Semi-linear mapping

In linear algebra, a semilinear mapping is a mapping of a vector space over a body to another vector space over the same body, which is linear except for a body automorphism , so in this sense it is “almost” a linear mapping . In geometry , in the same sense, more general semilinear mappings between left vector spaces over possibly also different oblique bodies are defined as images that are linear except for one oblique body monomorphism. ${\displaystyle K}$ ${\displaystyle \alpha }$

Every linear mapping is semilinear. Then every semilinear mapping over a vector space (or left vector space  ) is even linear if the body (or oblique body) is the only automorphism that allows identity. All prime fields , the field of real numbers and all Euclidean fields , especially the real closed fields , have this property . A semilinear function (also semilinear form ) is a semilinear mapping of a - (left) vector space into the (oblique) body itself as a one-dimensional vector space. ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K}$

If fixed bases of the vector spaces are chosen , each semilinear mapping can be unambiguously as a sequential execution of a linear mapping , i.e. H. a matrix , and the application of the respective (skew) body automorphism to each coordinate.

The most important cases for applications outside of geometry in the narrower sense, e.g. for sesquilinear forms , are the semilinear mappings between complex spaces , i.e. between -vector spaces, with regard to complex conjugation . For these cases, the term described in the present article is also referred to as anti- linear mapping or conjugate linear mapping , in the projective case a bijective , semilinear self-mapping is then also called antiprojectivity , with these terms the mapping must be semi- linear, but not linear , with In other words: the associated body automorphism must not be the identical mapping. ${\displaystyle \mathbb {C} }$

In synthetic geometry, each semi-linear mapping provides a representation of the homogeneous part of a straight line mapping of an at least two-dimensional Desarguese affine geometry with more than two points on each straight line to another affine geometry or a matrix representation of an at least two-dimensional, Desarguese projective geometry to another projective Geometry in relation to a coordinate system that is fixed in the value and target space . Here, the morphism from the definition and the representation can also be an oblique body monomorphism, i.e. an injective ring homomorphism between oblique bodies. The image space can then also be a left vector space over a "larger" oblique body and the value space over a body that is isomorphic to a partial body . ${\displaystyle \alpha }$ ${\displaystyle L}$ ${\displaystyle L}$ ${\displaystyle K}$ ${\displaystyle \alpha (K)<L}$

Bijective , semilinear self -images of an at least two-dimensional, Desarguean affine or projective space are in this sense precisely the matrix representations for the collineations of this space, possibly together with an oblique body automorphism.

definition

An illustration of a - (left) vector space above the body (or skew) to a -Linksvektorraum is semi-linear mapping , if a (Skew-) Körperautomorphismus exists with which it satisfies the following two conditions. The following applies to everyone and everyone : ${\displaystyle f\colon V\longrightarrow W}$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle W}$ ${\displaystyle \alpha \in \mathrm {Aut} (K)}$ ${\displaystyle x,y\in V}$ ${\displaystyle \lambda \in K}$

1. Additivity:, in other words: is a group homomorphism of the Abelian group . ${\displaystyle f(x+y)=f(x)+f(y)}$ ${\displaystyle f}$ ${\displaystyle (V,+)}$
2. ${\displaystyle f(\lambda \cdot x)=\alpha (\lambda )\cdot f(x).}$

presentation

It is a division ring and , be - or  dimensional left vector spaces over . Let be a semi-linear map. Then for an arbitrary vector space basis of and an arbitrary vector space basis of there exist unique matrices and an inclined body automorphism , so that for any coordinate vector in the coordinate representation with respect to the base ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle n}$ ${\displaystyle m}$ ${\displaystyle K}$ ${\displaystyle f\colon V\rightarrow W}$ ${\displaystyle B_{V}=(v_{1},v_{2},\ldots v_{n})}$ ${\displaystyle V}$ ${\displaystyle B_{W}=(w_{1},w_{2},\ldots w_{m})}$ ${\displaystyle W}$ ${\displaystyle n\times m}$ ${\displaystyle A,B}$ ${\displaystyle \alpha }$ ${\displaystyle {\overrightarrow {v}}\in V}$ ${\displaystyle B_{V}}$

${\displaystyle {\overrightarrow {w}}=\alpha \left(A\cdot {\overrightarrow {v}}\right)}$ applies or ${\displaystyle {\overrightarrow {w}}=B\cdot \alpha ({\overrightarrow {v}}),}$

when the image vector is represented as a coordinate vector with respect to the base . The matrices , are defined by the bases and said relationship to uniquely determined in each case, but different from each other in general. The same automorphism can be used in both representations, regardless of the bases selected. It is clearly determined by the relationship to , provided that the semilinear mapping applies to the image . Compare also collineation . ${\displaystyle {\overrightarrow {w}}=f({\overrightarrow {v}})\in W}$ ${\displaystyle B_{W}}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle f}$ ${\displaystyle \alpha }$ ${\displaystyle f}$ ${\displaystyle f(V)\neq \{0\}}$

Examples and counterexamples

• Let there be vector spaces over the complex numbers. An illustration ${\displaystyle V,W}$

${\displaystyle S\colon \;V\times W\to \mathbb {C} ,\quad (v,w)\mapsto S(v,w)=\langle v,w\rangle }$

is a sesquilinear form if and only if the mapping for every fixed vector is linear and the mapping for every fixed vector is semilinear with the conjugation as body automorphism. ${\displaystyle w\mapsto S(v,w)}$ ${\displaystyle v\in V}$ ${\displaystyle v\mapsto S(v,w)}$ ${\displaystyle w\in V}$

• Be it . The nonidentical, involutorial automorphism ${\displaystyle K=\mathbb {Q} ({\sqrt {2}})}$

${\displaystyle \alpha \colon a+{\sqrt {2}}b\mapsto a-{\sqrt {2}}b;\;a,b\in \mathbb {Q} }$

together with an arbitrary matrix induces a semilinear mapping ${\displaystyle n\times n}$ ${\displaystyle A}$

${\displaystyle f({\overrightarrow {v}})=A\cdot \alpha ({\overrightarrow {v}})}$

des -vector space with respect to its standard basis. If regular, this mapping geometrically represents a collineation of the affine space above . ${\displaystyle K}$ ${\displaystyle K^{n}}$ ${\displaystyle A}$ ${\displaystyle K^{n}}$

• A collineation of a projective translation plane of Lenz class IV cannot be represented by a semilinear mapping because the plane cannot be coordinated by an oblique body.
• An antiunitary operator is a semilinear mapping on a complex Hilbert space with respect to the complex conjugation, which results from the execution of a unitary operator and the coordinate-wise complex conjugation. Alternatively, anti-unit operators can be characterized as semilinear, surjective isometrics . As symmetries, they play a role in the mathematical description of quantum mechanics , albeit less important than unitary operators ( see also Wigner's theorem ). The time-reversal is an example of such symmetry.

The group of semilinear maps

General semilinear group

The group of invertible semilinear maps of a vector space is called the general semilinear group . It can be seen as a semi-direct product ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle \Gamma L(V)}$

${\displaystyle \Gamma L(V)=GL(V)\rtimes \operatorname {Gal} (K/k)}$

the general linear group with the Galois group of as the field extension of a prime field . (The second factor is precisely the field automorphisms of , because every field automorphism must fix the prime field.) ${\displaystyle GL(V)}$ ${\displaystyle K}$ ${\displaystyle k\subset K}$ ${\displaystyle K}$

Projective semilinear group

The projective semilinear group of a vector space is the semidirect product ${\displaystyle K}$ ${\displaystyle V}$

${\displaystyle P\Gamma L(V)=PGL(V)\rtimes \operatorname {Gal} (K/k)}$,

the projective linear group with the group of body automorphisms. It works on the projective space . ${\displaystyle PGL(V)}$ ${\displaystyle P(V)}$

generalization

If, more generally, is a ring and an endomorphism , then an additive mapping is called -semilinear, if ${\displaystyle R}$ ${\displaystyle \sigma \colon R\to R}$ ${\displaystyle f\colon V\to W}$ ${\displaystyle \sigma }$

${\displaystyle f(\lambda v)=\sigma (\lambda )\cdot f(v)}$

for everyone and applies. ${\displaystyle \lambda \in R}$ ${\displaystyle v\in V}$

See also

• Shape (algebra)

literature

• Günter Pickert : Analytical Geometry . 6th, revised edition. Academic publishing company Geest & Portig, Leipzig 1967.
• Hermann Schaal: Linear Algebra and Analytical Geometry, Volume II . 2nd revised edition. Vieweg, Braunschweig 1980, ISBN 3-528-13057-1 .
• Günter Scheja, Uwe Storch: Textbook of Algebra: including linear algebra . 2., revised. and exp. Edition. Teubner, Stuttgart 1994, ISBN 3-519-12203-0 ( table of contents [accessed on January 14, 2012]).
• Uwe Storch, Hartmut Wiebe: Textbook of Mathematics, Volume II: Linear Algebra . In: Textbook of mathematics for mathematicians, computer scientists and physicists: in 4 volumes . BI-Wissenschafts-Verlag, Mannheim / Leipzig / Vienna / Zurich 1990, ISBN 3-411-14101-8 .

Individual evidence

1. ↑ ^{a b c d} Scheja and Storch (1994)
2. ↑ Storch, Wiebe (1990)
3. ↑ Schaal (1980) p. 198
4. ^ Edson de Faria, Welinton de Melo: Mathematical Aspects of Quantum Field Theory . 1st edition. Cambridge University Press , 2010, ISBN 978-0-521-11577-3 , pp. 19 .