| 词条 | Springer resolution |
| 释义 |
In mathematics, the Springer resolution is a resolution of the variety of nilpotent elements in a semisimple Lie algebra,[1][2] or the unipotent elements of a reductive algebraic group, introduced by Tonny Albert Springer in 1969.[3] The fibers of this resolution are called Springer fibers.[4] 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.[5] 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.[6][7] References1. ^{{citation|mr=1433132|last1=Chriss|first1= Neil|authorlink1=Neil Chriss|last2= Ginzburg|first2= Victor|authorlink2=Victor Ginzburg | title=Representation theory and complex geometry|publisher=Birkhäuser Boston, Inc.|place= Boston, MA|year= 1997|isbn= 0-8176-3792-3 |url=https://books.google.com/books?id=lwS59rR78eIC&dq}} {{abstract-algebra-stub}}2. ^{{citation|mr=0739636|last1=Dolgachev|first1= Igor| authorlink1=Igor Dolgachev |last2= Goldstein|first2= Norman|title=On the Springer resolution of the minimal unipotent conjugacy class|journal=Journal of Pure and Applied Algebra|volume= 32 |year=1984|issue= 1|pages= 33–47|doi=10.1016/0022-4049(84)90012-4}} 3. ^{{citation|mr=0263830|last=Springer|first=Tonny A.| authorlink=T. A. Springer|chapter=The unipotent variety of a semi-simple group|year= 1969 |title=Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) |pages=373–391 |publisher=Oxford Univ. Press, London|url=https://books.google.com/books?ei=m5T1Tbz_A8TmiALD2KWUBw|isbn=978-0-19-635281-7}} 4. ^{{citation|mr=1649626|last=Ginzburg|first= Victor|authorlink=Victor Ginzburg|chapter=Geometric methods in the representation theory of Hecke algebras and quantum groups|series= NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences|volume= 514|title= Representation theories and algebraic geometry (Montreal, PQ, 1997)|pages= 127–183|publisher=Kluwer Acad. Publ., Dordrecht|year= 1998|isbn=0-7923-5193-2|arxiv= math/9802004|bibcode=1998math......2004G}} 5. ^{{citation|last=Springer|first= Tonny A.|authorlink=T. A. Springer |title=Trigonometric sums, Green functions of finite groups and representations of Weyl groups|journal=Inventiones Mathematicae|volume= 36 |year=1976|pages=173–207|mr=0442103 |doi=10.1007/BF01390009|bibcode=1976InMat..36..173S}} 6. ^{{citation|mr=0352279|last=Steinberg|first= Robert|authorlink=Robert Steinberg|title=Conjugacy classes in algebraic groups|series=Lecture Notes in Mathematics|volume=366|publisher= Springer-Verlag|place= Berlin-New York|year= 1974|doi=10.1007/BFb0067854|isbn=978-3-540-06657-6}} 7. ^{{citation|mr=0430094|last=Steinberg|first= Robert|authorlink=Robert Steinberg |title=On the desingularization of the unipotent variety|journal=Inventiones Mathematicae |volume=36 |year=1976|pages= 209–224|doi=10.1007/BF01390010|bibcode=1976InMat..36..209S}} 3 : Lie algebras|Singularity theory|Algebraic groups |
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