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In mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables. Specifically, given a homogeneous polynomial, polarization produces a unique symmetric multilinear form from which the original polynomial can be recovered by evaluating along a certain diagonal.
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm. The ...
In representation theory, polarization is the maximal totally isotropic subspace of a certain skew-symmetric bilinear form on a Lie algebra.The notion of polarization plays an important role in construction of irreducible unitary representations of some classes of Lie groups by means of the orbit method [1] as well as in harmonic analysis on Lie groups and mathematical physics.
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry ...
Hodge structures have been generalized for all complex varieties (even if they are singular and non-complete) in the form of mixed Hodge structures, defined by Pierre Deligne (1970). A variation of Hodge structure is a family of Hodge structures parameterized by a manifold, first studied by Phillip Griffiths (1968).
In mathematics, the Jacobian variety J(C) of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of C , hence an abelian variety .
Polarization of an Abelian variety, in the mathematics of complex manifolds; Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables; Polarization identity, expresses an inner product in terms of its associated norm; Polarization (Lie algebra)
An algebraic form, or simply form, is a function defined by a homogeneous polynomial. [ notes 1 ] A binary form is a form in two variables. A form is also a function defined on a vector space , which may be expressed as a homogeneous function of the coordinates over any basis .