In mathematics, the Springer resolution is a
resolution of the variety of
nilpotent elements in a
semisimple Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
, or the
unipotent elements of a reductive algebraic group, introduced by
Tonny Albert Springer in 1969.
The fibers of this resolution are called Springer fibers.
If ''U'' is the variety of unipotent elements in a
reductive group ''G'', and ''X'' the variety of
Borel subgroups ''B'', then the Springer resolution of ''U'' is the variety of pairs (''u'',''B'') of ''U''×''X'' such that ''u'' is in the Borel subgroup ''B''. The map to ''U'' is the projection to the first factor. The Springer resolution for Lie algebras is similar, except that ''U'' is replaced by the nilpotent elements of the Lie algebra of ''G'' and ''X'' replaced by the variety of Borel subalgebras.
The Grothendieck–Springer resolution is defined similarly, except that ''U'' is replaced by the whole group ''G'' (or the whole Lie algebra of ''G''). When restricted to the unipotent elements of ''G'' it becomes the Springer resolution.
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]
Examples
When ''G=SL(2)'', the Lie algebra Springer resolution is ''T
*P
1 → n'', where ''n'' are the nilpotent elements of ''sl(2)''. In this example, ''n'' are the matrices ''x'' with ''tr(x
2)=0'', which is a two dimensional conical subvariety of ''sl(2)''. ''n'' has a unique singular point ''0'', the fibre above which in the Springer resolution is the zero section ''P
1 ''.
References
Lie algebras
Singularity theory
Algebraic groups
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