On the Cohomology of Quiver Grassmannians for Acyclic Quivers
Abstract For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_{q}^{-}(\mathfrak {g})$, where $U_{q}^{-}(\mathfrak {g})$ is the negative half of the quantized enveloping algebra associated with the quiver. In or...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Bi, Yingjin [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2021 |
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| Schlagwörter: |
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| Anmerkung: |
© The Author(s), under exclusive licence to Springer Nature B.V. 2021 |
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| Übergeordnetes Werk: |
Enthalten in: Algebras and representation theory - Dordrecht [u.a.] : Springer Science + Business Media B.V, 1998, 25(2021), 5 vom: 29. Mai, Seite 1323-1343 |
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| Übergeordnetes Werk: |
volume:25 ; year:2021 ; number:5 ; day:29 ; month:05 ; pages:1323-1343 |
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| DOI / URN: |
10.1007/s10468-021-10069-3 |
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| Katalog-ID: |
SPR048098558 |
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| 520 | |a Abstract For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_{q}^{-}(\mathfrak {g})$, where $U_{q}^{-}(\mathfrak {g})$ is the negative half of the quantized enveloping algebra associated with the quiver. In order to achieve this goal, we study the cohomology of quiver Grassmannians by Lusztig’s category. As a consequence, we describe explicitly the Poincaré polynomials of rigid quiver Grassmannians in terms of the coefficients of dual canonical bases, which are viewed as elements of quantum shuffle algebras. By this result, we give another proof of the odd cohomology vanishing theorem for quiver Grassmanians. Meanwhile, for Dynkin quivers, we show that the Poincaré polynomials of rigid quiver Grassmannians are the coefficients of dual PBW bases of the algebra $U_{q}^{-}(\mathfrak {g})$. | ||
| 650 | 4 | |a Quiver Grassmannian |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Dual canonical basis |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Poincaré polynomial |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Quantum group |7 (dpeaa)DE-He213 | |
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10.1007/s10468-021-10069-3 doi (DE-627)SPR048098558 (SPR)s10468-021-10069-3-e DE-627 ger DE-627 rakwb eng Bi, Yingjin verfasserin (orcid)0000-0003-0153-3274 aut On the Cohomology of Quiver Grassmannians for Acyclic Quivers 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature B.V. 2021 Abstract For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_{q}^{-}(\mathfrak {g})$, where $U_{q}^{-}(\mathfrak {g})$ is the negative half of the quantized enveloping algebra associated with the quiver. In order to achieve this goal, we study the cohomology of quiver Grassmannians by Lusztig’s category. As a consequence, we describe explicitly the Poincaré polynomials of rigid quiver Grassmannians in terms of the coefficients of dual canonical bases, which are viewed as elements of quantum shuffle algebras. By this result, we give another proof of the odd cohomology vanishing theorem for quiver Grassmanians. Meanwhile, for Dynkin quivers, we show that the Poincaré polynomials of rigid quiver Grassmannians are the coefficients of dual PBW bases of the algebra $U_{q}^{-}(\mathfrak {g})$. Quiver Grassmannian (dpeaa)DE-He213 Dual canonical basis (dpeaa)DE-He213 Poincaré polynomial (dpeaa)DE-He213 Quantum group (dpeaa)DE-He213 Enthalten in Algebras and representation theory Dordrecht [u.a.] : Springer Science + Business Media B.V, 1998 25(2021), 5 vom: 29. Mai, Seite 1323-1343 (DE-627)320501647 (DE-600)2012288-3 1572-9079 nnns volume:25 year:2021 number:5 day:29 month:05 pages:1323-1343 https://dx.doi.org/10.1007/s10468-021-10069-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 25 2021 5 29 05 1323-1343 |
| spelling |
10.1007/s10468-021-10069-3 doi (DE-627)SPR048098558 (SPR)s10468-021-10069-3-e DE-627 ger DE-627 rakwb eng Bi, Yingjin verfasserin (orcid)0000-0003-0153-3274 aut On the Cohomology of Quiver Grassmannians for Acyclic Quivers 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature B.V. 2021 Abstract For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_{q}^{-}(\mathfrak {g})$, where $U_{q}^{-}(\mathfrak {g})$ is the negative half of the quantized enveloping algebra associated with the quiver. In order to achieve this goal, we study the cohomology of quiver Grassmannians by Lusztig’s category. As a consequence, we describe explicitly the Poincaré polynomials of rigid quiver Grassmannians in terms of the coefficients of dual canonical bases, which are viewed as elements of quantum shuffle algebras. By this result, we give another proof of the odd cohomology vanishing theorem for quiver Grassmanians. Meanwhile, for Dynkin quivers, we show that the Poincaré polynomials of rigid quiver Grassmannians are the coefficients of dual PBW bases of the algebra $U_{q}^{-}(\mathfrak {g})$. Quiver Grassmannian (dpeaa)DE-He213 Dual canonical basis (dpeaa)DE-He213 Poincaré polynomial (dpeaa)DE-He213 Quantum group (dpeaa)DE-He213 Enthalten in Algebras and representation theory Dordrecht [u.a.] : Springer Science + Business Media B.V, 1998 25(2021), 5 vom: 29. Mai, Seite 1323-1343 (DE-627)320501647 (DE-600)2012288-3 1572-9079 nnns volume:25 year:2021 number:5 day:29 month:05 pages:1323-1343 https://dx.doi.org/10.1007/s10468-021-10069-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 25 2021 5 29 05 1323-1343 |
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10.1007/s10468-021-10069-3 doi (DE-627)SPR048098558 (SPR)s10468-021-10069-3-e DE-627 ger DE-627 rakwb eng Bi, Yingjin verfasserin (orcid)0000-0003-0153-3274 aut On the Cohomology of Quiver Grassmannians for Acyclic Quivers 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature B.V. 2021 Abstract For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_{q}^{-}(\mathfrak {g})$, where $U_{q}^{-}(\mathfrak {g})$ is the negative half of the quantized enveloping algebra associated with the quiver. In order to achieve this goal, we study the cohomology of quiver Grassmannians by Lusztig’s category. As a consequence, we describe explicitly the Poincaré polynomials of rigid quiver Grassmannians in terms of the coefficients of dual canonical bases, which are viewed as elements of quantum shuffle algebras. By this result, we give another proof of the odd cohomology vanishing theorem for quiver Grassmanians. Meanwhile, for Dynkin quivers, we show that the Poincaré polynomials of rigid quiver Grassmannians are the coefficients of dual PBW bases of the algebra $U_{q}^{-}(\mathfrak {g})$. Quiver Grassmannian (dpeaa)DE-He213 Dual canonical basis (dpeaa)DE-He213 Poincaré polynomial (dpeaa)DE-He213 Quantum group (dpeaa)DE-He213 Enthalten in Algebras and representation theory Dordrecht [u.a.] : Springer Science + Business Media B.V, 1998 25(2021), 5 vom: 29. Mai, Seite 1323-1343 (DE-627)320501647 (DE-600)2012288-3 1572-9079 nnns volume:25 year:2021 number:5 day:29 month:05 pages:1323-1343 https://dx.doi.org/10.1007/s10468-021-10069-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 25 2021 5 29 05 1323-1343 |
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10.1007/s10468-021-10069-3 doi (DE-627)SPR048098558 (SPR)s10468-021-10069-3-e DE-627 ger DE-627 rakwb eng Bi, Yingjin verfasserin (orcid)0000-0003-0153-3274 aut On the Cohomology of Quiver Grassmannians for Acyclic Quivers 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature B.V. 2021 Abstract For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_{q}^{-}(\mathfrak {g})$, where $U_{q}^{-}(\mathfrak {g})$ is the negative half of the quantized enveloping algebra associated with the quiver. In order to achieve this goal, we study the cohomology of quiver Grassmannians by Lusztig’s category. As a consequence, we describe explicitly the Poincaré polynomials of rigid quiver Grassmannians in terms of the coefficients of dual canonical bases, which are viewed as elements of quantum shuffle algebras. By this result, we give another proof of the odd cohomology vanishing theorem for quiver Grassmanians. Meanwhile, for Dynkin quivers, we show that the Poincaré polynomials of rigid quiver Grassmannians are the coefficients of dual PBW bases of the algebra $U_{q}^{-}(\mathfrak {g})$. Quiver Grassmannian (dpeaa)DE-He213 Dual canonical basis (dpeaa)DE-He213 Poincaré polynomial (dpeaa)DE-He213 Quantum group (dpeaa)DE-He213 Enthalten in Algebras and representation theory Dordrecht [u.a.] : Springer Science + Business Media B.V, 1998 25(2021), 5 vom: 29. Mai, Seite 1323-1343 (DE-627)320501647 (DE-600)2012288-3 1572-9079 nnns volume:25 year:2021 number:5 day:29 month:05 pages:1323-1343 https://dx.doi.org/10.1007/s10468-021-10069-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 25 2021 5 29 05 1323-1343 |
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10.1007/s10468-021-10069-3 doi (DE-627)SPR048098558 (SPR)s10468-021-10069-3-e DE-627 ger DE-627 rakwb eng Bi, Yingjin verfasserin (orcid)0000-0003-0153-3274 aut On the Cohomology of Quiver Grassmannians for Acyclic Quivers 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature B.V. 2021 Abstract For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_{q}^{-}(\mathfrak {g})$, where $U_{q}^{-}(\mathfrak {g})$ is the negative half of the quantized enveloping algebra associated with the quiver. In order to achieve this goal, we study the cohomology of quiver Grassmannians by Lusztig’s category. As a consequence, we describe explicitly the Poincaré polynomials of rigid quiver Grassmannians in terms of the coefficients of dual canonical bases, which are viewed as elements of quantum shuffle algebras. By this result, we give another proof of the odd cohomology vanishing theorem for quiver Grassmanians. Meanwhile, for Dynkin quivers, we show that the Poincaré polynomials of rigid quiver Grassmannians are the coefficients of dual PBW bases of the algebra $U_{q}^{-}(\mathfrak {g})$. Quiver Grassmannian (dpeaa)DE-He213 Dual canonical basis (dpeaa)DE-He213 Poincaré polynomial (dpeaa)DE-He213 Quantum group (dpeaa)DE-He213 Enthalten in Algebras and representation theory Dordrecht [u.a.] : Springer Science + Business Media B.V, 1998 25(2021), 5 vom: 29. Mai, Seite 1323-1343 (DE-627)320501647 (DE-600)2012288-3 1572-9079 nnns volume:25 year:2021 number:5 day:29 month:05 pages:1323-1343 https://dx.doi.org/10.1007/s10468-021-10069-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 25 2021 5 29 05 1323-1343 |
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Bi, Yingjin misc Quiver Grassmannian misc Dual canonical basis misc Poincaré polynomial misc Quantum group On the Cohomology of Quiver Grassmannians for Acyclic Quivers |
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On the Cohomology of Quiver Grassmannians for Acyclic Quivers Quiver Grassmannian (dpeaa)DE-He213 Dual canonical basis (dpeaa)DE-He213 Poincaré polynomial (dpeaa)DE-He213 Quantum group (dpeaa)DE-He213 |
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on the cohomology of quiver grassmannians for acyclic quivers |
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On the Cohomology of Quiver Grassmannians for Acyclic Quivers |
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Abstract For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_{q}^{-}(\mathfrak {g})$, where $U_{q}^{-}(\mathfrak {g})$ is the negative half of the quantized enveloping algebra associated with the quiver. In order to achieve this goal, we study the cohomology of quiver Grassmannians by Lusztig’s category. As a consequence, we describe explicitly the Poincaré polynomials of rigid quiver Grassmannians in terms of the coefficients of dual canonical bases, which are viewed as elements of quantum shuffle algebras. By this result, we give another proof of the odd cohomology vanishing theorem for quiver Grassmanians. Meanwhile, for Dynkin quivers, we show that the Poincaré polynomials of rigid quiver Grassmannians are the coefficients of dual PBW bases of the algebra $U_{q}^{-}(\mathfrak {g})$. © The Author(s), under exclusive licence to Springer Nature B.V. 2021 |
| abstractGer |
Abstract For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_{q}^{-}(\mathfrak {g})$, where $U_{q}^{-}(\mathfrak {g})$ is the negative half of the quantized enveloping algebra associated with the quiver. In order to achieve this goal, we study the cohomology of quiver Grassmannians by Lusztig’s category. As a consequence, we describe explicitly the Poincaré polynomials of rigid quiver Grassmannians in terms of the coefficients of dual canonical bases, which are viewed as elements of quantum shuffle algebras. By this result, we give another proof of the odd cohomology vanishing theorem for quiver Grassmanians. Meanwhile, for Dynkin quivers, we show that the Poincaré polynomials of rigid quiver Grassmannians are the coefficients of dual PBW bases of the algebra $U_{q}^{-}(\mathfrak {g})$. © The Author(s), under exclusive licence to Springer Nature B.V. 2021 |
| abstract_unstemmed |
Abstract For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_{q}^{-}(\mathfrak {g})$, where $U_{q}^{-}(\mathfrak {g})$ is the negative half of the quantized enveloping algebra associated with the quiver. In order to achieve this goal, we study the cohomology of quiver Grassmannians by Lusztig’s category. As a consequence, we describe explicitly the Poincaré polynomials of rigid quiver Grassmannians in terms of the coefficients of dual canonical bases, which are viewed as elements of quantum shuffle algebras. By this result, we give another proof of the odd cohomology vanishing theorem for quiver Grassmanians. Meanwhile, for Dynkin quivers, we show that the Poincaré polynomials of rigid quiver Grassmannians are the coefficients of dual PBW bases of the algebra $U_{q}^{-}(\mathfrak {g})$. © The Author(s), under exclusive licence to Springer Nature B.V. 2021 |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">SPR048098558</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230509111519.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">220915s2021 xx |||||o 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s10468-021-10069-3</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR048098558</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s10468-021-10069-3-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Bi, Yingjin</subfield><subfield code="e">verfasserin</subfield><subfield code="0">(orcid)0000-0003-0153-3274</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">On the Cohomology of Quiver Grassmannians for Acyclic Quivers</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2021</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© The Author(s), under exclusive licence to Springer Nature B.V. 2021</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_{q}^{-}(\mathfrak {g})$, where $U_{q}^{-}(\mathfrak {g})$ is the negative half of the quantized enveloping algebra associated with the quiver. In order to achieve this goal, we study the cohomology of quiver Grassmannians by Lusztig’s category. As a consequence, we describe explicitly the Poincaré polynomials of rigid quiver Grassmannians in terms of the coefficients of dual canonical bases, which are viewed as elements of quantum shuffle algebras. By this result, we give another proof of the odd cohomology vanishing theorem for quiver Grassmanians. 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