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Department of Mathematics, Royal Institute of Technology (KTH), Lindstedts v¨ag 25, S-100 44 Stockholm, Sweden

- *Corresponding Author:
*Daniel MONDOC*

Department of Mathematics Royal

Institute of Technology (KTH) Lindstedts v¨ag 25

S-100 44 Stockholm, Sweden

**E-mail:**[email protected]

**Received date:** November 27, 2006

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

Let A be the realification of the matrix algebra determined by Jordan algebra of hermitian matrices of order three over a complex composition algebra. We define an involutive automorphism on A with a certain action on the triple system obtained from A which give models of simple compact Kantor triple systems. In addition, we give an explicit formula for the canonical trace form and the classification for these triples and their corresponding exceptional real simple Lie algebras. Moreover, we present all realifications of complex exceptional simple Lie algebras as Kantor algebras for a compact simple Kantor triple system defined on a structurable algebra of skew-dimension one

Models of Kantor triple systems defined on the 2 × 2-matrix algebra determined by the Jordan algebra of hermitian 3 × 3-matrices over complex composition algebras considered over the field of complex numbers appeared in a unified formula given by I. L. Kantor [17,18] in connection with exceptional Lie algebras and a classification theorem over .

The notion of (simple) structurable algebras was given by B. N. Allison [1] who studied in particular those of skew-dimension one [3]. Moreover, the connection between Kantor triple systems and structurable algebras was studied by H. Asano and S. Kaneyuki [7] who also defined and studied [5,15] compact Kantor triple systems in connection with classical real Lie algebras and a classification theorem over the field

In this paper we continue the work on compact simple Kantor triple systems of [5] and [20,21,22] giving, by a unified formula (Theorem 1), the classification of exceptional compact simple Kantor triple systems defined on the realification of the 2×2-matrix algebra determined by Jordan algebra of hermitian 3×3-matrices over a complex composition algebra corresponding to realifications of complex exceptional simple Lie algebras (Theorem 2). In addition, we give an explicit formula for the quadratic canonical trace form for these Kantor triple systems (Corollary 1). Further, we present all realifications of complex exceptional simple Lie algebras as Kantor algebras for a compact simple Kantor triple system defined on a structurable algebra of skew-dimension one (Theorem 2, Proposition 5).

The results presented here are a continuation of [20] where models of exceptional compact
simple Kantor triple systems defined on the 2×2-matrix algebra determined by Jordan algebra of hermitian 3 × 3-matrices over a real composition algebra have been given.
Related results are those of [10] where a construction of exceptional simple 5-graded Lie algebras and an explicit realization of the subspaces *U _{l}* have been given by different methods.
Moreover, the notion of Kantor triple systems and their structure theory have been generalized
by (∈, δ)-Freudenthal-Kantor triple systems [13,24] such that Kantor triple systems coincide
with (−1, 1)-Freudenthal-Kantor triple systems. A realization of exceptional simple 5-graded
Lie algebras in terms of Freudenthal-Kantor triple systems have been given by N. Kamiya [14].

The models of compact simple Kantor triple systems considered here start with a structurable
algebra A, its associated Kantor triple system and an involutive
automorphism . Then the new triple product is considered, which gives
again a Kantor triple system. Suitable elections of *A* and where − is the
standard involution and ˜denotes a certain involution on A) give compact simple models.

The structure of this paper is as follows. Section 2 serves a preliminary purpose; we give a short overview of the basic definitions and known results on triple systems, graded Lie algebras and structurable algebras. The main results mentioned above, on the corresponding canonical trace form and the corresponding exceptional real simple Lie algebras are proved in section 3.

Let U be a Lie algebra over a field F of characteristic zero. U is called a graded Lie algebra (abbreviated as GLA) if it is a Lie algebra of the form such that

A GLA is called 5-graded if U_{±n} = 0 for any integer n > 2.

Let U be a finite dimensional vector space over the field F and B : U × U × U ! U be a trilinear map. The pair (B,U) is called a triple system over F.

For x, y 2 U define the linear endomorphisms L_{x,y},R_{x,y} and S_{x,y} on U by

(2.1a)

(2.1b)

A triple system (B,U) is called a generalized Jordan triple system (abbreviated as GJTS) if the following identity is valid [7] (§1):

(2.2)

Let (B,U) and be two GJTS’s. We say that a linear map F of U into is a homomorphism if F satisfies the identity , for all Moreover, if F is bijective, then F is called an isomorphism. In this case the GJTS’s (B,U) and are said to be isomorphic.

Let (B,U) be a GJTS and *V _{k}, k* = 1, 2, 3, be subspaces of U. We denote by B(V

Starting from a given GJTS (B,U), I. L. Kantor [17] constructed a certain GLA L(B) = such that U−1 = U. The Lie algebra L(B) is called the Kantor algebra for (B,U) [5]. A GJTS (B,U) is called of the n-th order if its Kantor algebra is of the form We shall call a GJTS of the second order for short a Kantor triple system [4] (abbreviated as KTS). By [17] Proposition 10, a GJTS (B,U) is a KTS if and only if

(2.3)

**Remark.** Many authors [4,18] define a KTS to be a triple system (B,U) satisfying the identities
(2.2), (2.3) instead of the identity (2.2) together with the fact that its Kantor algebra is 5-graded.
The definitions are equivalent.

A GJTS is called exceptional (classical) if its Kantor algebra is exceptional (classical) Lie algebra.

For we define a bilinear map Bz on U by We say that
(B,U) satisfies the condition (A) if B_{z} = 0 implies z = 0.

Let (B,U) be a finite dimensional KTS. We consider the symmetric bilinear form on U [5]

(2.4)

where Tr(f) means the trace of a linear endomorphism f. We shall call the form γB defined by (2.4) the canonical (trace) form for the KTS (B,U).

Let A be an algebra over F. Let left (right) multiplication be
defined by and denote by *Hom*F(A) the associative algebra
over F of all linear transformations on A. If A is finite dimensional we denote by dimFA the
dimension of A over F. For any extension field K of F we denote

**Proposition 1** ([11]). Let A be a finite dimensional algebra over an algebraically closed field Τ
and let Φ be a subfield of Τ. If A is simple over Τ then A is simple as algebra considered over F.

**Remark. **A direct proof of Proposition 1 is available in [22].

Let (A,^{−} ) be a unital non-associative algebra over F with involution (involutive anti-automorphism)
^{−}. We define and the triple system by

(2.5a)

(2.5b)

is called the triple system obtained from the algebra (A,^{−} ) [7] (§2). We shall write for short B_{A} for (B_{A},A).

A unital non-associative algebra with involution (A,^{−} ) is called a structurable algebra if the
following identity is fulfilled [3]:

Let (A,^{−} ) be a structurable algebra. Then, by [3], , where

are the spaces of skew-hermitian and hermitian elements of A, respectively and dimS is called
the skew-dimension of (A,^{−} ).

To a structurable algebra (A,^{−} ) Allison [2] associated a 5-GLA K(A) as follows

(2.6a)

(2.6b)

is an isomorphic copy of K_{−l} (2.6c)

By [7] Theorem 2.5, Allison’s 5-GLA K(A) coincides with Kantor’s 5-GLA L(*B _{A}*), where

Let J be a finite dimensional separable degree 3 Jordan algebra over F. Let N, T and # be
the norm form, trace form and adjoint map on J respectively [12] (§6.3). Define × : J × J → J
by the algebra *M*(J) with multiplication and standard involution
− defined [3] (§1) by

(2.7a)

(2.7b)

(2.7c)

is called the 2 × 2-*matrix algebra determined by Jordan algebra* J.

Let denote the real algebras of real and complex numbers, quaternions and octonions, respectively. They are called division composition algebras and are defined by their explicit multiplication tables [23].

Let be any of the division composition algebras and let where 8}, denote the standard units of . We define conjugation − on a standard unit u of by

(2.8)

and extend conjugation − by linearity on .

**Remark. **By [23] §3, it is known that − is an involution on any composition algebra above.

Let now be any of the complex composition algebras i.e. the division composition algebras regarded as algebras over We define conjugation − and scalar extended conjugation ^, called pseudoconjugation, on the complex algebra by

(2.9)

(2.10)

where lul is an arbitrary element of , ul are the standard units and ul is defined
by (2.8) hence is the standard complex conjugate of α_{1}.

**Remark.** Then clearly − and ^ are involutions of the complex algebra .

Let be each one of the division composition algebras

Let denote any of the real or complex composition algebras and let denote the set of matrices of order 3 with entries in A. Then, by definition, the conjugation − on the algebra and the conjugation − and pseudoconjugation ^ on the algebra are induced on by − and ^, respectively, on each entry.

**Remark. **Clearly − and ^, respectively, are involutive on

Let denote the Jordan algebra of hermitian 3 × 3-matrices over a composition algebra [8] (§6) with the product

where in the right hand side we have usual matrix multiplication and denotes the conjugate
transposed of Then, by [9] (p. 218), the trace form and ×-operation in formulas
(2.7) are defined on H_{3}(A) by

(2.11a)

(2.11b)

for all where I_{3} is the unit matrix of order 3 and Tr(x) denotes the trace of x.

**Lemma 1** ([3]). Let be the 2 × 2-matrix algebra determined by the
Jordan algebra of hermitian 3 × 3-matrices over a complex composition algebra ∈ where (−) is the standard involution on A. Then, over he algebras (A,^{−} )
are simple structurable of skew-dimension 1.

**Proof.** The assertion follows directly from [3] (§1 Proposition 1.10).

Let now A be a K-algebra for any extension field of K of F. We denote by A_{F} the algebra
A considered as algebra over F. If A is an algebra over then we call the algebra the
realification of the complex algebra A. Further, if BA is the triple system obtained from a
complex algebra (A,^{−} ) then we call the triple system BAR obtained from algebra (,^{−} ) the
realification of B_{A}.

**Proposition 2 **([22] Proposition 1.4). Let (A,− ) be a structurable algebra over . Then, over
, the triple system is a KTS satisfying the condition (A), and is simple if and only
if (,− ) is simple.

We show first a property of the trace form on the Jordan algebra

Let εlm denote in the square matrix with entry 1 where the l-th row and the m-th column meet, all other entries being 0, and denote

(3.1a)

(3.1b)

(3.1c)

Let denote any of the complex composition algebras From now on an arbitrary element of the Jordan algebra is of the form

(3.2)

**Lemma 2. **Let x be an arbitrary element in of the form (3.2) and let the trace form
be defined by (2.11). Let − and ^ be the conjugation and pseudoconjugation
defined on by (2.9) and (2.10), respectively. Then

where denotes the norm of

**Proof. **Let be an arbitrary element in of the form (3.2), where ∈ Let We write for short Then by (2.9), and by (2.10), where − is defined by (2.8) so is the standard complex conjugate of c_{lm}. Hence and by (3.2) we have

(3.3)

and

(3.4)

Then, by (2.11), (3.3) and (3.4), straightforward calculations give

where ||x_{lm}|| denotes the norm of xlm, 1 ≤ l,m ≤ 3.

**The exceptional simple Lie algebras **

We define now models of compact simple Kantor triple systems.

Let denote any of the complex composition algebras

Let be the 2×2-matrix algebra determined by the Jordan algebra defined by (2.7) with standard involution −. Let us define a second involution by

(3.5)

where denotes the pseudo-conjugate of x_{i} conjugate, is the standard conjugate
of Hence the following involutive automorphism is defined on

Let be the realification of the algebra and let denote the triple system defined by formula

(3.7)

We prove now the main results in the Theorems 1, 2 and Corollary 1.

**Proposition 3.** The triple systems defined by (3.5), (3.7) are KTS’s satisfying
the condition (A).

**Proof.** From Lemma 1 and Propositions 1, 2 follows that the triple systems are (simple) KTS’s satisfying the condition (A). Further, we remark that by (3.7) and (2.5). Then the assertions follow from [6] Lemma
1.5 since the map is an involutive automorphism on the algebra

**Theorem 1. **The KTS’s defined by (3.5), (3.7) on the realification of the
2 × 2-matrix algebra determined by Jordan algebra of hermitian 3 × 3-matrices over are compact and simple.

**Proof. **We prove first compactness. We must show that the canonical (trace) form
defined by
(2.4) for the KTS’s is positive definite. Since the canonical form is symmetric
let us consider the corresponding quadratic form which, by (2.4), is equal to

. (3.8)

We remark first that

(3.9)

such that, by (3.7) and (2.1),

(3.10)

where in the last equality we have used that − is an involution on

(3.12)

where O_{3} denotes the zero matrix of order 3, the identity y − y =
(η1 − η2)s_{0} follows from (2.7). Hence

by (2.7), and then the identity

follows from (3.11). Then, by (2.7), (3.5) and (3.12), we have

(3.13)

Recall that for any linear map f : U → F, U a vector space over the field F, yields Tr(f(·)v) = f(v) and let f_{x} : → be the linear map

Then by (3.13) follows g(x, x) = fx(·)v with

(3.14)

for all where the factor 2 in (3.14) follows from the fact that the trace is calculated over the realification

We calculate now For this, we remark that by (3.10) and the fact that − and ˜ are involutive on follows

(3.15)

Further, we calculate . By (2.7), (3.5) and (3.12) we have

Let us denote the units of by

(3.16a)

(3.16b)

(3.16c)

(3.16d)

(3.16e)

where is the basis of such that el , fl, gl are defined by (3.1), i denotes the complex unit and

Let denote the coefficient of a generic unit μ of Then, by (3.16), straightforward calculations give

(3.17)

for all

We show now that for all where h_{x} is defined by (3.15).

We remark first that where is an involutive automorphism on Then

for allTherefore so are similar and hence they have the same trace.

Finally, by (3.18), (3.15), (3.14), (3.9) and the last line follows

(3.19)

for all Then, by (3.19) and Lemma 2, is positive definite for all

We prove now simplicity.

Since the KTS’s are are isomorphic as GLA’s, by [6] Proposition 1.6.compact then they are simple if and only if the corresponding Kantor algebras L((x, y, z)) are simple,by [5] Theorem 3.7. Moreover, since are KTS’s satisfying the condition (A) then the algebras and are isomorphic as GLA’s, by [6] Proposition 1.6. But the Kantor algebras are simple if and only if the structurable algebras are simple, by [2] Corollary 6 and [7] Theorem 25. Then the simplicity assertion follows from Lemma 1.

**Corollary 1.** Let be the compact KTS’s defined by (3.5), (3.7), where . Then the canonical quadratic form has the form

**Proof.** The assertion follows from (3.19) since clearly

**Remark. **By similarity to [22] §2, define triple systems

where ˜ is the involution on defined by formula (3.5). Then the triple systems are simple compact KTS’s, since it can be easily proved that and are isomorphic under the map

We give the classification theorem. Let Lie algebras be denoted as in [16].

Let denote any of the complex composition algebras be the 2 × 2-matrix algebra determined by the Jordan algebra defined by (2.7) with the involutions − and ˜ defined by (3.5).

**Theorem 2. **All compact realifications of exceptional simple KTS’s defined on the 2 × 2-
matrix algebra determined by the Jordan algebra are the KTS’s defined by (3.7) and the corresponding Kantor algebras are the following
realifications of complex simple Lie algebras

Proof. By [15] (Theorem 3.14 and §4.1), in order to classify all compact simple KTS’s we have to find one such model for each 5-grading of each real simple Lie algebra. Moreover, by [16] (Theorem 3.3, Table I), all 5-gradings of realifications of complex exceptional simple Lie algebras are such that

Let now be the simple compact KTS’s defined by (3.7). By the proof of Theorem1, the Kantor algebras and are isomorphic as GLA’s, hence isomorphic to Allison’s 5-GLA by [7] Theorem 2.5. Then the assertions follow from (2.6) and [16] (Table I) since it can be easily seen that the only possible are those for which

**The exceptional simple Lie algebras and G _{2}**

We give now a close related structure to the one of the previous chapter which leads to models
of compact KTS’s such that the corresponding Kantor algebra is the real exceptional simple
Lie algebra and moreover the real split G_{2}. The approach is closer related to the models
of compact KTS’s defined in [20] and the presentation of [14], by defining the KTS’s on a
structurable algebra of skew-dimension one (over ) i.e. KTS’s defined on a 2 × 2-matrix
algebra, than the presentation of (complex) KTS’s defined on symmetric tensors of [18].

From now on let be the algebra with multiplication and
standard involution − defined by formula (2.7) [19] (§4). The algebra M(F) is called (in the
terminology of [2] (§8) the 2 × 2-matrix algebra constructed from an admissible non-degenerate
cubic form N (with basepoint 1 and scalar 1), for short here, the 2×2-matrix algebra determined
by F (where N(x) = x^{3}, x 2 F).

**Remark.** As a direct consequence of the embedding F → H_{3}(F), x 7! xI_{3}, where I_{3} is the unit
matrix of order 3, follows N(x) = x^{3}, Tr(x) = 3x, hence

T(x, y) = 3xy and x × y = xy, for all x, y 2 F (3.20)

by the formulas (2.11).

**Lemma 3** ([3]). Let be the 2×2-matrix algebra determined by where
(−) is the standard involution on A. Then, over is simple structurable
of skew-dimension 1, if F =

Proof. The assertions follow from [2] (§7, Theorem 11) and [3] (§1, Proposition 1.10).

We define now models of compact simple Kantor triple systems.

Let be the 2 × 2-matrix algebra determined by where − is the standard involution on A. We define a second involution ˜ on M(F) by

(3.21)

where is the standard conjugate of

**Remark.** Clearly, − is the identity map in the right hand side of formula (3.21) if F = R.
Moreover, the definition (3.21) is consistent with (3.5), if F = C, as well as with the definition
(2.16) of [20] (§2.2), if F = R.

Then the following involutive automorphism is defined on M(F)

(3.22)

where are the standard conjugates of

Remark. As above, − is the identity map in the right hand side of formula (3.22) if F = R.

Let denote the realification of the algebra Then we have

**Proposition 4. **The triple systems defined by (3.7), (3.21) are
KTS’s satisfying the condition (A).

**Proof.** For the case the proof is identical to the proof of Proposition 3, by replacing
in the proof of Proposition 3 the algebra with and Lemma 1 with Lemma 3,
respectively. Further, for the case the proof is identical to the proof of Proposition 3, by replacing
in the proof of Proposition 3 the algebra with and Lemma 1 with Lemma 3,
respectively. Further, for the case the proof is identical to the proof of [20] Proposition
2.4, by replacing in the proof of [20] Proposition 2.4 the algebra with and [20]
Lemma 1.2 with Lemma 3, respectively.

We give now the analog of Theorem 1 and [20] Theorem 2.1.

**Theorem 3. **The KTS’s defined by (3.7),(3.21) are compact, simple.

**Proof. **We prove first compactness. We must show that the canonical (trace) form defined
by (2.4) for the KTS’s and, respectively, is positive definite. Since the
canonical form is symmetric we consider the corresponding quadratic form (3.8). Then, by (3.19)
and (3.20),

(3.23)

where ||c|| denotes the norm of c 2 C. Then, by (3.23), hence is positive definite for all Analogously, by [20] (2.29) and (3.20),

(3.24)

for all Then, by (3.24), hence is positive definite for all

We prove now simplicity.

For the case the proof is identical to the proof of the simplicity assertion of Theorem 1, by replacing in the proof of Theorem 1 the algebra with and Lemma 1 with Lemma 3, respectively. Further, for the case the proof is identical to the proof of the simplicity assertion of [20] Theorem 2.1, by replacing in the proof of [20] Theorem 2.1 the algebra with and [20] Lemma 1.2 with Lemma 3, respectively.

**Remark. **By similarity to [22] §2, define triple systems by

where ˜ is the involution on defined by (3.21). Then the triple systems are simple compact KTS’s, since it can be easily checked that and are isomorphic under the map

Analogously, the triple systems are simple compact KTS’s.

**Proposition 5**. Let and be the KTS’s defined by (3.7), (3.21). Then
the corresponding Kantor algebras are the exceptional simple Lie algebras and

**Proof. **The proof is based on dimensional reasons. Consider first the simple compact KTS By [6] Proposition 1.6, the Kantor algebras are isomorphic
as GLA’s, hence isomorphic to Allison’s 5-GLA by [7] Theorem 2.5. Then the
assertion follow from (2.6) and [16] (Table I) since it can be easily seen that the only possible

Analogously, consider the simple compact KTS By [6] Proposition 1.6, the Kantor
algebras are isomorphic as GLA’s, hence isomorphic to Allison’s 5-GLA by [7] Theorem 2.5. Then the assertion follow from (2.6) and [16] (Table I) since it can
be easily seen that the only possible with is G_{2}.

**Remark.** The identity follows also from [2] (§8, p. 1871).

The author is grateful to Professor A. Elduque and the Department of Mathematics, University of Zaragoza, Spain, for a research visit during June 2005 and acknowledges partial support for this research from the Spanish Ministerio de Educati´on y Ciencia (MTM 2004-081159-C04-02).

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