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Bracket Algebra
In mathematics, a bracket algebra is an algebraic system that connects the notion of a supersymmetry algebra with a symbolic representation of projective invariants. Given that ''L'' is a proper signed alphabet and Super 'L''is the supersymmetric algebra, the bracket algebra Bracket 'L''of dimension ''n'' over the field ''K'' is the quotient of the algebra Brace obtained by imposing the congruence relations below, where ''w'', ''w, ..., ''w''" are any monomials in Super 'L'' # = 0 if length(''w'') ≠ ''n'' # ... = 0 whenever any positive letter ''a'' of ''L'' occurs more than ''n'' times in the monomial .... # Let ... be a monomial in Brace in which some positive letter ''a'' occurs more than ''n'' times, and let ''b'', ''c'', ''d'', ''e'', ..., ''f'', ''g'' be any letters in ''L''. See also * Bracket ring In mathematics invariant theory, the bracket ring is the subring of the ring of polynomials ''k'' 'x''11,...,''x'dn''generated by the ''d''-by-''d'' minors of a generic ''d' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supersymmetry Algebra
In theoretical physics, a supersymmetry algebra (or SUSY algebra) is a mathematical formalism for describing the relation between bosons and fermions. The supersymmetry algebra contains not only the Poincaré algebra and a compact subalgebra of internal symmetries, but also contains some fermionic supercharges, transforming as a sum of ''N'' real spinor representations of the Poincaré group. Such symmetries are allowed by the Haag–Łopuszański–Sohnius theorem. When ''N''>1 the algebra is said to have extended supersymmetry. The supersymmetry algebra is a semidirect sum of a central extension of the super-Poincaré algebra by a compact Lie algebra ''B'' of internal symmetries. Bosonic fields commute while fermionic fields anticommute. In order to have a transformation that relates the two kinds of fields, the introduction of a Z2-grading under which the even elements are bosonic and the odd elements are fermionic is required. Such an algebra is called a Lie superalgebra. J ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Invariant
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, their cross ratio is defined as : (A,B;C,D) = \frac where an orientation of the line determines the sign of each distance and the distance is measured as projected into Euclidean space. (If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.) The point ''D'' is the harmonic conjugate of ''C'' with respect to ''A'' and ''B'' precisely if the cross-ratio of the quadruple is −1, called the ''harmonic ratio''. The cross-ratio can therefore be regarded as measuring the quadruple's deviation from this ratio; hence the name ''anharmonic ratio''. The cross-ratio is preserved by linear fractional transformations. It is essentially the only projective inv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bracket Ring
In mathematics invariant theory, the bracket ring is the subring of the ring of polynomials ''k'' 'x''11,...,''x''''dn''generated by the ''d''-by-''d'' minors of a generic ''d''-by-''n'' matrix (''x''''ij''). The bracket ring may be regarded as the ring of polynomials on the image of a Grassmannian under the Plücker embedding. For given ''d'' ≤ ''n'' we define as formal variables the ''brackets'' 1 λ2 ... λ''d''with the λ taken from , subject to 1 λ2 ... λ''d''= − 2 λ1 ... λ''d''and similarly for other transpositions. The set Λ(''n'',''d'') of size \binom generates a polynomial ring ''K'' (''n'',''d'')over a field ''K''. There is a homomorphism Φ(''n'',''d'') from ''K'' (''n'',''d'')to the polynomial ring ''K'' 'x''''i'',''j''in ''nd'' indeterminates given by mapping 1 λ2 ... λ''d''to the determinant of the ''d'' by ''d'' matrix consisting of the columns of the ''x''''i'',''j'' indexed by the λ. The ''bracket ring'' ''B''(''n'',''d'') is the image of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant Theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are ''invariant'', under the transformations from a given linear group. For example, if we consider the action of the special linear group ''SLn'' on the space of ''n'' by ''n'' matrices by left multiplication, then the determinant is an invariant of this action because the determinant of ''A X'' equals the determinant of ''X'', when ''A'' is in ''SLn''. Introduction Let G be a group, and V a finite-dimensional vector space over a field k (which in classical invariant theory was usually assumed to be the complex numbers). A representation of G in V is a group homomorphism \pi:G \to GL(V), which induces a group action of G on V. If k /math> is the space of polynomial functions on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
