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A [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911901.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911902.png" /> endowed with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911903.png" />-grading <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911904.png" />. The elements of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911906.png" /> are said to be even and odd, respectively; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911907.png" />, the parity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911908.png" /> is defined to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911909.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119010.png" />. Each super-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119011.png" /> has associated to it another super-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119013.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119014.png" />. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119017.png" />, is called the dimension of the super-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119018.png" />. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119019.png" /> is usually considered as a super-space of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119020.png" />.
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For two super-spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119022.png" />, the structure of a super-space on the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119025.png" />, etc., is defined naturally. In particular, a linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119026.png" /> is even if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119027.png" />, and odd if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119028.png" />. A homogeneous bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119029.png" /> is said to be symmetric if
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119030.png" /></td> </tr></table>
+
A [[Vector space|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
 
and skew-symmetric if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119031.png" /></td> </tr></table>
+
$$
 +
\beta ( y, x)  = -(- 1) ^ {p( x) p( y)+ p( \beta )( p( x)+ p( y)) } \beta ( x, y).
 +
$$
  
All these concepts apply equally to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119032.png" />-graded free modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119033.png" /> over an arbitrary commutative [[Superalgebra|superalgebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119034.png" />. The basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119035.png" /> is usually selected so that its first vectors are even and its last ones odd. Any endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119036.png" /> of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119037.png" /> is denoted in this basis by a block matrix
+
All these concepts apply equally to $  \mathbf Z / 2 $-graded free modules $  V $
 +
over an arbitrary commutative [[Superalgebra|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119038.png" /></td> </tr></table>
+
$$
 +
\alpha  = \left (
 +
\begin{array}{cc}
 +
X  & Y  \\
 +
Z  & T  \\
 +
\end{array}
 +
\right ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119040.png" />, such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119041.png" /> is even, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119043.png" /> consist of even elements and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119045.png" /> consist of odd elements, whereas if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119046.png" /> is odd, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119048.png" /> consist of odd elements and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119050.png" /> consist of even elements (in the former case the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119051.png" /> is even, in the latter, odd).
+
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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F.A. Berezin,  "Introduction to superanalysis" , Reidel  (1987)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.A. Leites (ed.) , ''Seminar on super-manifolds'' , Kluwer  (1990)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F.A. Berezin,  "Introduction to superanalysis" , Reidel  (1987)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.A. Leites (ed.) , ''Seminar on super-manifolds'' , Kluwer  (1990)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.A. Berezin,  M.A. Shubin,  "The Schrödinger equation" , Kluwer  (1991)  (Translated from Russian)  (Supplement 3: D.A. Leites, Quantization and supermanifolds)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.A. Berezin,  M.A. Shubin,  "The Schrödinger equation" , Kluwer  (1991)  (Translated from Russian)  (Supplement 3: D.A. Leites, Quantization and supermanifolds)</TD></TR></table>

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=11499
This article was adapted from an original article by D.A. Leites (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article