Subalgebras of the Fomin–Kirillov algebra
Abstract The Fomin–Kirillov algebra $${\mathcal {E}}_n$$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define $${{\mathcal {E}}_G}$$ to be the subalgebra of $${\mathcal {E}}_n$$ generated by the edges of G...
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| Autor*in: |
Blasiak, Jonah [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2016 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media New York 2016 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of algebraic combinatorics - Springer US, 1992, 44(2016), 3 vom: 11. Mai, Seite 785-829 |
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| Übergeordnetes Werk: |
volume:44 ; year:2016 ; number:3 ; day:11 ; month:05 ; pages:785-829 |
| Links: |
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| DOI / URN: |
10.1007/s10801-016-0688-4 |
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| 520 | |a Abstract The Fomin–Kirillov algebra $${\mathcal {E}}_n$$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define $${{\mathcal {E}}_G}$$ to be the subalgebra of $${\mathcal {E}}_n$$ generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, $${\mathcal {E}}_G$$ is a free $${\mathcal {E}}_H$$-module for any $$H\subseteq G$$, and if $${\mathcal {E}}_G$$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for $${\mathcal {E}}_G$$ when G is a simply laced finite Dynkin diagram or a cycle, in particular showing that $${\mathcal {E}}_G$$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of $${\mathcal {E}}_{\tilde{D}_n}$$, $${\mathcal {E}}_{\tilde{E}_6}$$, and $${\mathcal {E}}_{\tilde{E}_7}$$, as well as the graphs G on six vertices for which $$\mathcal {E}_G$$ is finite-dimensional. | ||
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| 650 | 4 | |a nil-Coxeter algebra | |
| 650 | 4 | |a Nichols algebra | |
| 700 | 1 | |a Liu, Ricky Ini |4 aut | |
| 700 | 1 | |a Mészáros, Karola |4 aut | |
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10.1007/s10801-016-0688-4 doi (DE-627)OLC2034252675 (DE-He213)s10801-016-0688-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Blasiak, Jonah verfasserin aut Subalgebras of the Fomin–Kirillov algebra 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2016 Abstract The Fomin–Kirillov algebra $${\mathcal {E}}_n$$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define $${{\mathcal {E}}_G}$$ to be the subalgebra of $${\mathcal {E}}_n$$ generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, $${\mathcal {E}}_G$$ is a free $${\mathcal {E}}_H$$-module for any $$H\subseteq G$$, and if $${\mathcal {E}}_G$$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for $${\mathcal {E}}_G$$ when G is a simply laced finite Dynkin diagram or a cycle, in particular showing that $${\mathcal {E}}_G$$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of $${\mathcal {E}}_{\tilde{D}_n}$$, $${\mathcal {E}}_{\tilde{E}_6}$$, and $${\mathcal {E}}_{\tilde{E}_7}$$, as well as the graphs G on six vertices for which $$\mathcal {E}_G$$ is finite-dimensional. Fomin–Kirillov algebra Coxeter group nil-Coxeter algebra Nichols algebra Liu, Ricky Ini aut Mészáros, Karola aut Enthalten in Journal of algebraic combinatorics Springer US, 1992 44(2016), 3 vom: 11. Mai, Seite 785-829 (DE-627)131180770 (DE-600)1143271-8 (DE-576)033043701 0925-9899 nnns volume:44 year:2016 number:3 day:11 month:05 pages:785-829 https://doi.org/10.1007/s10801-016-0688-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_70 GBV_ILN_2088 GBV_ILN_4323 AR 44 2016 3 11 05 785-829 |
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Subalgebras of the Fomin–Kirillov algebra |
| abstract |
Abstract The Fomin–Kirillov algebra $${\mathcal {E}}_n$$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define $${{\mathcal {E}}_G}$$ to be the subalgebra of $${\mathcal {E}}_n$$ generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, $${\mathcal {E}}_G$$ is a free $${\mathcal {E}}_H$$-module for any $$H\subseteq G$$, and if $${\mathcal {E}}_G$$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for $${\mathcal {E}}_G$$ when G is a simply laced finite Dynkin diagram or a cycle, in particular showing that $${\mathcal {E}}_G$$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of $${\mathcal {E}}_{\tilde{D}_n}$$, $${\mathcal {E}}_{\tilde{E}_6}$$, and $${\mathcal {E}}_{\tilde{E}_7}$$, as well as the graphs G on six vertices for which $$\mathcal {E}_G$$ is finite-dimensional. © Springer Science+Business Media New York 2016 |
| abstractGer |
Abstract The Fomin–Kirillov algebra $${\mathcal {E}}_n$$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define $${{\mathcal {E}}_G}$$ to be the subalgebra of $${\mathcal {E}}_n$$ generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, $${\mathcal {E}}_G$$ is a free $${\mathcal {E}}_H$$-module for any $$H\subseteq G$$, and if $${\mathcal {E}}_G$$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for $${\mathcal {E}}_G$$ when G is a simply laced finite Dynkin diagram or a cycle, in particular showing that $${\mathcal {E}}_G$$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of $${\mathcal {E}}_{\tilde{D}_n}$$, $${\mathcal {E}}_{\tilde{E}_6}$$, and $${\mathcal {E}}_{\tilde{E}_7}$$, as well as the graphs G on six vertices for which $$\mathcal {E}_G$$ is finite-dimensional. © Springer Science+Business Media New York 2016 |
| abstract_unstemmed |
Abstract The Fomin–Kirillov algebra $${\mathcal {E}}_n$$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define $${{\mathcal {E}}_G}$$ to be the subalgebra of $${\mathcal {E}}_n$$ generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, $${\mathcal {E}}_G$$ is a free $${\mathcal {E}}_H$$-module for any $$H\subseteq G$$, and if $${\mathcal {E}}_G$$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for $${\mathcal {E}}_G$$ when G is a simply laced finite Dynkin diagram or a cycle, in particular showing that $${\mathcal {E}}_G$$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of $${\mathcal {E}}_{\tilde{D}_n}$$, $${\mathcal {E}}_{\tilde{E}_6}$$, and $${\mathcal {E}}_{\tilde{E}_7}$$, as well as the graphs G on six vertices for which $$\mathcal {E}_G$$ is finite-dimensional. © Springer Science+Business Media New York 2016 |
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3 |
| title_short |
Subalgebras of the Fomin–Kirillov algebra |
| url |
https://doi.org/10.1007/s10801-016-0688-4 |
| remote_bool |
false |
| author2 |
Liu, Ricky Ini Mészáros, Karola |
| author2Str |
Liu, Ricky Ini Mészáros, Karola |
| ppnlink |
131180770 |
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| doi_str |
10.1007/s10801-016-0688-4 |
| up_date |
2024-07-03T20:15:58.766Z |
| _version_ |
1803590312941060096 |
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