Radical of a Lie Algebra
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In the mathematical field of Lie theory, the radical of a Lie algebra mathfrak{g} is the largest solvable ideal of mathfrak{g}. Let k be a field and let mathfrak{g} be a finite-dimensional Lie algebra over k. A maximal solvable ideal, which is called the radical, exists for the following reason. Firstly let mathfrak{a} and mathfrak{b} be two solvable ideals of mathfrak{g}. Then mathfrak{a}+mathfrak{b} is again an ideal of mathfrak{g}, and it is solvable because it is…mehr

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In the mathematical field of Lie theory, the radical of a Lie algebra mathfrak{g} is the largest solvable ideal of mathfrak{g}. Let k be a field and let mathfrak{g} be a finite-dimensional Lie algebra over k. A maximal solvable ideal, which is called the radical, exists for the following reason. Firstly let mathfrak{a} and mathfrak{b} be two solvable ideals of mathfrak{g}. Then mathfrak{a}+mathfrak{b} is again an ideal of mathfrak{g}, and it is solvable because it is an extension of (mathfrak{a}+mathfrak{b})/mathfrak{a}simeqmathfrak{b}/(mathfrak{a}capmathfrak{b}) by mathfrak{a}. Therefore we may also define the radical of mathfrak{g} as the sum of all the solvable ideals of mathfrak{g}, hence the radical of mathfrak{g} is unique. Secondly, as {0} is always a solvable ideal of mathfrak{g}, the radical of mathfrak{g} always exists.