Mem. Fac. Sci. Eng. Shimane Univ.
Series B: Mathematical Science
30 (1997), pp. 37–47

GENERALIZATION OF LIE TRIPLE ALGEBRAMichihiko Kikkawa(Received December 25, 1996)

Abstract. We generalize the concept of Lie triple algebra, introduced astangent algebra of geodesic homogeneous left Lie loop [19], to some algebraic
systems equipped with some more multilinear operations, under an idea basedon purely geometric point of view. That is, the operations of Lie triple algebras deﬁned by the parallel torsion and curvature tensors of the canonicalconnection of homogeneous left Lie loops will be extended to ones deﬁned bysome connection whose torsion is not assumed to be parallel. The new algebraic system thus obtained will be called Lie triple multi-algebra. We present
a method of constructing a Lie triple multi-algebra from double Lie algebras.

1. Introduction
The concept of Lie triple algebra has been introduced by Yamaguti [60] under the name of general Lie triple system, related with the canonical connectionof reductive homogeneous space of Nomizu [51]. It is an algebraic system on avector space equipped with a bilinear multiplication and a trilinear multiplication, which contains both of the concepts of Lie algebra and Lie triple system as
special cases.
In 1975, the author has introduced the concept of homogeneous Lie loops(cf. Kikkawa [19] as a non-associative generalization of the concept of Liegroups, that is, a kind of diﬀerentiable algebraic binary systems on manifolds.
Then, he deﬁned the tangent Lie triple algebra of the homogeneous Lie loop,
or of homogeneous left Lie loop (cf. Kikkawa [19], [38]) with the bilinear andtrilinear multiplications on the tangent space at the unit element, by the values of
the torsion tensor and curvature tensor of the canonical connection, respectively.
This is a generalization of the Lie algebras of Lie groups. In fact, any Lie groupis a homogeneous Lie loop with the (−)-connection as the canonical connection,and the bracket operation of the Lie algebra is given by the value of the torsion
tensor of this connection. For diﬀerentiable homogeneous loops and tangent Lie
triple algebras, see also Hofmann-Strambach [10], [11] and Miheev-Sabinin
1991 Mathematics Subject Classiﬁcation. 17A40(20N05).
Key words and phrases. Ternary compositions, Lie triple algebra, Lie algebra.
37

38

MICHIHIKO KIKKAWA[50]. On the same line, the theory of generalization of Lie groups and tangent
algebraic systems related with linear connections has been developed by Sagle
and others in [12], [55], [56] and [57].
In this paper, we generalize the concept of Lie triple algebra from the viewpoint of algebraic systems whose multiplications are given by the values of torsion
and curvature tensors of some linear connection. To do this, in section 2, we recall some identities for the covariant diﬀerentiation of linear connections. Then,in section 3, we introduce the new algebraic system called Lie triple multi-algebraon any vector space which is a generalized concept of that of Lie triple algebras.
The concept of diﬀerentiable local loops was originated in 1936 by S. S.
Chern [7] as diﬀerentiable webs on manifolds. For the relations between loops
and webs, see e.g. Bol [5]. In 1964, the concept of geodesic local loops on
linearly connected manifolds was introduced by Kikkawa [13] and, independently, by Sabinin [53] in 1972. See also Akivis [3], Miheev-Sabinin [50] andSabinin [54]. For locally reductive spaces, that is, linearly connected manifolds
whose torsion and curvature are parallel, the geodesic local loops are reduced
to homogeneous local Lie loops and their tangent algebras are reduced to Lietriple algebras. In [19] - [46] the author has developed the extensive theory ofgeodesic homogeneous (left) Lie loops and their tangent Lie triple algebras andshown that it is a full generalization of the theory of Lie groups and Lie algebras. The tangent algebraic system of analytic loops in general was considered by
Akivis [2], which is called Akivis Algebra. The related theory has been developedin Akivis [1] - [3], Akivis-Shelekhov [4] and Hofmann-Strambach [10].
See also Hofmann-Strambach [11], Goldberg [9]. The concept of Akivis
algebras was generalized for analytic n-loops in Goldberg [8] and Smith [59].
On the other hand, in [37], the author has introduced the concept of projectivity of geodesic local loops, and found the method to get from a geodesic
homogeneous left Lie loops a new one, by means of changing of the canonical connection to the other locally reductive connection. By this method, the theory of
projectivity has been developed in Kikkawa [37] - [45] and Sanami-Kikkawa[58]. Especially, the structural theory of projectivity of simple Lie groups has
been investigated in [58], where the tangent Lie triple algebras of homogeneousleft Lie loops in projective relation with a simple Lie group G are determined byprojective double Lie algebras of the Lie algebra of G. In section 4, motivatedby this method, we give a method of constructing a Lie triple multi-algebra gfrom two Lie algebras s and t on the same vector space.
The problem of ﬁnding algebraic properties of geodesic local loops on linearly
connected manifolds whose tangent algebras form Lie triple multi-algebras is stillopen.

2. Covariant differentiation of linear connections
In this section, we recall some identities on covariant diﬀerentiation of linear
connections, that are familiar to diﬀerential geometers.

GENERALIZATION OF LIE TRIPLE ALGEBRA

39

Let M be a diﬀerentiable manifold of class C ∞ with a linear connection ∇.
In what follows, we assume that all functions, vector ﬁelds and tensor ﬁelds areof class C ∞ . We denote the torsion tensor ﬁeld and the curvature tensor ﬁeld of∇ by S and R, respectively, which are given by the following formulas:
(2.1)
(2.2)

S(X, Y ) := [X, Y ] − ∇X Y + ∇Y X
R(X, Y )Z := ∇[X,Y ] Z − ∇X ∇Y Z + ∇Y ∇X Z,

for any vector ﬁelds X, Y, Z on M . We denote here the Lie bracket of vectorﬁelds by the big bracket [X, Y ] to distinguish from the usual bracket notation
used later for a multilinear algebraic operation.
For any tensor ﬁeld K of type (1, p), the covariant derivative ∇K of K is atensor ﬁeld of type (1, p + 1) given by the following equations:(cf. KobayashiNomizu [47] pp.124-125)(2.3)(∇K)(X1 , X2 , . . . , Xp ; Y ) := ∇Y (K(X1 , X2 , . . . , Xp ))− K(∇Y X1 , X2 , . . . , Xp ) − K(X1 , ∇Y X2 , . . . , Xp )− · · · − K(X1 , X2 , . . . , ∇Y Xp ),for any vector ﬁelds X1 , X2 , . . . , Xp and Y on M . A tensor ﬁeld K on M issaid to be parallel if ∇K = 0 . Moreover, for any integer m, the m-th covariant
derivative ∇m K of any tensor ﬁeld K is deﬁned inductively to be a tensor ﬁeldof type (1, p + m) given by the equation(2.4)(∇m K)(X1 , X2 , . . . , Xp ; Y1 ; Y2 ; . . . ; Ym ):= (∇(∇m−1 K))(X1 , X2 , . . . , Xp ; Y1 ; Y2 ; . . . ; Ym−1 ; Ym ).

In particular, for any vector ﬁelds Y1 , Y2 , we set(∇Y1 K)(X1 , X2 , . . . , Xp ) := (∇K)(X1 , X2 , . . . , Xp ; Y1 ),(2.5)(∇2 K)(; Y1 ; Y2 )(X1 , X2 , . . . , Xp ):= (∇2 K)(X1 , X2 , . . . , Xp ; Y1 ; Y2 ).

Then, we get tensor ﬁelds ∇Y1 K = ∇K(; Y1 ) and (∇2 K)(; Y1 ; Y2 ) of type (1, p),which satisfy the following identity:(2.6)(∇2 K)(; X; Y ) = ∇Y (∇X K) − ∇∇Y X K.

From this equation, we get the following ;(2.7)(∇2 K)(; X; Y ) − (∇2 K)(; Y ; X) = R(X, Y )K − ∇S(X,Y ) K(Ricci’s identity),where(R(X, Y )K)(X1 , X2 , . . . , Xp ) := R(X, Y )(K(X1 , X2 , . . . , Xp ))− K(R(X, Y )X1 , X2 , . . . , Xp ) − K(X1 , R(X, Y )X2 , . . . , Xp )(2.8)
− · · · − K(X1 , X2 , . . . , R(X, Y )Xp ).

40

MICHIHIKO KIKKAWA

In the following we recall some identities for the torsion tensor ﬁeld S andthe curvature tensor ﬁeld R. Here, we denote by SX,Y,Z the cyclic sum withrespect to X, Y, Z :(2.9)

SX,Y,Z {R(X, Y )Z + S(S(X, Y ), Z) − (∇X S)(Y, Z)} = 0(Bianchi’s 1st identity)(2.10)

SX,Y,Z {R(S(X, Y ), Z)W − ((∇X R)(Y, Z)W )} = 0(Bianchi’s 2nd identity)(2.11)(∇2 S)(; X; Y ) − (∇2 S)(; Y ; X) = R(X, Y )S − ∇S(X,Y ) S(Ricci’s identity for S)(2.12)(∇2 R)(; X; Y ) − (∇2 R)(; Y ; X) = R(X, Y )R − ∇S(X,Y ) R(Ricci’s identity for R)(2.13)(∇2 (∇S))(; X; Y ) − (∇2 (∇S))(; Y ; X)
= R(X, Y )((∇S)) − ∇S(X,Y ) (∇S)
(Ricci’s identity for ∇S)

and so on.
3. Lie triple multi-algebras
In this section, we introduce a new concept of algebraic systems on vector
spaces which is induced from the relations of torsion and curvature tensors of
linearly connected manifolds mentioned in the last section.
Let M be a manifold with a linear connection ∇. If the torsion tensor S andthe curvature tensor R are parallel, we see that the real vector space X(M ) of
all vector ﬁelds on M forms a Lie triple algebra (X(M ); [X, Y ], hX, Y, Zi), where
the bilinear and trilinear operations [X, Y ] and hX, Y, Zi are given by ;[X, Y ] := S(X, Y ),hX, Y, Zi := R(X, Y )Z,

for any vector ﬁelds X, Y, Z. Notice that the bracket [X, Y ] does not mean the
Lie bracket of vector ﬁelds.
Assume that M is a geodesic homogeneous (left) Lie loop and ∇ is its canonical connection. Then, by restricting these operations to the tangent space Te (M )at the unit element e of M, we get the tangent Lie triple algebra(g := (Te (M ); [X, Y ], hX, Y, Zi)

GENERALIZATION OF LIE TRIPLE ALGEBRA

41

for any elements X = Xe , Y = Ye , Z = Ze ∈ Te (M ) and(3.1)[X, Y ] := Se (Xe , Ye ),hX, Y, Zi := Re (Xe , Ye )Ze ,that is, it satisﬁes the following relations:
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)

[X, Y ] = −[Y, X]
hX, Y, Zi = −hY, X, Zi
SX,Y,Z {hX, Y, Zi + [[X, Y ], Z]} = 0
SX,Y,Z {h[X, Y ], Z, W i} = 0
hU, V, [X, Y ]i = [hU, V, Xi, Y ] + [X, hU, V, Y i]

(3.7)

hU, V, hX, Y, Zii = hhU, V, Xi, Y, Zi+ hX, hU, V, Y i, Zi + hX, Y, hU, V, Zii

for any X, Y, Z, U, V, W ∈ g. In fact, under the assumptions ∇S = 0 and ∇R = 0,
the formulae (3.4), (3.5), (3.6) and (3.7) are obtained from Bianchi’s 1st identity,
2nd identity, Ricci’s identity for S and for R in §2, respectively.
Remark. As is well-known, a Lie triple algebra g is reduced to a Lie algebra if
the ternary operation hX, Y, Zi vanishes identically, while it is reduced to a Lietriple system if the binary operation [X, Y ] vanishes identically.
Now we introduce the new algebraic system, motivated by the direct correspondence above between the axiom of Lie triple algebra ((3.2) - (3.7)) and theidentities of Bianchi and Ricci for the parallel torsion tensor S and the parallel
curvature tensor R:
Deﬁnition 3.1. A Lie triple multi-algebra g = (V; [X, Y ], [X, Y ; Z], hX, Y, Zi)
is a vector space V over a ﬁeld of characteristic 0 equipped with a family of
V-valued multilinear operations on V :[X, Y ], [X, Y ; Z] and hX, Y, Zi
satisfying the following relations:
(3.8)
(3.9)
(3.10)(3.11)(3.12)[X, Y ] = −[Y, X][X, Y ; Z] = −[Y, X; Z]
hX, Y, Zi = −hY, X, Zi
SX,Y,Z {hX, Y, Zi + [[X, Y ], Z] − [X, Y ; Z]} = 0
SX,Y,Z {h[X, Y ], Z, W i} = 0(3.13)

hU, V, [X, Y ]i = [hU, V, Xi, Y ] + [X, hU, V, Y i]
− [X, Y ; [U, V ]]

42

MICHIHIKO KIKKAWA(3.14)

hU, V, [X, Y ; Z]i = [hU, V, Xi, Y ; Z] + [X, hU, V, Y i; Z]+ [X, Y ; hU, V, Zi](3.15)

hU, V, hX, Y, Zii = hhU, V, Xi, Y, Zi + hX, hU, V, Y i, Zi+ hX, Y, hU, V, Zii

Indeed, assume that a linear connection ∇ on a manifold M has the torsion
tensor S and the curvature tensor R satisfying ∇2 S = 0 and ∇R = 0. Thenthe inﬁnite-dimensional vector space X(M ) of all vector ﬁelds on M forms a Lietriple multi-algebra, because (3.8) - (3.10) are evident by anti-symmetricity of S
and R, and Bianchi’s identities and Ricci’s identities for S, ∇S and R assure theformulas (3.11) - (3.15), respectively.
Remark. If the ternary operation [X, Y ; Z] of a Lie triple multi-algebra vanishes
identically, it is reduced to a Lie triple algebra. On the other hand, we can deﬁne
more general concepts of Lie triple multi-algebras of order k for the operation[X, Y ] and of order m for the operation hX, Y, Zi as [X, Y ; Z1 ; Z2 ; . . . ; Zk ] andhX, Y, Z; W1 ; W2 ; . . . ; Wm i, respectively. In this paper, we do not consider thembecause we want to investigate the ﬁrst stage of generalization of the tangent Lie
triple multi-algebras of analytic loops .
In what follows, we will consider Lie triple multi-algebras (of order 1 for[X, Y ]). Let g = (V; [X, Y ], [X, Y ; Z], hX, Y, Zi) be a Lie triple multi-algebra on
a vector space V. We can deﬁne the concepts of Lie triple multi-subalgebras ofg and homomorphisms of Lie triple multi-algebras in a natural manner.
Deﬁnition 3.2. A Lie triple multi-subalgebra h of g given on a subspace H of
V will be called an ideal of g if it satisﬁes the following relations:(3.16)(3.17)(3.18)[g, h] ⊂ h[g, h; g] ⊂ h and [g, g; h] ⊂ hhg, h, gi ⊂ h.

A Lie triple multi-algebra g is said to be simple if it has no non-trivial ideal.

Remark. If the Lie triple multi-algebra g is reduced to a Lie triple algebra,
the ideals of g are reduced to those of the Lie triple algebra (general Lie triple
system) which was introduced by Yamaguti in [61].
From this deﬁnition, we can easily obtain the following results:
Proposition 3.1. Let h be a Lie triple multi-subalgebra of a Lie triple multialgebra g. Then, h is an ideal of g if and only if it is a kernel of a homomorphismof g into some Lie triple multi-algebra.

GENERALIZATION OF LIE TRIPLE ALGEBRA

43

Proposition 3.2. Let h be an ideal of a Lie triple multi-algebra g. Then, thequotient multi-algebra g/h is deﬁned in a natural manner which forms a Lietriple multi-algebra under the natural multilinear operations [X̃, Ỹ ] , [X̃, Ỹ ; Z̃]and hX̃, Ỹ , Z̃i deﬁned for X̃ = X + h, Ỹ = Y + h and Z̃ = Z + h.

4. Construction of Lie triple multi-algebras from double Liealgebras
In this section, we construct a Lie triple multi-algebra
g = (V; [X, Y ], [X, Y ; Z], hX, Y, Zi)
from two Lie algebras (double Lie algebras) given on the same vector space V.
Theorem 4.1. Let s = (V; S(X, Y )) and t = (V; T (X, Y )) be two Lie algebras
on the same vector space V whose bracket operations are given by S(X, Y ) andT (X, Y ), respectively. Consider an algebraic systemg = (V; [X, Y ], [X, Y ; Z], hX, Y, Zi)
on V given by
(4.1)
(4.2)
(4.3)

[X, Y ] := S(X, Y ) + 2T (X, Y )[X, Y ; Z] := T (S(X, Y ), Z) − S(T (X, Z), Y ) − S(X, T (Y, Z))
hX, Y, Zi := −T (S(X, Y ) + T (X, Y ), Z)

for X, Y, Z ∈ V.
Assume that the two Lie algebras s and t satisfy the following:
(4.4)
(4.5)
(4.6)
(4.7)

T (W, [X, Y ; Z]) = 0[T (W, X), Y ; Z] + [X, T (W, Y ); Z] = 0[X, Y ; S(U, V )] = 0[X, Y ; T (U, V )] = 0

for X, Y, Z, U, V, W ∈ V. Then g forms a Lie triple multi-algebra on V.

Proof. We show that the operations (4.1), (4.2) and (4.3) satisfy the relations(3.8) - (3.15) under the assumptions (4.4) - (4.7). By deﬁnition of the operations,the relations (3.8), (3.9) and (3.10) are clear. By a straightforward calculation
we get the relation (3.11). We see that the (3.12) is equivalent to
SX,Y,Z T (W, [X, Y ; Z]) = 0,

44

MICHIHIKO KIKKAWA

which is valid because (4.4) is assumed. On the other hand, we can show thatthe relation (3.13) is equivalent to the following condition:[X, Y ; 2S(U, V ) + T (U, V )]) = 0,
which is assured by (4.6) and (4.7). We can also show that the relations (3.14) isdeduced by the assumptions (4.5), (4.6) and (4.7). Finally, we can see that therelation (3.15) is equivalent to the following equation:
T (W, [X, Y ; S(U, V ) + T (U, V )]) = 0,
which is clear by (4.4).

Remark. As a special case of the results in the theorem above, we have alreadyknown the case of projective double Lie algebras discussed in [58], where thecondition(4.8)

T (U, S(X, Y )) = S(T (U, X), Y ) + S(X, T (U, Y ))is given. In this case, we have
[X, Y ; Z] = 0 for all X, Y, Z ∈ g,
and we see that the Lie triple multi-algebra g is reduced to a Lie triple algebra.
As for the Lie triple multi-algebras constructed by ideals of double Lie algebras, we have the following
Theorem 4.2. Let g = (V; [X, Y ], [X, Y ; Z], hX, Y, Zi) be a Lie triple multialgebra constructed by double Lie algebras s = (V; S(X, Y )) and t = (V; T (X, Y ))on a vector space V. Assume that a subspace H of V induces ideals of bothof the Lie algebras s and t, that is, the restrictions sH = (H; SH (X, Y )) andtH = (H; TH (X, Y )) to H are ideals of s and t, respectively. Then, the Lie triple
multi-algebrah = (H; [X, Y ]H , [X, Y ; Z]H , hX, Y, ZiH )constructed by sH and tH on H is an ideal of g.
Proof. Since S and T satisfy the relations
S(V, H) ⊂ H and T (V, H) ⊂ H
the deﬁnitions (4.1), (4.2) and (4.3) of the operations of the constructed Lie triple
multi-algebra imply the following relations:[g, h] ⊂ h, [g, h; g] ⊂ h, [g, g; h] ⊂ h and hg, h, gi ⊂ hwhich show the assertion of the theorem.
From this theorem we have the following corollary:GENERALIZATION OF LIE TRIPLE ALGEBRA

45

Corollary 4.3. If the Lie triple multi-algebra costructed by double Lie algebras
is simple, then the two Lie algebras have no ideals on any non-trivial underlying
subspace in common.

References
1. M.Akivis, Local diﬀerentiable quasigroups and three webs of multidimensional surfaces(Russian), Shtiinsta Kishinev (1973), 3–12
2.
, Local algebras of a multidimensional web (Russian), Sib. Mat. Zh. 17 (1976), 5–11.
, Geodesic loops and local triple systems in aﬃnely connected spaces (Russian),
3.
Sibir. Mat. Z. 19 (1978), 243–253.
4. M.Akivis - A.Shelekhov, Geometry and Algebra of Multidimensional Three-Webs,
Kluwer Academic Publ. 1992.
5. G.Bol, Gewebe und Gruppen, Math. Ann.144(1937), 414–431.
6. O.Chein - H.O.Pflugfelder - J.D.H.Smith, Quasigroups and Loops: Theory andApplications, Heldermann Verlag 1990.
7. S.S.Chern, Eine Invariantentheorie der Dreigewebe aus r-dimensionalen Mannigfaltigkeiten in R2r , Abh. Math. Sem. Univ. Hamburg 11(1963), 333–358.
8. V.V.Goldberg, Theory of multicodimensional (n + 1)-webs, KluwerAcademic publ. 1988.
9.
, Local diﬀerentiable quasigroups and webs, Chapter X, pp.263–312 in Quasigroups
and Loops, Theory and Application, Heldermann Verlag 1990.
10. K.H.Hofmann - K.Strambach, The Akivis algebra of a homogeneous loop, Mathematika
33(1986), 87–95.
11.
, Topological and analytic loops, Chapter IX pp.205–262 in Quasigroups and Loops,
Theory and Application, Heldermann Verlag 1990.
12. J.P.Holmes - A.A.Sagle, Analytic H-spaces, Campbell-Hausdorﬀ formula and alternative algebra, Paciﬁc J. Math. 91(1980), 105–134.
13. M.Kikkawa, On local loops in aﬃne manifolds, J. Sci. Hiroshima Univ. A-I, 28(1964),
199–207.
14.
, System of local loops on manifold and aﬃne connection, Bull. Shimane Univ. (Nat.
Sci.), 16(1966), 12–14.
15.
, On the decomposition of a linearly connected manifold with torsion, J. Sci. Hiroshima Univ. A-I(1969), 1–9.
, On locally reductive spaces and tangent algebras, Mem. Fac. Lit. Sci. Shimane
16.
Univ. Nat. Sci. 5(1972), 1–13.
17.
, On some quasigroups of algebraic models of symmetric spaces, Mem. Fac. Lit. Sci.
Shimane Univ. Nat. Sci. 6(1973), 9–13.
, On some quasigroups of algebraic models of symmetric spaces II, Mem. Fac. Lit.
18.
Sci. Shimane Univ. Nat. Sci. 7(1974), 29–35.
,Geometry of homogeneous Lie loops, Hiroshima Math. J. 5 1975, 141–179.
19.
20.
, A note on subloops of a homogeneous Lie loop and subsystems of its Lie triple
algebra, Hiroshima Math. J. 5(1975), 439–446.
, On some quasigroups of algebraic models of symmetric spaces III, Mem. Fac. Lit.
21.
Sci. Shimane Univ. Nat. Sci. 9(1975), 7–12.
22.
, On the left translations of homogeneous loops, Mem. Fac. Lit. Sci. Shimane Univ.
Nat. Sci. 10(1976), 19–25.
, On homogeneous systems I, Mem. Fac. Lit. Sci. Shimane Univ. 11(1977), , 9–17.
23.
24.
, On ternary systems(homogeneous systems) on manifolds(Japanese), R.I.M.S.
Kokyuroku 308(1977), 12–31.

46

MICHIHIKO KIKKAWA, On homogeneous systems II, Mem. Fac. Lit. Sci. Shimane Univ. 12(1978), 5–13.
, Remarks on solvability of Lie triple algebras, Mem. Fac. Sci. Shimane Univ.
13(1979), 17–22.
, On homogeneous systems III, Mem. Fac. Sci. Shimane Univ. bf 14(1980), 41–46.
27.
28.
, On Killing-Ricci forms of Lie triple algebras, Paciﬁc J. Math. 96(1981), 153–161.
, On the decomposition of homogeneous systems with nondegenerate Killing-Ricci
29.
tensor, Hiroshima Math. J. 11(1981), 525-531.
30.
, On homogeneous systems IV, Mem. Fac. Sci. Shimane Univ. 15(1981), 1–7.
31.
, On the Killing radical of Lie triple algebras, Proc. Japan Acad. 58-A(1982), 212–
215.
, Remarks on invariant forms of Lie triple algebras, Mem. Fac. Sci. Shimane Univ.
32.
16(1982), 23–27.
33.
, On homogeneous systems V, Mem. Fac. Sci. Shimane Univ. 17(1983), 9–13.
34.
, Totally geodesic imbeddings of homogeneous systems into their enveloping Liegroups, Mem. Fac. Sci. Shimane Univ. 18(1984), 1–8.
35.
, Canonical connections of homogeneous Lie loops and 3-webs, Mem. Fac. Sci. Shimane Univ. 19(1985), 37–55.
36.
, Remarks on canonical connections of loops with the left inverse property, Mem.
Fac. Sci. Shimane Univ. 20(1986), 9–19.
37.
, Aﬃne homogeneous structures on analytic loops, Mem. Fac. Sci. Shimane Univ.
21(1987), 1–15.
38.
, Projectivity of left loops on Rn , Mem. Fac. Sci. Shimane Univ. 22(1988), 33–41.
39.
, Projectivity of homogeneous left loops on Lie groups I (Algebraic framework),
Mem. Fac. Sci. Shimane Univ. 23(1989), 17–22.
40.
, Projectivity of homogeneous left loops on Lie groups II (Local theory), Mem. Fac.
Sci. Shimane Univ. 24(1990), 1–16.
, Projectivity of homogeneous left loops, Nonassociative Algebras and Related Top41.
ics, World Scientiﬁc 1991, 77–99.
42.
, Symmetrizability of geodesic homogeneous left Lie loops, Proc. 3rd Congr. Geom.
Thessaloniki, 230–238(1991).
, On tangent algebras of symmetrizable homogeneous left Lie loops, Mem. Fac. Sci.
43.
Shimane Univ. 26(1992), 55–63.
44.
, On normal left Lie subloops of homogeneous left Lie loops, Mem. Fac. Sci. Shimane
Univ.27(1993), 1–10.
, Remarks on projectivity of subsystems of homogeneous systems, Mem. Fac. Sci.
45.
Shimane Univ. 28(1994), 27–31.
46.
, Remarks on Akivis left loops, Mem. Fac. Sci. Shimane Univ. 29(1995), 1–9.
47. S.Kobayashi - K.Nomizu, Foundation of Diﬀerential Geomatry I, Interscience 1963.
48.
, Foundation of Diﬀerential Geometry II, Interscience 1969.
49. O.Loos, Symmetric Spaces I, Benjamin 1969.
50. Miheev-Sabinin, Quasigroups and diﬀerential geometry, Chapter XII pp. 357–430 inQuasigroups and Loops, Theory and Application, Heldermann Verlag 1990.
51. K.Nomizu, Invariant aﬃne connections on homogeneous spaces, Amer. J. Math. 76(1954),
33–65.
52. H.O.Pflugfelder, Quasigroups and Loops: Introduction, Heldermann Verlag 1990.
53. L.V.Sabinin, Geometry of loops(Russian), Mat. Zametki 12(1972), 605–616.
, Methods of nonassociative algebras in diﬀerential geometry(Russian), an added
54.
chapter to the Russian translation of Kobayashi-Nomizu ”Foundation of DiﬀerentialGeometry I”, Nauka 1981.
55. A.A.Sagle - J.Schumi, Multiplications on homogeneous spaces, nonassociative algebras
and connections, Paciﬁc J. Math. 48(1973), 247–266.
56.
, Anti-commutative algebras and homogeneous spaces with multiplications, Paciﬁc
25.
26.

GENERALIZATION OF LIE TRIPLE ALGEBRA

47

J. Math. 68(1977),255–269.
57. A.A.Sagle - R.E.Walde, Introduction to Lie Groups and Lie Algebras, Academic Press
1973.
58. M.Sanami-M.Kikkawa, A class of double Lie algebras on simple Lie algebras and projectivity of simple Lie groups, Mem. Fac. Sci. Shimane Univ. 25(1991), 39–44.
59. J. D. H. Smith, Multilinear algebras and Lie’s theorem for normal n-loops, Arch.
Math.(Basel) 51(1988), 169–177.
60. K.Yamaguti, On the Lie triple system and its generalization, J. Sci. Hiroshima Univ.
A-21 (1958), 155–160.
61.
, On cohomology groups of general Lie triple systems, Kumamoto J. Sci. A8(1969),
135–164.
Department of Mathematics, Shimane Univarsity, Matsue 690 JAPANE-mail address: kikkawa@riko.shimane-u.ac.jp