Graded Lie algebras of maximal class of type $n$
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arXiv
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| Format: | Preprint |
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2019
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| author | Mattarei, Sandro Ugolini, Simone |
| author_facet | Mattarei, Sandro Ugolini, Simone |
| contents | Let $n>1$ be an integer. The algebras of the title, which we abbreviate as algebras of type $n$, are infinite-dimensional graded Lie algebras $L= \bigoplus_{i=1}^{\infty}L_i$, which are generated by an element of degree $1$ and an element of degree $n$, and satisfy $[L_i,L_1]=L_{i+1}$ for $i\ge n$. Algebras of type $2$ were classified by Caranti and Vaughan-Lee in 2000 over any field of odd characteristic. In this paper we lay the foundations for a classification of algebras of arbitrary type $n$, over fields of sufficiently large characteristic relative to $n$. Our main result describes precisely all possibilities for the first constituent length of an algebra of type $n$, which is a numerical invariant closely related to the dimension of its largest metabelian quotient. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1911_00970 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Graded Lie algebras of maximal class of type $n$ Mattarei, Sandro Ugolini, Simone Rings and Algebras 17B70 (Primary), 17B65, 17B05 (Secondary) Let $n>1$ be an integer. The algebras of the title, which we abbreviate as algebras of type $n$, are infinite-dimensional graded Lie algebras $L= \bigoplus_{i=1}^{\infty}L_i$, which are generated by an element of degree $1$ and an element of degree $n$, and satisfy $[L_i,L_1]=L_{i+1}$ for $i\ge n$. Algebras of type $2$ were classified by Caranti and Vaughan-Lee in 2000 over any field of odd characteristic. In this paper we lay the foundations for a classification of algebras of arbitrary type $n$, over fields of sufficiently large characteristic relative to $n$. Our main result describes precisely all possibilities for the first constituent length of an algebra of type $n$, which is a numerical invariant closely related to the dimension of its largest metabelian quotient. |
| title | Graded Lie algebras of maximal class of type $n$ |
| topic | Rings and Algebras 17B70 (Primary), 17B65, 17B05 (Secondary) |
| url | https://arxiv.org/abs/1911.00970 |