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Difference between revisions of "Super-space"

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A [[Vector space|vector space]]  $  V $
 
A [[Vector space|vector space]]  $  V $
 
over a field  $  k $
 
over a field  $  k $
endowed with a  $  \mathbf Z / 2 $-
+
endowed with a  $  \mathbf Z / 2 $-grading  $  V = V _ {\overline{0} }  \oplus V _ {\overline{1} }  $.  
grading  $  V = V _ {\overline{0}\; }  \oplus V _ {\overline{1}\; }  $.  
+
The elements of the spaces  $  V _ {\overline{0} }  $
The elements of the spaces  $  V _ {\overline{0}\; }  $
+
and  $  V _ {\overline{1} }  $
and  $  V _ {\overline{1}\; }  $
 
 
are said to be even and odd, respectively; for  $  x \in V _ {i} $,  
 
are said to be even and odd, respectively; for  $  x \in V _ {i} $,  
 
the parity  $  p( x) $
 
the parity  $  p( x) $
 
is defined to be  $  i $
 
is defined to be  $  i $
$  ( i \in \mathbf Z / 2 = \{ \overline{0}\; , \overline{1}\; \} ) $.  
+
$  ( i \in \mathbf Z / 2 = \{ \overline{0} , \overline{1} \} ) $.  
 
Each super-space  $  V $
 
Each super-space  $  V $
 
has associated to it another super-space  $  \Pi ( V) $
 
has associated to it another super-space  $  \Pi ( V) $
such that  $  \Pi ( V) _ {i} = V _ {i+ \overline{1}\; }  $
+
such that  $  \Pi ( V) _ {i} = V _ {i+ \overline{1} }  $
 
$  ( i \in \mathbf Z / 2 ) $.  
 
$  ( i \in \mathbf Z / 2 ) $.  
 
The pair  $  ( m, n) $,  
 
The pair  $  ( m, n) $,  
where  $  m =  \mathop{\rm dim}  V _ {\overline{0}\; }  $,  
+
where  $  m =  \mathop{\rm dim}  V _ {\overline{0} }  $,  
$  n =  \mathop{\rm dim}  V _ {\overline{1}\; }  $,  
+
$  n =  \mathop{\rm dim}  V _ {\overline{1} }  $,  
 
is called the dimension of the super-space  $  V $.  
 
is called the dimension of the super-space  $  V $.  
 
The field  $  k $
 
The field  $  k $
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etc., is defined naturally. In particular, a linear mapping  $  \phi :  V \rightarrow W $
 
etc., is defined naturally. In particular, a linear mapping  $  \phi :  V \rightarrow W $
 
is even if  $  \phi ( V _ {i} ) \subset  W _ {i} $,  
 
is even if  $  \phi ( V _ {i} ) \subset  W _ {i} $,  
and odd if  $  \phi ( V _ {i} ) \subset  W _ {i+ \overline{1}\; }  $.  
+
and odd if  $  \phi ( V _ {i} ) \subset  W _ {i+ \overline{1} }  $.  
 
A homogeneous bilinear form  $  \beta :  V \otimes V \mapsto k $
 
A homogeneous bilinear form  $  \beta :  V \otimes V \mapsto k $
 
is said to be symmetric if
 
is said to be symmetric if
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$$
 
$$
  
All these concepts apply equally to  $  \mathbf Z / 2 $-
+
All these concepts apply equally to  $  \mathbf Z / 2 $-graded free modules  $  V $
graded free modules  $  V $
 
 
over an arbitrary commutative [[Superalgebra|superalgebra]]  $  C $.  
 
over an arbitrary commutative [[Superalgebra|superalgebra]]  $  C $.  
 
The basis in  $  V $
 
The basis in  $  V $

Latest revision as of 08:37, 16 June 2022


A vector space $ V $ over a field $ k $ endowed with a $ \mathbf Z / 2 $-grading $ V = V _ {\overline{0} } \oplus V _ {\overline{1} } $. The elements of the spaces $ V _ {\overline{0} } $ and $ V _ {\overline{1} } $ are said to be even and odd, respectively; for $ x \in V _ {i} $, the parity $ p( x) $ is defined to be $ i $ $ ( i \in \mathbf Z / 2 = \{ \overline{0} , \overline{1} \} ) $. Each super-space $ V $ has associated to it another super-space $ \Pi ( V) $ such that $ \Pi ( V) _ {i} = V _ {i+ \overline{1} } $ $ ( i \in \mathbf Z / 2 ) $. The pair $ ( m, n) $, where $ m = \mathop{\rm dim} V _ {\overline{0} } $, $ n = \mathop{\rm dim} V _ {\overline{1} } $, is called the dimension of the super-space $ V $. The field $ k $ is usually considered as a super-space of dimension $ ( 1, 0) $.

For two super-spaces $ V $ and $ W $, the structure of a super-space on the spaces $ V \oplus W $, $ \mathop{\rm Hom} _ {k} ( V, W) $, $ V ^ \star $, etc., is defined naturally. In particular, a linear mapping $ \phi : V \rightarrow W $ is even if $ \phi ( V _ {i} ) \subset W _ {i} $, and odd if $ \phi ( V _ {i} ) \subset W _ {i+ \overline{1} } $. A homogeneous bilinear form $ \beta : V \otimes V \mapsto k $ is said to be symmetric if

$$ \beta ( y, x) = (- 1) ^ {p( x) p( y)+ p( \beta )( p( x)+ p( y)) } \beta ( x, y), $$

and skew-symmetric if

$$ \beta ( y, x) = -(- 1) ^ {p( x) p( y)+ p( \beta )( p( x)+ p( y)) } \beta ( x, y). $$

All these concepts apply equally to $ \mathbf Z / 2 $-graded free modules $ V $ over an arbitrary commutative superalgebra $ C $. The basis in $ V $ is usually selected so that its first vectors are even and its last ones odd. Any endomorphism $ \phi $ of the module $ V $ is denoted in this basis by a block matrix

$$ \alpha = \left ( \begin{array}{cc} X & Y \\ Z & T \\ \end{array} \right ) , $$

where $ X \in M _ {n} ( C) $, $ T \in M _ {m} ( C) $, such that if $ \phi $ is even, then $ X $ and $ T $ consist of even elements and $ Y $ and $ Z $ consist of odd elements, whereas if $ \phi $ is odd, then $ X $ and $ T $ consist of odd elements and $ Y $ and $ Z $ consist of even elements (in the former case the matrix $ \alpha $ is even, in the latter, odd).

References

[1] F.A. Berezin, "Introduction to superanalysis" , Reidel (1987) (Translated from Russian)
[2] D.A. Leites (ed.) , Seminar on super-manifolds , Kluwer (1990)

Comments

References

[a1] F.A. Berezin, M.A. Shubin, "The Schrödinger equation" , Kluwer (1991) (Translated from Russian) (Supplement 3: D.A. Leites, Quantization and supermanifolds)
How to Cite This Entry:
Super-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Super-space&oldid=49616
This article was adapted from an original article by D.A. Leites (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article