Metabolic square space

In mathematics , metabolic square spaces are a term from algebra , more specifically from the theory of square forms .

definition

A square room is metabolically when a subspace with are, ${\displaystyle (V,q)}$ ${\displaystyle U\subset V}$ ${\displaystyle U=U^{\perp }}$

Example: the hyperbolic plane

Be a body of characteristic . The square space with and (or equivalent ) is called the hyperbolic plane (not to be confused with the hyperbolic plane of hyperbolic geometry). ${\displaystyle K}$ ${\displaystyle \operatorname {char} (K)\neq 2}$ ${\displaystyle (V,q)}$ ${\displaystyle V=K^{2}}$ ${\displaystyle q(x,y)=xy}$ ${\displaystyle q(x,y)=x^{2}-y^{2}}$

classification

Be a body of characteristic . Then every metabolic square space is a direct sum of hyperbolic levels. ${\displaystyle K}$ ${\displaystyle \operatorname {char} (K)\neq 2}$

literature

• John Milnor , Dale Husemöller : Symmetric bilinear forms . Springer, 1973. ISBN 3-540-06009-X