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Journal of Generalized Lie Theory and Applications
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On compact realifications of exceptional simple Kantor triple systems

Daniel MONDOC*

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

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Abstract

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

Introduction

Models of Kantor triple systems defined on the 2 × 2-matrix algebra determined by the Jordan algebra equation of hermitian 3 × 3-matrices over complex composition algebras equation considered over the field equation 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 equation.

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 equation

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 equation of hermitian 3×3-matrices over a complex composition algebra equation 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 equation of hermitian 3 × 3-matrices over a real composition algebra equation have been given. Related results are those of [10] where a construction of exceptional simple 5-graded Lie algebrasequation and an explicit realization of the subspaces Ul 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 equation and an involutive automorphism equation. Then the new triple productequation is considered, which gives again a Kantor triple system. Suitable elections of A and equation 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 equation the corresponding canonical trace form and the corresponding exceptional real simple Lie algebras are proved in section 3.

Triple systems, graded Lie algebras and structurable algebras

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 equation such thatequation

A GLA equation 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 Lx,y,Rx,y and Sx,y on U by

equation (2.1a)

equation (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):

equation (2.2)

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

Let (B,U) be a GJTS and Vk, k = 1, 2, 3, be subspaces of U. We denote by B(V1, V2, V3) the subspace of U spanned by elements equation A subspace V of U is called an ideal of (B,U) if the following relations hold B(V,U,U) ⊆ V, B(U, V,U ) ⊆ V, B(U,U, V ) ⊆ V . The GJTS (B,U) is called simple if B is not a zero map and (B,U) has no non-trivial ideal.

Starting from a given GJTS (B,U), I. L. Kantor [17] constructed a certain GLA L(B) =equation 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 equation 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

equation (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 equation we define a bilinear map Bz on U by equation We say that (B,U) satisfies the condition (A) if Bz = 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 equation be defined by and denote by HomF(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 equation

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 defineequation and the triple systemequation by

equation (2.5a)

equation (2.5b)

equation is called the triple system obtained from the algebra (A, ) [7] (§2). We shall write for short BA for (BA,A).

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

equation

Let (A, ) be a structurable algebra. Then, by [3], equation, where

equation

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

equation (2.6a)

equation (2.6b)

equation 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(BA), where BA is the triple system obtained from the algebra (A,− ).

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 equation the algebra M(J) with multiplication and standard involution − defined [3] (§1) by

equation (2.7a)

equation(2.7b)

equation (2.7c)

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

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

equation (2.8)

and extend conjugation − by linearity on equation.

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

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

equation (2.9)

equation (2.10)

where equation lul is an arbitrary element of equation, ul are the standard units and ulequation is defined by (2.8) hence equationis the standard complex conjugate of α1.

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

Let equation be each one of the division composition algebras equation

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

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

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

equation

where in the right hand side we have usual matrix multiplication and equation denotes the conjugate transposed of equation Then, by [9] (p. 218), the trace form and ×-operation in formulas (2.7) are defined on H3(A) by

equation (2.11a)

equation (2.11b)

for all equation where I3 is the unit matrix of order 3 and Tr(x) denotes the trace of x.

Lemma 1 ([3]). Let equation be the 2 × 2-matrix algebra determined by the Jordan algebra equation of hermitian 3 × 3-matrices over a complex composition algebra equationequation where (−) is the standard involution on A. Then, over equation 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 AF the algebra A considered as algebra over F. If A is an algebra over equation then we call the algebra equation 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 (equation, ) the realification of BA.

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

On compact realifications of exceptional simple Kantor triple systems

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

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

equation (3.1a)

equation (3.1b)

equation (3.1c)

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

equation (3.2)

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

equation

where equation denotes the norm ofequation

Proof. Let equation be an arbitrary element in equation of the form (3.2), whereequationequationequation Letequation We write for shortequation Thenequation by (2.9), andequation by (2.10), where − is defined by (2.8) so equation is the standard complex conjugate of clm. Hence equation and by (3.2) we have

equation (3.3)

and

equation (3.4)

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

equation

where ||xlm|| denotes the norm of xlm, 1 ≤ l,m ≤ 3.

The exceptional simple Lie algebras equation

We define now models of compact simple Kantor triple systems.

Let equation denote any of the complex composition algebrasequation

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

equation (3.5)

where equation denotes the pseudo-conjugate of xi conjugate, equation is the standard conjugate of equation Hence the following involutive automorphism is defined onequation

equation

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

equation (3.7)

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

Proposition 3. The triple systems equation 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 equationequation are (simple) KTS’s satisfying the condition (A). Further, we remark that equation by (3.7) and (2.5). Then the assertions follow from [6] Lemma 1.5 since the mapequation is an involutive automorphism on the algebraequation

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

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

equation. (3.8)

We remark first that

equation

equation (3.9)

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

equation (3.10)

equation

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

equation (3.12)

equation where O3 denotes the zero matrix of order 3, the identity y − y = (η1 − η2)s0 follows from (2.7). Hence

equation

by (2.7), and then the identity

equation

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

equation (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 fx : equationequation be the linear map

equation

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

equation(3.14)

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

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

equation (3.15)

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

equation

Let us denote the units of equation by

equation (3.16a)

equation (3.16b)

equation (3.16c)

equation (3.16d)

equation (3.16e)

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

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

equation (3.17)

for allequation

equation

We show now that equation for allequation where hx is defined by (3.15).

We remark first that equation whereequation is an involutive automorphism onequation Then

equation

for allequationTherefore equation soequation are similar and hence they have the same trace.

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

equation (3.19)

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

We prove now simplicity.

Since the KTS’s equation 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 equation are KTS’s satisfying the condition (A) then the algebras equation andequation are isomorphic as GLA’s, by [6] Proposition 1.6. But the Kantor algebrasequation are simple if and only if the structurable algebras equation are simple, by [2] Corollary 6 and [7] Theorem 25. Then the simplicity assertion follows from Lemma 1.

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

equation

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

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

equation

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

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

Let equation denote any of the complex composition algebrasequation be the 2 × 2-matrix algebra determined by the Jordan algebra equation 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 equation are the KTS’s equationequation defined by (3.7) and the corresponding Kantor algebras are the following realifications of complex simple Lie algebras equationequation

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 equation of realifications of complex exceptional simple Lie algebras are such that equationequation

Let now equation be the simple compact KTS’s defined by (3.7). By the proof of Theorem1, the Kantor algebras equation andequation are isomorphic as GLA’s, hence isomorphic to Allison’s 5-GLA equation 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 equationequation are those for whichequation

The exceptional simple Lie algebras equation and G2

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 equation and moreover the real split G2. 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 equation) 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 equation 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) = x3, x 2 F).

Remark. As a direct consequence of the embedding F → H3(F), x 7! xI3, where I3 is the unit matrix of order 3, follows N(x) = x3, 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 equation be the 2×2-matrix algebra determined by equation where (−) is the standard involution on A. Then, over equationequation is simple structurable of skew-dimension 1, if F = equation

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

equation (3.21)

where equationis the standard conjugate of equation

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)

equation (3.22)

where equation are the standard conjugates of equation

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

Let equation denote the realification of the algebraequation Then we have

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

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

equation (3.23)

equationwhere ||c|| denotes the norm of c 2 C. Then, by (3.23),equationequation henceequation is positive definite for allequation Analogously, by [20] (2.29) and (3.20),

equation (3.24)

for all equation Then, by (3.24),equation henceequation is positive definite for all equation

We prove now simplicity.

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

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

equation

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

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

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

Proof. The proof is based on dimensional reasons. Consider first the simple compact KTS equation By [6] Proposition 1.6, the Kantor algebras equation are isomorphic as GLA’s, hence isomorphic to Allison’s 5-GLA equation 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 possibleequation

Analogously, consider the simple compact KTS equation By [6] Proposition 1.6, the Kantor algebras equation are isomorphic as GLA’s, hence isomorphic to Allison’s 5-GLA equation 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 equation withequation is G2.

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

Acknowledgements

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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