduced in order to study analytic hypoellipticity and were called the Lie algebrassatisfying the hypothesis H. The Lie algebras were also studied in [14, Definition 1.3] under the name non-singular, in [29, 32] asLie algebras of the M´etivier group or in [26] as fat algebras since they are a source of fat distributions.

Let us observe that if a 2-step nilpotent Lie algebra ncarries a positive definite symmetric bilinear formh. , .i, which we call an inner product, and the mapJ_z:V →V is defined by

hJ_zx, yi=hz,[x, y]i=hz,ad_x(y)i, (1) thenJz is a non-singular linear map for any non-zero z ∈Z, if and only if the Lie algebra n is non-singular. The presence of an inner product is not restrictive at all, because Eberlein showed in [15] that any 2-step nilpotent Lie algebra is isomorphic to a standard metric form (N,h. , .i_N), where N = Rⁿ⊕W, with W ⊂ so(n) and the inner product is defined by h. , .i_N =h. , .iRⁿ+h. , .iso(n). Thus any 2-step nilpotent Lie algebra can be considered as a metric Lie algebra with the inner product defined as above.

We are interested in 2-step nilpotent Lie algebras, for which the map J:Z → End(V) is a representation of the Clifford algebra Cl(R^r,s). The map J and the Lie bracket are related by (1) by making use of an indefinite non-degenerate symmetric bilinear form, which we call scalar product for short. The corresponding Lie algebras, denoted byNr,s(V), were called pseudo H-type Lie algebras in [10]. For the Clifford algebras Cl(R^r,0), generated by

2010 Mathematics Subject Classification. Primary 17B60, 17B30, 17B70, 22E15.

Key words and phrases. Clifford module, nilpotent 2-step Lie algebra, pseudo H-type Lie algebras, Lie algebra isomorphism, scalar product.

The first author was partially supported by the NCTS at National Taiwan University, Taipei, and both of authors were partially supported by ISP project 239033/F20 and SPIRE 710022, University of Bergen, Norway.

1

the Euclidean space R^r, the H-type Lie algebras Nr,0(V) were introduced by Kaplan [23]

and attracted a lot of attention [7, 13, 12, 24, 25, 34, 35]. The Lie algebras Nr,0(V) is a typical example of a standard metric form. The pseudo H-type Lie algebras N_r,s(V), related to representations of the Clifford algebras Cl(R^r,s), were introduced in [10], and studied in [11, 16, 18, 20, 21].

We are interested in the isomorphism properties between the Lie algebras Nr,s(V). We show that the Lie algebrasNr,s(V) can not be isomorphic toNu,t(V) unlessr=tand s=u or r=u ands=t. The present paper is the first part of the complete classification of these Lie algebras. Here we concentrate on the classification of the Lie algebras constructed from the Clifford modulesV of minimal possible dimensions (which are not necessarily irreducible), admitting a non-degenerate symmetric bilinear form making the representation mapJ_z skew symmetric. We also show that the Lie algebras based on non-equivalent irreducible Clifford modules are isomorphic. We stress, that the isomorphic relation between Clifford algebras and the associated pseudo H-type Lie algebras is not functorial: in some cases isomorphic Clifford algebras lead to isomorphic Lie algebras, in other cases not.

Being motivated first of all by an interesting mathematical problem of classification of Lie algebras, we also want to stress here possible applications in other areas of mathematics.

It was shown in [13, 16] that the pseudo H-type Lie algebras admit the integer structure constants that in turn, according to the Mal´cev theorem [30], guarantees the existence of lattices on the corresponding Lie groups. The factorization of pseudo H-type Lie groups by lattices gives many new examples of nilmanifolds, which type strongly depends on the classification of pseudo H-type Lie algebras [8, 9, 11, 21]. Nilmanifolds are related to the Grushin type differential operators descended from elliptic and sub-elliptic type operators on the corresponding pseudo H-type Lie groups. This kind of nilmanifolds allows precise construction of the spectral zeta function for the Grushin operator, [5, 4] and gives new examples of iso-spectral but non-diffeomorphic nilmanifolds [6].

It was noticed recently that Tanaka prolongations of some pseudo H-type Lie algebras coincide with Tanaka prolongations of the negative nilpotent part in 2-grading of simple Lie algebras. This relation was studied for a class of algebraic structures, including pseudo H- type Lie algebras in [1]. It shows a close relation between the classification of pseudoH-type algebras and the theory of simple Lie algebras. The fact that Clifford algebras are pretty much in use in the orthogonal design, signal processing, space-time block coding, or computer vision, is well known [19, 22]. The structure of pseudo H-type Lie algebras allows a new construction of orthogonal designs with possible applications in wireless communications, as was shown in [18].

The article is organized in the following way. After the introduction we give necessary definitions and notation in Section 2, including the notion of admissible modules and relations between Clifford algebras and pseudo H-type Lie algebras. We also describe the scheme of classification, that includes 4 steps. In the rest of sections we realize the first 3 steps of the classification.

Finally, we would like to thank the referees for the attentive reading of our paper, for the suggestion of a new proof of Theorem 12, and for attracting our attention to the results in [27].

2. Clifford algebras, modules, and pseudo H-type Lie algebras

2.1. Clifford algebras and representations. We use the notationR^r,s for the spaceR^r+s equipped with the non-degenerate symmetric bilinear form

hx, yi_r,s= Xi=r i=1

x_iy_i− Xj=s j=1

x_r+jy_r+j, x, y∈R^r,s.

We denote by{z1, . . . , z_r+s}an orthonormal basis for R^r,s. Thus, hz_i, z_ji_r,s=ǫ_i(r, s)δ_i,j with ǫ_i(r, s) =

(1, if i= 1, . . . , r,

−1, if i=r+ 1, . . . , r+s, (2) whereδ_i,j is the Kronecker symbol. By Cl_r,swe denote the Clifford algebra generated by R^r,s, that is, the quotient algebra of the tensor algebra

T(R^r+s) =R⊕ R^r+s

⊕ 2

⊗R^r+s

⊕ 3

⊗R^r+s

⊕ · · · ,

divided by the two-sided ideal I_r,s generated by the elements of the form z⊗z+hz, zi_r,s·1, z∈R^r,s and 1 is the unit of the Clifford algebra. The explicit determination of the Clifford algebras as isomorphic to matrix algebras was given in [2], see also [27, 28]. We mention in formula (3) useful isomorphisms of Clifford algebras, related to 8-periodicity, established in [2]

and (4-4)-periodicity, see [27, 28]. Here the symbol “∼=” stands for an isomorphism.

Cl_r,s⊗Cl_µ,ν ∼= Cl_r+µ,s+ν ∼= Cl_r,s⊗R(16), (µ, ν)∈ {(8,0),(0,8),(4,4)}. (3) An algebra homomorphismJb: Cl_r,s→End(U) is called a representation map and the vector spaceU is said to be the representation space. The representation spaceU becomes a Clifford Cl_r,s-module, where the multiplication is given by φu =Jb_φu, u ∈U, φ ∈ Cl_r,s. To define a representation of the Clifford algebra Clr,sit is enough to find a linear mapJ:R^r,s →End(U) satisfyingJ_z²=− hz, zi_r,sIdU for an arbitraryz∈R^r,s. ThenJ can be uniquely extended to the representation Jbby the universal property, see, for instance [2, 27, 28]. We present the table of the Clifford algebras Cl_r,s, 0≤r, s≤8, for completeness. The symbol F(k) denotes a (k×k) matrix algebra over the fieldF.

Table 1. Representation of Clifford algebras Cl_r,s

8 R(16) C(16) H(16) H(16)⊕H(16) H(32) C(64) R(128) R(128)⊕R(128) R(256)

7 C(8) H(8) H(8)⊕H(8) H(16) C(32) R(64) R(64)⊕R(64) R(128) C(128)

6 H(4) H(4)⊕H(4) H(8) C(16) R(32) R(32)⊕R(32) R(64) C(64) H(64)

5 H(2)⊕H(2) H(4) C(8) R(16) R(16)⊕R(16) R(32) C(32) H(32) H(32)⊕H(32)

4 H(2) C(4) R(8) R(8)⊕R(8) R(16) C(16) H(16) H(16)⊕H(16) H(32)

3 C(2) R(4) R(4)⊕R(4) R(8) C(8) H(8) H(8)⊕H(8) H(16) C(32)

2 R(2) R(2)⊕R(2) R(4) C(4) H(4) H(4)⊕H(4) H(8) C(16) R(32)

1 R⊕R R(2) C(2) H(2) H(2)⊕H(2) H(4) C(8) R(16) R(16)⊕R(16)

s = 0 R C H H⊕H H(2) C(4) R(8) R(8)⊕R(8) R(16)

s/r r = 0 1 2 3 4 5 6 7 8

2.2. Admissible modules. LetU be a Clifford Cl_r,s-module. We call the module U admis- sible following [10], if there is a non-degenerate symmetric bilinear form h. , .i_U on U such that the representation mapJ satisfies the condition:

hJ_zx, yi_U+hx, J_zyi_U = 0 for all z∈R^r,s, x, y∈U.

We say that the map J_z ∈ End(U) is skew symmetric with respect to h. , .i_U. We write U = (U,h. , .i_U) for an admissible module and call the non-degenerate symmetric bilinear

form h. , .i_U the admissible scalar product. If (U,h. , .i_U) is an admissible module, then it decomposes into the orthogonal sum of minimal dimensional admissible modules [16], since the orthogonal complement to an admissible submodule is an admissible module.

If U is a Cl_r,0-module, then there always exists a positive definite scalar product h. , .i_U such thatU becomes an admissible module. Particularly, any irreducible module is admissible with respect to some positive definite scalar product. This fact allowed Kaplan to introduce H-type Lie algebras in [23].

If s > 0, and if (U,h. , .i_U) is an admissible Cl_r,s-module, then the scalar product space (U,h. , .i_U) has to be a neutral space [10], that is an even dimensional space, where the symmetric bilinear form has an equal number of positive and negative eigenvalues. In this case, an irreducible module is not necessarily admissible.

Recall that the Clifford algebras Cl_r,s with r−s 6= 3 mod 4 admit only one irreducible module up to equivalence. Some irreducible modules V can be supplied with an admissible scalar product and become admissible. In other cases the direct sum V ⊕V must be taken in order to define an admissible scalar product, see [10]. In both cases we call the obtained admissible modules minimal admissible modules. Thus, for the Clifford algebras Cl_r,s with r−s 6= 3 mod 4 the minimal admissible module is either (V,h. , .i_V) or (V ⊕V,h. , .i_V⊕V), whereV is the irreducible module. We denote the minimal admissible module of the Clifford algebra Clr,s by V^r,s.

We clarify now the structure of minimal admissible modules for Cl_r,s withr−s= 3 mod 4.

In this case, there are two non-equivalent irreducible modules. Let {z1, . . . , z_r+s} be an orthonormal basis forR^r,s. The product Ω^r,s=Qr+s

j=1z_j is called the volume form. Ifr−s= 3 mod 4, then Ω^r,s belongs to the centre of the Clifford algebra Cl_r,s and (Ω^r,s)² = 1. Two non-equivalent irreducible modules are distinguished by the action of Ω^r,s. We denote byV+

the irreducible module, where the volume form acts as the identity operator and byV₋ the non-equivalent irreducible Cl_r,s-module, where the volume form Ω^r,sacts as minus the identity operator. If none of the irreducible modules is admissible, then the minimal admissible module takes one of the following forms V₊⊕V₊, V₋⊕V₋ or V₊⊕V₋. A possible choice of form depends on the value of the index s and is explained in Proposition 1. The summary of possible structures of the minimal admissible modules is given in Table 2.

Table 2. Structure of possible minimal admissible modulesV^r,s r−s6= 3 mod 4 r−s= 3 mod 4

V^r,s=V or V^r,s=V ⊕V

sis even sis even sis odd V+^r,s=V+ or

V−^r,s=V−

V+^r,s=V+⊕V+ or V−^r,s=V−⊕V−

V^r,s=V+⊕V−

Proposition 1. Let Cl_r,s be a Clifford algebra with r−s= 3 mod 4.

1. If s is odd, then none of the irreducible modules V± is admissible. The minimal admissible module is unique, up to an isomorphism, and has the formV^r,s=V₊⊕V−; 2. If s is even and if the irreducible moduleV+ is admissible, then V− is also admissible and vice versa. Thus, we have two minimal admissible modules: V₊^r,s = V₊ and V₋^r,s=V₋;

3. Ifsis even and if one of irreducible modules is not admissible, then the other one is also not admissible. The minimal admissible module takes one of the forms: V₊^r,s=V₊⊕V+

or V₋^r,s=V−⊕V−.

Proof. To prove the first claim we assume that (V₊,h. , .iV+) is admissible. Then hx, xiV+ =hΩ^r,s(x),Ω^r,s(x)iV+ =

r+sY

i=1

hzi, z_iir,shx, xiV+ = (−1)^shx, xiV+ (4) for any x ∈ V₊. This shows that all the vectors x ∈ V₊ are null vectors and the scalar producth. , .iV+ is identically zero. Thus the irreducible moduleV₊can not be equipped with an admissible scalar product. Similar arguments are valid for V−. Thus if (V^r,s,h. , .i_V^r,s) is a minimal admissible module, then V^r,s has to contain both of irreducible modulesV_±.

The second statement is obvious. For the last statement, we note that if r−s= 3 mod 4, thenr+s= 2s+ 3 mod 4 is odd and ^r+s−1₂ =s+ 1 mod 2 is also odd for even s. We assume now that none of two non-equivalent irreducible modulesV_± is admissible and we consider a minimal admissible module (V^r,s,h. , .i_V^r,s). Then the volume form is an isometry by (4) and a symmetric operator because

hΩ^r,s(x), yiV^r,s= (−1)^r+shx, Jzr+s. . . J_z1yiV^r,s= (−1)^r+s(−1)^{r+s−12 (r+s)}hx,Ω^r,s(y)iV^r,s. Thus, ifV^r,s contains two eigenspaces V+ and V− of Ω^r,s, thenV+ andV− have to be orthog- onal non-degenerate subspaces of (V^r,s,h. , .iV^r,s), and therefore, admissible modules. This contradicts the assumption that none of the irreducible modulesV_± is admissible.

In Table 3 we give the dimensions of minimal admissible modules V^r,s, r, s ≤ 8. By the black colour we denote the dimensions of minimal admissible modules, that are also irreducible Clifford modules. The red colour is used for the minimal admissible modules which are the direct sum of two irreducible Clifford modules. The notation N×2 means that there are two minimal admissible modules.

Table 3. Dimensions of minimal admissible modules 8 16 32 64 64×2 128 128 128 128×2 256 7 16 32 64 64 128 128 128 128 256 6 16 16×2 32 32 64 64×2 128 128 256

5 16 16 16 16 32 64 128 128 256

4 8 8 8 8_×2 16 32 64 64_×2 128

3 8 8 8 8 16 32 64 64 128

2 4 4_×2 8 8 16 16_×2 32 32 64

1 2 4 8 8 16 16 16 16 32

0 1 2 4 4_×2 8 8 8 8_×2 16

s/r 0 1 2 3 4 5 6 7 8

Lemma 1. [16, Lemma 2.9] Let (U,h. , .iU) be an admissible module, and let Λ₁, . . . ,Λ_l be symmetric or anti-symmetric linear operators on U such that

1) Λ²_k =−Id_U, k= 1, . . . , l;

2) Λ_kΛ_j =−ΛjΛ_k for all k, j = 1, . . . , l.

Then for any v∈U withhv, viU = 1 there is a vector v˜satisfying:

h˜v,Λ_k˜viU = 0, and h˜v,˜viU = 1, k= 1, . . . , l.

IfP is a linear operator onU such thatP² = Id,PΛ_k= Λ_kP,k= 1, . . . , l, and ifv∈U with hv, viU = 1, satisfies P v=v, then the vector v˜is also an eigenvector of P: Pv˜= ˜v.

2.3. Pseudo H-type Lie algebras. We give the definition of pseudo H-type Lie algebras that is convenient for the present paper. Equivalent definitions and their relations to Clifford algebras can be found in [10, 12, 13, 16, 20, 23, 24, 25].

Definition 1. Let (U,h. , .i_U) be an admissible module of the representation J: Clr,s → End(U). We call the space U ⊕R^r,s a pseudo H-type Lie algebra Nr,s(U) if the Lie bracket [. , .] : U×U →R^r,s is defined by the relation

hJzx, yi_U =hz,[x, y]i_r,s, z∈R^r,s, x, y∈U.

If U =V^r,s is minimal admissible, then we simply write Nr,s, if there are no confusion.

Definition 1 implies that pseudoH-type Lie algebras Nr,s(U) are 2-step nilpotent andR^r,s is the centre. In Section 6 we prove that the Lie algebraNr,sis unique up to an isomorphism;

that is N_r,s(U) ∼= N_r,s( ˜U) if U and ˜U are minimal admissible modules. One of particular consequences of Definition 1 is the relationhJ_zx, J_z^′xi_U =hz, z^′i_r,shx, xi_U. Thus, the maps J_zj:U → U are isometries ifj = 1, . . . , r, and anti-isometries if j =r+ 1, . . . , r+s for the basis (2). The following result can be found in [13, Theorem 1.2], [15, Corollary, page 37], [16, Theorem 1.3].

Theorem 1. Let us fix an orthonormal basis{zk}^r+s_k=1 for R^r,s as in (2). We assume that the module (V^r,s,h. , .i_Vr,s,) is minimal admissible and has dimension 2N. Then there exists an orthonormal basis{xi}^2N_i=1 for V^r,s such that

1. hx_i, x_ji_Vr,s =ǫ_i(N, N)δ_i,j;

2. For each k, the operator Jzk maps xi to ±xj for some j 6=i;

3. There is a vector v ∈ V^r,s, hv, vi_V^r,s = ±1, such that all the basis {xi} is obtained from v by the actions of J_zj, j= 1, . . . , r+s, or their products.

We call the basis{xi, z_j} forNr,s satisfying the properties of Theorem 1 theintegral basis.

Let (W,h. , .i_W) be a vector space with a scalar product. We say that a vector w ∈ W is positive if hw, wi_W > 0, negative if hw, wi_W < 0, and a null-vector, if hw, wi_W = 0. We formulate some consequences of Theorem 1.

Corollary 1. If there exists an index i∈ {1, . . . ,2N} such that J_zkx_i =±Jzlx_i, then k=l.

Hence any basis vector x_i is mapped tox_j or −x_j by at most one J_zk.

Proof. If k ≤ r, then Jzk preserves positive and negative elements. If k > r, then Jzk

interchange the positive and negative elements. Therefore, under the assumption of the corollary, only the casesk, l≤r or k, l > rare possible. Assume k6=l. Then, from one hand

±x_i=J_zkJ_zlx_i, but from the other hand

(J_zkJ_zl)²=−J_z²_kJ_z²_l =− hz_k, z_ki_r,shz_l, z_li_r,sId =−Id,

which contradicts the existence of the eigenvalue 1 or−1 of the operatorJ_zkJ_zl. Corollary 2. Let N_r,s be a pseudo H-type Lie algebra and let {x_i, z_k} be an integral basis.

Set [x_i, x_j] =P

c^k_ijz_k, then the coefficients c^k_ij for fixed i and j vanish, for all but one k and in the latter case c^k_ij =±1.

Proof. The proof follows from hJ_zkx_i, x_ji_Vr,s=hz_k,[x_i, x_j]i_r,s=

(c^k_ij ifk≤r

−c^k_ij ifk > r. Corollary 3. Let Nr,s be a pseudo H-type Lie algebra, let {xi, z_k} be an integral basis, and let[x_i, x_j] =±zk, i, j= 1, . . . ,2N, k= 1, . . . , r+s.

1. If either 1≤i, j≤N or N < i, j ≤2N, then z_k is positive, that isk≤r.

2. If 1≤i≤N < j ≤2N, then z_k is negative, i.e., k > r.

Proof. We prove only the first statement, since the second one can be shown similarly. If we assume, on the contrary, thatk > r, thenJ_zk should be an anti-isometry and

0 =hJ_zkx_i, x_ji_Vr,s=hz_k,[x_i, x_j]i_r,s=±1,

which is a contradiction.

2.4. Scheme for the classification. Step 1. We study the isomorphic Lie algebras Nr,s

and Ns,r, r, s ≤ 8, r 6= s of equal dimensions. If dim(V^r,s) = 2 dim(V^s,r), then the Lie algebrasN_r,s and N_s,r are not isomorphic simply because they have different dimension. We call these algebras trivially non-isomorphic. We show that the Lie algebras Nr,s(V^r,s) and N_s,r(V^s,r⊕V^s,r) are isomorphic, whereV^r,sandV^s,r are the minimal admissible modules. We also construct an automorphism of N_r,r, r = 1,2,4 possessing some special property. Then the periodicity (3) is applied to extend these results to the Lie algebras Nr,s withr, s >8.

Step 2. We find non-isomorphic Lie algebras and show that there is no an automorphism ofNr,r,r = 3 mod 4 possessing the same property asNr,r,r= 1,2,4 mod 4.

Step 3. LetV₊^r,s=V₊or V₋^r,s=V−, whereV₊, V− are non-equivalent irreducible modules andr−s= 3 mod 4. We show that the Lie algebrasNr,s(V₊^r,s) andNr,s(V₋^r,s) are isomorphic.

An analogous question is considered whenV₊^r,s =V₊⊕V₊ or V₋^r,s = V−⊕V−. This result, in particular, shows the uniqueness of the pseudoH-type Lie algebras corresponding to two minimal admissible modulesV_±^r,s, see Section 6.

Step 4. The last step is devoted to the classification of Lie algebras constructed from the multiple sum of minimal admissible modules and will be given in the forthcoming paper [17], see also Section 2.5. The “classical” H-type Lie algebras were studied in [7].

In the present paper we complete the first 3 steps, finishing the classification of the Lie alge- bras whose complement to the centre is a minimal admissible Clifford module. We summarize the classification of the Step 1 among the basic pairs in Table 4. Here “d” stands for “double”,

Table 4. Classification result after the second step 8 ∼= ∼= ∼= h

7 d d d 6∼= 6 d ∼= ∼= h 5 d ∼= ∼= h

4 ∼= h h h

3 d 6∼= 6∼= 6 d d d 6∼= d 2 ∼= h 6∼= d ∼= ∼= h ∼= 1 ∼= d 6∼= d ∼= ∼= h ∼= 0 ∼= ∼= h ∼= h h h ∼=

s/r 0 1 2 3 4 5 6 7 8

meaning that dimV^r,s= 2 dimV^s,r, and “h” (half) means that dimV^r,s= ¹₂dimV^s,r (r > s).

The symbol∼= on the position (r, s) indicates thatNr,s∼=Ns,r, and6∼= shows thatNr,s6∼=Ns,r. The symbol denotes the Lie algebra Nr,r admitting a special type of automorphisms, and

6

denotes the Lie algebra which does not have this type of automorphism.

2.5. Remarks on Step 4 and further development. In the forthcoming paper [17] we will consider arbitrary sums U = ⊕_iV_i^r,s of minimal admissible modules V_i^r,s = (V,h. , .i_V) and plan to complete the last step of the classification. The different minimal admissible modulesV_i^r,s can have a common vector space V but scalar products of opposite signs. The minimal admissible modules can differ also by the choice of the irreducible modules for their construction: V^r,s=V₊ or V^r,s=V₋.

We also aim at studying the automorphism groups of the pseudo H-type Lie algebras Nr,s(U), whereU is not necessary a minimal admissible module. The results on automorphism groups forH-type Lie algebras N_r,0(U) can be found in [3, 33, 36]. The authomorphisms are determined by solving the equations arising during the construction of the mapA:U → U. The present paper indicates that the first part of this problem is to determine the sequence

{0} →K →Aut(Nr,s(U))→O(r, s)→ {0}.

The map Aut(Nr,s(U)) → O(r, s) is defined in (5), and this map is distinguished by the properties of C. In some cases C^τC = Id, as for instance, for Aut(Nr,r(U)), r = 3 mod 4, meanwhile for Aut(Nr,r(U)), r = 1,2,4 mod 4 one has C^τC =±Id. The mapC determines the mapA and the freedom in the construction ofA gives the kernel K. In the forthcoming papers we aim to classify all Lie algebrasNr,s(U) and describe their automorphism groups.

3. Lie algebras of minimal dimensions

3.1. Necessary condition for the existence of an isomorphism. Let Λ : V → Ve be a linear map between the spaces with non-degenerate symmetric bilinear forms h. , .i_V and h. , .i_Ve, respectively. We denote by Λ^τ the transposed map:

hΛx, yi_Ve =hx,Λ^τyi_V, x∈V, y∈V .e

Theorem 2. Let{U,h. , .i_U;J}and{U ,e h. , .i_Ue; ˜J}be admissible modules with representation mapsJ andJ˜of the Clifford algebrasCl_r,s andCl_r,˜˜s, respectively. Assume that r+s= ˜r+ ˜s, dimU = dimUe, and that there is a Lie algebra isomorphism Φ : Nr,s(U) → Nr,˜˜s(Ue). Then, necessarily, one of the cases (r, s) = (˜r,s)˜ or (r, s) = (˜s,r)˜ holds. Moreover, Φ has to be of the form

Φ =

A 0 B C

:

U

⊕⊥

R^r,s

−→

Ue

⊕⊥

R^˜r,˜s

, (5)

where A:U →Ue and C: R^r,s→R^r,˜˜s are linear bijective maps satisfying the relation

A^τJ˜_wA=J_C^τ_(w) for any w∈R^˜r,˜s. (6) There is no condition on B:U →R^r,˜˜s and we may set B = 0. In addition, multiplying A by a suitable constant, we may assume that |det (AA^τ)|= 1 and CC^τ =±Id.

Proof. If there exists a Lie algebra isomorphism Φ : Nr,s(U) → N˜r,˜s(Ue), then it must be of the form (5), since it maps the centre to the centre. Relation (6) follows from the definition of Lie bracket

hA^τJ˜_wA(x), yi_U = hJ˜_wA(x), A(y)i_Ue =hw,[A(x), A(y)]i_˜r,˜s=hw, C([x, y])i_r,˜˜s

= hC^τ(w),[x, y]i_r,s=hJ_C^τ_(w)x, yi_U, (7) for allx, y∈U and w∈R^r,˜˜s. Conversely, if (6) holds, then from (7) we obtain [A(x), A(y)] = C([x, y]), and therefore, the map Φ = A⊕C is a Lie algebra isomorphism. Note that (6) implies that J_C^τ_(w) is singular, if and only if ˜J_w is singular.

Let z₊ and z₋ be a positive and a negative vector in R^r,s, respectively. We set a_t = (1−t)z++tz−, 0≤t≤1. Then

ha₀, a₀i_r,s=hz₊, z₊i_r,s>0 and ha₁, a₁i_r,s =hz−, z−i_r,s<0.

Hence, there ist₀ ∈(0,1) withha_t0, a_t0i_r,s= 0, andJ_at0 is singular.

On the other hand, assume that z₁ and z₂ are orthonormal and both positive (or negative) vectors in R^r,s and put b_t = (1−t)z₁+tz₂, 0 ≤ t ≤ 1. Then hb_t, b_ti_r,s = (1−t)²+t² >

0 (or < 0) for all t ∈ [0,1]. This implies that J_bt is non-singular for all t ∈ [0,1]. Hence, the operatorC (and alsoC^τ) either preserves or reverses the sign of elements in R^r,s. These observations prove that only the cases (r, s) = (˜r,s) or (r, s) = (˜˜ s,r) are possible.˜

For the remaining part of the proof we assume that r 6= s and Φ : Nr,s(U) → Ns,r(Ue) is a Lie algebra isomorphism. Then (A^τJ˜_zA)² = J_C²τ(z) = − hC^τ(z), C^τ(z)i_r,sId_U by (6), and therefore,

det (A^τJ˜zA)²

= (detAA^τ)²hz, zi^2N_s,r =hC^τ(z), C^τ(z)i^2N_r,s,

where 2N = dimU. Since the operatorC^τ:R^s,r →R^r,sreverses the sign of vectors we obtain

|(detAA^τ)|^1/N · hz, zi_s,r =− hC^τ(z), C^τ(z)i_r,s=− hz, CC^τ(z)i_s,r.

MultiplyingAby a suitable constant we assume that |detAA^τ|= 1 and CC^τ =−Id.

We write Φ =A⊕C for isomorphisms of pseudoH-type Lie algebras.

Corollary 4. Letr6=sand letΦ =A⊕C:Nr,s(U)→ Nr,s( ˆU)be a Lie algebra isomorphism.

ThenCC^τ = Id. Ifr =sboth casesCC^τ =±Idare possible, see Theorem 5 and Corollary 6.

Lemma 2. Let {U,h. , .i_U;J} and {U ,ˆ h. , .iUˆ; ˆJ} be admissible modules of Cl_r,s with the representation maps J and Jˆ. Let also U ,˜ h. , .iU˜; ˜J} be an admissible module and the repre- sentation map of Cl_s,r. If

Ψ = ¯A⊕C¯:N_r,s(U)→ N_r,s( ˆU) and Φ =A⊕C:N_r,s(U)→ N_s,r( ˜U), are Lie algebra isomorphisms, then the maps defined by

Ψ^τ = ¯A^τ ⊕C¯^τ:Nr,s( ˆU)→ Nr,s(U) and Φ^τ =A^τ ⊕C^τ:Ns,r( ˜U)→ Nr,s(U), are Lie algebra isomorphisms as well.

Proof. We have ( ¯A^τJˆ_zA)¯ ² = − hC¯^τ(z),C¯^τ(z)i_r,sId = ± hz, zi_r,sId, according to (6) and Corollary 4. This implies that ¯AA¯^τJˆ_zA¯A¯^τ =±Jˆ_z. Hence ¯AA¯^τJˆ_zA¯A¯^τ = ¯AJC¯^τ(z)A¯^τ = ±Jˆ_z. We put ¯C^τ(z) =w and obtain ¯AJ_wA¯^τ = ˆJC(w)¯ , for any case of ¯C^τC¯ =±Id. Hence the map Ψ^τ = ¯A^τ⊕C¯^τ:Nr,s(U)→ Nr,s( ˆU) is a Lie algebra isomorphism.

The statement for Φ and Φ^τ is proved by a similar way.

The structure of a Lie algebra isomorphism inheritsZ₂-grading property of the underlying Clifford algebras as shows the following lemma.

Lemma 3. Let {z_i}^r+s_i=1 be an orthonormal basis for R^r,s as in (2) and let Φ =A⊕C be as in Lemma 2. Then the following relations hold:

1. If p= 2m, m∈N, then A

Yp j=1

Jzj = (−1)^m Yp j=1

Je_C(zj)A, A^τ Yp j=1

Je_zj = (−1)^m Yp j=1

J_C^τ_(zj)A^τ. (8)

A^τA Yp j=1

Jzj = Yp j=1

JzjA^τA, AA^τ Yp j=1

e

J_C(zj)= Yp j=1

e

J_C(zj)AA^τ. (9) 2. If p= 2m+ 1, m∈N∪ {0}, then

A Yp j=1

J_zjA^τ = (−1)^m Yp j=1

e

J_C(zj) A^τ Yp j=1

e

J_zjA= (−1)^m Yp j=1

J_C^τ_(zj). (10)

A^τA Yp j=1

J_zjA^τA=− Yp j=1

J_zj, AA^τ Yp j=1

e

J_zjAA^τ =− Yp j=1

e

J_zj (11)

Proof. We show only the second parts of the equalities, since the first parts can be obtained from them by transposition. We assume that C^τC = −Id and apply induction. If m = 0 (p= 1) then (10) is reduced to (6). Assume now that (10) holds for p= 2m+ 1. Choosez^∗ from the orthonormal basis{zi}^r+s_i=1 and calculate

A^τ Yp j=1

Je_zj

Je_z^∗ =A^τ Yp j=1

Je_zjAA⁻¹Je_z^∗(A^τ)⁻¹A^τ = (−1)^m+1Y^p

j=1

J_C^τ_(zj)

J_C^τ_(z^∗₎A^τ.

Thus, we proved (8) forp= 2(m+ 1). The last equality follows from

A^τJe⁻¹_z∗ A=− hz^∗, z^∗i_s,rA^τJe_z^∗A=hC^τ(z^∗), C^τ(z^∗)i_r,sJ_C^τ_(z^∗₎.

Hence, in the previous step we particularly showed that (8) is true form= 1 (p= 2). Assume now that (8) holds forp= 2m,m= 0,1, . . ., then

A^τY^p

j=1

Je_zj

Je_z^∗A= (−1)^m Yp j=1

J_C^τ_(zj)A^τJe_z^∗A= (−1)^mY^p

j=1

J_C^τ_(zj)

J_C^τ_(Z^∗₎. Thus, assertion (10) holds forp= 2m+ 1.

It is sufficient to show (9) for p = 2. We have A^τAJ_z1J_z2 = −A^τeJ_C(z1)Je_C(z2)A = J_C^τ_C(z1)J_C^τ_C(z2)A^τA=J_z1J_z2A^τA by (8). Identity (11) is deduced from (10). We obtain

A^τA Yp j=1

J_zjA^τA= (−1)^mA^τ Yp j=1

Je_C(zj)A= (−1)^2m Yp j=1

J_C^τ_C(zj) = (−1)^p Yp j=1

J_zj.

and sincep is odd equality (11) follows.

Remark 1. We emphasize that the existence of an isomorphism Φ = A⊕C: N_r,s(U) → Ns,r(Ue) is equivalent to the requirement that relation (6) holds. Moreover, (6) implies all the properties listed in Lemmas 2 and 3. The relation of type (6) and analogous to those in Lemma 3 were used in[33, 36] for the study of automorphisms ofH-type algebras N_r,0(U).

3.2. Observations on general structure of an isomorphism. Let (V^r,s, J) and (V^s,r,J˜) be minimal admissible modules of Cl_r,s and Cl_s,r, respectively. Let P:V^r,s → V^r,s be an involution; that is a linear map possessing the property P² = Id_V^r,s. We denote by E_P^k, k ∈ {1,−1} the eigenspace of the involution P with the eigenvalue k = ±1. In order to denote the intersection of eigenspaces of several involutions P_j, j = 1, . . . , N, we use the multi-indexI = (k₁, . . . , k_N), k_j =±1 and writeE^I =∩^N_j=1E_P^kj

j. We denote by{z₁, . . . , z_r+s} the orthonormal basis forR^r,s as in (2), and by{ws+r, . . . , w_r+1, w_r, . . . , w₁}the orthonormal basis for R^s,r with the firsts elements being positive and the last r vectors being negative.

Thus, the representation maps ˜J: Cls,r →End(V^s,r) satisfy

J˜_w²_j =−Id_V^s,r, j=s+r, . . . , r+ 1, J˜_w²_j = Id_V^s,r, j=r, . . . ,1.

In general, operators and other objects related to the Clifford algebra Cl_r,s will be denoted by letters P, E, R, . . ., meanwhile the operators, associated to the Clifford algebra Cl_s,r will carry the tilde on the top: Pe,Ee,R, . . .. We formulate an immediate corollary of Lemma 3e that will be used frequently in the paper.

Corollary 5. Let r 6= s and assume that there is a Lie algebra isomorphism Φ = A ⊕ C: N^r,s → N^s,r with A: V^r,s → V^s,r, C: R^r,s → R^s,r, where we set C(z_j) = w_j, which implies C^τ(w_j) = −z_j. Let P_l, l = 1, . . . , N be mutually commuting isometric involutions on V^r,s obtained as a product of the representation maps J_zj. Let Pe_l be mutually commuting isometric involutions on V^s,r obtained from P_l by changing J_zj to Je_wj = Je_C(zj) for each

j= 1, . . . , r+s, and such that AP_ℓ =Pe_ℓA, ℓ= 1, . . . , N. We denote by E^I the eigenspaces of P_ℓ, ℓ= 1, . . . , N, and by Ee^I the eigenspaces of Pe_ℓ,ℓ= 1, . . . , N, respectively. Then 1. A=⊕AI, where A_I:E^I →Ee^I for any choice of I = (k₁, . . . , k_N);

2. if Qp j=1

J_zj:E^I →E^I for some I, then Qp j=1

Je_wj: Ee^I →Ee^I, and

A_I Yp j=1

J_zj =







 (−1)^m

Qp j=1

Je_wj(A^τ_I)⁻¹(x_I), if p= 2m+ 1, (−1)^m

Qp j=1

Je_wjA_I(x_I), if p= 2m,

x_I ∈E_I,

A^τ_I Yp j=1

Je_wj =









(−1)^m+1 Qp j=1

J_zj(A_I)⁻¹(y_I), if p= 2m,+1 (−1)^m

Qp j=1

J_zjA^τ_I(y_I), if p= 2m,

y_I ∈Ee_I.

Corollary 5 gives an idea for a possible construction of an isomorphism Φ =A⊕C:N^r,s→ N^s,r. Choosing the bases{zj}^r+s_j=1 forR^r,s and{wj}^s+r_j=1 forR^s,r we define the mapC:R^r,s→ R^s,r, by setting C(z_j) = w_j and C^τ(w_j) = −zj. Further, if we find a maximal collection of mutually commuting isometric involutions P_l and Pe_l,l = 1, . . . , N, acting on V^r,s and V^s,r, respectively, we can reduce the construction of the map A:V^r,s → V^s,r to the construction of the maps A^I:E^I → Ee^I. Finally, we set A = ⊕AI. Theorem 3 states that, under some conditions, the construction of all maps A_I can be obtained from the map A₁:E¹ → Ee¹, where we denoteE¹=TN

l=1E_P¹

l.

Theorem 3. Let us suppose that conditions of Corollary 5 are satisfied. We set E¹ = TN

l=1E_P¹

l and Ee¹=TN

l=1Ee¹_Pel. We assume also that

(a)there are maps G_I:E¹ →E^I for all multi-indicesI, written in the form G_I =Q

J_zi, and (b) there exists a map A₁:E¹ →Ee¹ such that

A₁ Yp j=1

J_zj =







 (−1)^m

Qp j=1

eJ_C(zj)(A^τ₁)⁻¹, if p= 2m+ 1, (−1)^m

Qp j=1

eJ_C(zj)A₁, if p= 2m, for any choice of the product Qp

j=1Jzj that leaves the space E¹ invariant.

Then there is a map A:V^r,s→V^s,r such thatΦ =A⊕C:N^r,s→ N^s,r is an isomorphism.

Proof. We define the maps A_I:E^I →Ee^I by

A_I =











(−1)^mGe_I(A⁻¹₁ )^τG⁻¹_I , if GI =

p=2m+1Q

j=1

Jzj, Ge_I =

p=2m+1Q

j=1

Je_wj, (−1)^mGe_IA₁G⁻¹_I , if G_I =

p=2mQ

j=1

J_zj, Ge_I =

p=2mQ

j=1

e J_wj.

(12)

Here and furtherJe_wk =eJ_C(zk). For the convenience we also write the transposed maps:

A^τ_I =











(−1)^m+1G_IA⁻¹₁ Ge⁻¹_I , if G_I =

p=2m+1Q

j=1

J_zj, Ge_I =

p=2m+1Q

j=1

Je_wj, (−1)^mG_IA^τ₁Ge⁻¹_I , if G_I =

p=2mQ

j=1

J_zj, Ge_I =

p=2mQ

j=1

e J_wj.

(13)