Subalgebra
Subalgebra In universal algebra, a subalgebra of an algebra A is a subset ... every operation must be checked. The term subalgebra is also used in the context of ...
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Cartan subalgebra
Cartan subalgebra In mathematics, a Cartan subalgebra is a certain kind of subalgebra of a Lie algebra. The subalgebra $\backslash mathfrak\{h\}$ of a Lie ... math>\mathfrak{g} is a Cartan subalgebra if $\backslash mathfrak\{h\}$ is ...
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Image:Lorentz group subalgebra lattice.png
Image:Lorentz group subalgebra lattice.png Image File history File links ...
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Lorentz group
... math>X_1 generates a one-parameter subalgebra of parabolics SO(0,1), $X\_3$ generates a one-parameter subalgebra of boosts SO(1,1), $X\_4\; ...\; neq\; 0$) generates a one-parameter subalgebra of loxodromic transformations. (Strictly speaking the last ... math>X_1, X_2 generate an abelian subalgebra consisting entirely of parabolics, $X\_1,\; X\_3$ generate a nonabelian subalgebra isomorphic to the Lie algebra of ...
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Generalized Kac-Moody algebra
... that the degree zero piece (the Cartan subalgebra) is abelian They have a (Cartan) involution ... from the diagonal" so that the Cartan subalgebra is the diagonal elements. Conversely one can ... a Weyl group, Weyl character formula, Cartan subalgebra, roots, weights, and so on. Definition A ... The degree zero piece is an abelian subalgebra spanned by the elements h i and is called the Cartan subalgebra. Properties Most properties of generalized Kac- ...
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Octonion   (translated from French)
... note that each 7 line generates a subalgebra of \mathbb{O} isomorph with ... octonion of same co-ordinates in the subalgebra generated by octonions unit {\ 1, \ I ... easily that all the operations in the subalgebra of octonions the imaginary part is null ... of réels . In the same way the subalgebra of octonions all real dimensions except the ... of complexes . In the same way the subalgebra of octonionsall real dimensions except the ...
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Theorem of Stone-Weierstrass   (translated from French)
... whole of the polynomial functions forms a subalgebra of C([has,B]), and the theorem of Weierstrass approximation says that this subalgebra is dense in C([has,B]). The ... be considered. The crucial property, that the subalgebra must check, is that it separate the ... two points and if With is a subalgebra of the algebra of Banach C(X ... the polynomials on [has,B] form a subalgebra of C[has,B] which separates ...
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Quantum group
... math>t_{\lambda}, of the Cartan subalgebra satisfies $(t\_\{\backslash lambda\},h)\; =\; \backslash lambda(h\; ...\; math>\; for\; all\; h\; in\; the\; Cartan$subalgebra. There are various coassociative coproducts under which ... for the dual space to the Cartan subalgebra, and $\backslash \{\backslash mu\_j\backslash \}$ is the dual ... G can be used to generate a *-subalgebra of $C(G)$ which is ... in $C$ such that The *-subalgebra, $C\_0$, of $C$
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Clifford algebra
... Cℓ 0 (V,Q) forms a subalgebra of Cℓ(V,Q), called the even subalgebra. The piece Cℓ 1 (V,Q ... x2113;(V,Q) (it is not a subalgebra). This Z 2 -grading plays an important ... F is not 2 then the even subalgebra Cℓ 0 (V,Q) of a ... case, one can show that the even subalgebra of Cℓ n (C) is ...
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Stone–Weierstrass theorem
... set of all polynomial functions forms a subalgebra of C[a,b], and the content ... the Weierstrass approximation theorem is that this subalgebra is dense in C[a,b]. Stone ... out that the crucial property that a subalgebra must satisfy is that it separates points ... compact Hausdorff space and A is a subalgebra of C(K,R) which contains a ... the polynomials on [a,b] form a subalgebra of C[a,b] which separates ...
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