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Let E be a Euclidean space, (beta,alpha) be the dot product, and denote the reflection in the hyperplane P_alpha={beta in E|(beta,alpha)=0} by ...

The real part R[z] of a complex number z=x+iy is the real number not multiplying i, so R[x+iy]=x. In terms of z itself, R[z]=1/2(z+z^_), where z^_ is the complex conjugate of ...

That portion of mathematics dealing with functions of real variables. While this includes some portions of topology, it is most commonly used to distinguish that portion of ...

A number r is an nth root of unity if r^n=1 and a primitive nth root of unity if, in addition, n is the smallest integer of k=1, ..., n for which r^k=1.

The root lattice of a semisimple Lie algebra is the discrete lattice generated by the Lie algebra roots in h^*, the dual vector space to the Cartan subalgebra.

A reduced root system is a root system R satisfying the additional property that, if alpha in R, then the only multiples of alpha in R are +/-alpha.

A real vector space is a vector space whose field of scalars is the field of reals. A linear transformation between real vector spaces is given by a matrix with real entries ...

The nth roots of unity are roots e^(2piik/n) of the cyclotomic equation x^n=1, which are known as the de Moivre numbers. The notations zeta_k, epsilon_k, and epsilon_k, where ...

A principal nth root omega of unity is a root satisfying the equations omega^n=1 and sum_(i=0)^(n-1)omega^(ij)=0 for j=1, 2, ..., n. Therefore, every primitive root of unity ...

A simple root of a Lie algebra is a positive root that is not the sum of two positive roots.