In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these.
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Semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e. , non-abelian Lie algebras whose only ideals are {0} and itself. Throughout the article, unless otherwise stated, is a finite-dimensional Lie algebra over a field of characteristic 0. The following conditions are equivalent: is semisimple the Killing form, κ(x,y) = tr(adad), is non-degenerate, has no non-zero abelian ideals, has no non-zero solvable ideals, The radical of is zero.
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Weight (representation theory)
In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F – a linear functional – or equivalently, a one dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups.
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Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication.
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Italic type
In typography, italic type is a cursive typeface based on a stylized form of calligraphic handwriting. Owing to the influence from calligraphy, such typefaces often slant slightly to the right. Different glyph shapes from roman type are also usually used—another influence from calligraphy. True italics are therefore distinct from oblique type, in which the font is merely distorted into a slanted orientation.
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G
G is the seventh letter in the ISO basic Latin alphabet.
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