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Popular T-Girlz 2 2015 CX WOW O Shemale Amazon Destroys His Face With Her Big Booty 2015 pure-ts.com Smothered By A Shemale 2015 CX WOW O Transsexual Prostitutes 74 2013 Devil's Film Facial Bottom 1 DRO TS Femdom 2016 Severe Sex DRO TS Pussy Hunters 24089 2012 kink.com In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory (such as singularity theory). Finally, root systems are important for their own sake, as in spectral graph theory.[1]



Contents 1 Definitions and examples 1.1 Definition 1.2 Weyl group 1.3 Rank one example 1.4 Rank two examples 1.5 Root systems arising from semisimple Lie algebras 2 History 3 Elementary consequences of the root system axioms 4 Positive roots and simple roots 5 Dual root system, coroots, and integral elements 5.1 The dual root system 5.2 Integral elements 6 Classification of root systems by Dynkin diagrams 6.1 Constructing the Dynkin diagram 6.2 Classifying root systems 7 Weyl chambers and the Weyl group 8 Root systems and Lie theory 9 Properties of the irreducible root systems 10 Explicit construction of the irreducible root systems 10.1 An 10.2 Bn 10.3 Cn 10.4 Dn 10.5 E6, E7, E8 10.6 F4 10.7 G2 11 The root poset 12 See also 13 Notes 14 References 15 Further reading 16 External links Definitions and examples The six vectors of the root system A2. As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right; call them roots. These vectors span the whole space. If you consider the line perpendicular to any root, say , then the reflection of R2 in that line sends any other root, say a, to another root. Moreover, the root to which it is sent equals a + n, where n is an integer (in this case, n equals 1). These six vectors satisfy the following definition, and therefore they form a root system; this one is known as A2. Definition Let E be a finite-dimensional Euclidean vector space, with the standard Euclidean inner product denoted by {\displaystyle (\cdot ,\cdot )}(\cdot ,\cdot ). A root system {\displaystyle \Phi }\Phi in E is a finite set of non-zero vectors (called roots) that satisfy the following conditions:[2][3] The roots span E. The only scalar multiples of a root {\displaystyle \alpha \in \Phi }\alpha \in \Phi that belong to {\displaystyle \Phi }\Phi are {\displaystyle \alpha }\alpha itself and {\displaystyle -\alpha }-\alpha . For every root {\displaystyle \alpha \in \Phi }\alpha \in \Phi , the set {\displaystyle \Phi }\Phi is closed under reflection through the hyperplane perpendicular to {\displaystyle \alpha }\alpha . (Integrality) If {\displaystyle \alpha }\alpha and {\displaystyle \beta }\beta are roots in {\displaystyle \Phi }\Phi , then the projection of {\displaystyle \beta }\beta onto the line through {\displaystyle \alpha }\alpha is an integer or half-integer multiple of {\displaystyle \alpha }\alpha . An equivalent way of writing conditions 3 and 4 is as follows: For any two roots {\displaystyle \alpha ,\beta \in \Phi }{\displaystyle \alpha ,\beta \in \Phi }, the set {\displaystyle \Phi }\Phi contains the element {\displaystyle \sigma _{\alpha }(\beta ):=\beta -2{\frac {(\alpha ,\beta )}{(\alpha ,\alpha )}}\alpha .}{\displaystyle \sigma _{\alpha }(\beta ):=\beta -2{\frac {(\alpha ,\beta )}{(\alpha ,\alpha )}}\alpha .} For any two roots {\displaystyle \alpha ,\beta \in \Phi }{\displaystyle \alpha ,\beta \in \Phi }, the number {\displaystyle \langle \beta ,\alpha \rangle :=2{\frac {(\alpha ,\beta )}{(\alpha ,\alpha )}}}{\displaystyle \langle \beta ,\alpha \rangle :=2{\frac {(\alpha ,\beta )}{(\alpha ,\alpha )}}} is an integer. Some authors only include conditions 13 in the definition of a root system.[4] In this context, a root system that also satisfies the integrality condition is known as a crystallographic root system.[5] Other authors omit condition 2; then they call root systems satisfying condition 2 reduced.[6] In this article, all root systems are assumed to be reduced and crystallographic. In view of property 3, the integrality condition is equivalent to stating that and its reflection sa() differ by an integer multiple of a. Note that the operator {\displaystyle \langle \cdot ,\cdot \rangle \colon \Phi \times \Phi \to \mathbb {Z} }\langle \cdot ,\cdot \rangle \colon \Phi \times \Phi \to \mathbb {Z} defined by property 4 is not an inner product. It is not necessarily symmetric and is linear only in the first argument. Rank-2 root systems Root system A1 + A1 Root system D2 Root system {\displaystyle A_{1}\times A_{1}}A_{1}\times A_{1} Dyn-node n1.pngDyn-2.pngDyn-node n2.png Root system {\displaystyle D_{2}}D_{2} Dyn2-nodes.png Root system A2 Root system G2 Root system {\displaystyle A_{2}}A_{2} Dyn2-node n1.pngDyn2-3.pngDyn2-node n2.png Root system {\displaystyle G_{2}}G_{2} Dyn2-nodeg n1.pngDyn2-6a.pngDyn2-node n2.png Root system B2 Root system C2 Root system {\displaystyle B_{2}}B_{2} Dyn2-nodeg n1.pngDyn2-4a.pngDyn2-node n2.png Root system {\displaystyle C_{2}}C_{2} Dyn2-node n1.pngDyn2-4b.pngDyn2-nodeg n2.png The rank of a root system F is the dimension of E. Two root systems may be combined by regarding the Euclidean spaces they span as mutually orthogonal subspaces of a common Euclidean space. A root system which does not arise from such a combination, such as the systems A2, B2, and G2 pictured to the right, is said to be irreducible. Two root systems (E1, F1) and (E2, F2) are called isomorphic if there is an invertible linear transformation E1 ? E2 which sends F1 to F2 such that for each pair of roots, the number {\displaystyle \langle x,y\rangle }\langle x,y\rangle is preserved.[7] The root lattice of a root system F is the Z-submodule of E generated by F. It is a lattice in E. Weyl group Main article: Weyl group The Weyl group of the {\displaystyle A_{2}}A_{2} root system is the symmetry group of an equilateral triangle The group of isometries of E generated by reflections through hyperplanes associated to the roots of F is called the Weyl group of F. As it acts faithfully on the finite set F, the Weyl group is always finite. The reflection planes are the hyperplanes perpendicular to the roots, indicated for {\displaystyle A_{2}}A_{2} by dashed lines in the figure. The Weyl group is the symmetry group of an equilateral triangle, which has six elements. In this case, the Weyl group is not the full symmetry group of the root system (e.g., a 60-degree rotation is a symmetry of the root system but not an element of the Weyl group


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