View source for Anisotropic kernel

The subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125401.png" /> of a semi-simple [[Algebraic group|algebraic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125402.png" />, defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125403.png" />, which is the commutator subgroup of the centralizer of a maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125404.png" />-split torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125405.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125406.png" />. The anisotropic kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125407.png" /> is a semi-simple [[Anisotropic group|anisotropic group]] defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125408.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125409.png" />. The concept of the anisotropic kernel plays an important role in the study of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254010.png" />-structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254011.png" /> [[#References|[1]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254012.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254014.png" /> is anisotropic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254015.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254016.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254017.png" /> is called quasi-split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254018.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Tits,   "Classification of algebraic simple groups" , ''Algebraic Groups and Discontinuous Subgroups'' , ''Proc. Symp. Pure Math.'' , '''9''' , Amer. Math. Soc.  (1966)  pp. 33–62  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Borel,   J. Tits,   "Groupes réductifs"  ''Publ. Math. IHES'' , '''27'''  (1965)  pp. 55–150  {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR></table>