By Y Matsushima

Some time past 30 years, differential geometry has passed through a big swap with infusion of topology, Lie idea, advanced research, algebraic geometry and partial differential equations. Professor Matsushima performed a number one position during this transformation through bringing new thoughts of Lie teams and Lie algebras into the examine of actual and intricate manifolds. This quantity is a set of all of the forty six papers written by means of him.

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**Example text**

Be defined for A analogously. tt the eigen-space of 3* for a characteristic root X, of A and by E,' the matrix of projection of 91 On 91*,. Since Ba^siS&Ms 51 is invariant under £) and E, = E,' or £ , = 0 according as X, is a characteristic root of A or not. Hence, by ( 3 ) and Theorem of S 1 , 3t is invariaht under all replicas of A. i%X* j=> Let ( f = l . 2, ft), where r „ are elements of P and where r = 0 for fJ If we put A/ = Zr,jE/ ( > = 1 , 2, .... m), where E,' — 0 i f X, is not a characteristic root of A, then we may represent A i n the form (12) A = A,' + \A,'+^A/ + ---+K,AJ and every replica of A is represented in the form.

Conversely we see _ easily that any replica of A is induced by a replica of A. Lemma 7. Let 34 48 Y . MATSUSHIMA. [Vol. 23. 4. Conversely any replica of A is of the form 3, where B is a replica of A. Proof. For simplicity, we prove this lemma in the case =! •A ( A )• w If e represent A as in (4). +*< A,, V ^=(V) where The linear space on which A operates is ® =TO+ 2' cteristic roots of A are A,, \ x* and and the chara- ri 2 r * = - \ , - - - - V + X X J , + --+ J«= 1-1 (- ft, - • • • - tf,e -f- > s v + - + r )x v lt where (,, *•-, iV, }\, <•• , / .

Theorem 1 . We may construct a normal Lie algebra i , containing tfie given Lie algebra L such that every ideal in L is also an ideal in Z, and L = A + L, l Ar\L=0 , where A is an abelian subalgebra. Proof.