Construction of a bilinear antisymmetric form in symplectic vector space

Eugeniusz Kusak

Warsaw University, Bialystok

Wojciech Leonczuk

Warsaw University, Bialystok

Michal Muzalewski

Warsaw University, Bialystok

Summary.

In this text we will present unpublished results by Eu\-ge\-niusz Ku\-sak. It contains an axiomatic description of the class of all spaces $\langle V$; $\perp_\xi \rangle$, where $V$ is a vector space over a field F, $\xi: V \times V \to F$ is a bilinear antisymmetric form i.e. $\xi(x,y) = -\xi(y,x)$ and $x \perp_\xi y $ iff $\xi(x,y) = 0$ for $x$, $y \in V$. It also contains an effective construction of bilinear antisymmetric form $\xi$ for given symplectic space $\langle V$; $\perp \rangle$ such that $\perp = \perp_\xi$. The basic tool used in this method is the notion of orthogonal projection J$(a,b,x)$ for $a,b,x \in V$. We should stress the fact that axioms of orthogonal and symplectic spaces differ only by one axiom, namely: $x\perp y+\varepsilon z \>\&\> y\perp z+\varepsilon x \Rightarrow z\perp x+\varepsilon y. $ For $\varepsilon=+1$ we get the axiom characterizing symplectic geometry. For $\varepsilon=-1$ we get the axiom on three perpendiculars characterizing orthogonal geometry - see [5].

Supported by RPBP.III-24.C6.

MML Identifier: SYMSP_1

Contents (PDF format)

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