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semiski
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There are 7 letters in SEMISKI ( E1I1K5M3S1 )
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In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras * * * * * * g * * * * * {\displaystyle {\mathfrak {g}}} * whose only ideals are {0} and * * * * * * g * * * * * {\displaystyle {\mathfrak {g}}} * itself. It is important to emphasize that a one-dimensional Lie algebra (which is necessarily abelian) is by definition not considered a simple Lie algebra, even though such an algebra certainly has no nontrivial ideals. Thus, one-dimensional algebras are not allowed as summands in a semisimple Lie algebra. * Throughout the article, unless otherwise stated, * * * * * * g * * * * * {\displaystyle {\mathfrak {g}}} * is a non-zero finite-dimensional Lie algebra over a field of characteristic 0. The following conditions are equivalent:* * * * * * g * * * * * {\displaystyle {\mathfrak {g}}} * is semisimple * the Killing form, κ(x,y) = tr(ad(x)ad(y)), is non-degenerate, * * * * * * * g * * * * * {\displaystyle {\mathfrak {g}}} * has no non-zero abelian ideals, * * * * * * * g * * * * * {\displaystyle {\mathfrak {g}}} * has no non-zero solvable ideals, * The radical (maximal solvable ideal) of * * * * * * g * * * * * {\displaystyle {\mathfrak {g}}} * is zero. |