For Witt Lie Algebras over field of characterestic $p>3$ we know that $\operatorname{dim}W(n;m):=np^{|m|}$ , such that $|m|=m_1+⋯+m_n$ . I would like to know what is the dimension of Witt algebras over $\mathrm {GF}(2)$. Why $\operatorname{dim}W(2,1)= ?$
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5$\begingroup$ It wouldn't hurt to define W(n; m) for the unenlightened. Is it the $W\left(n, \overline{m}\right)$ in Definition 1.3 of Theresia Nolte's math.rwth-aachen.de/~Gerhard.Hiss/Students/… ? $\endgroup$– darij grinbergCommented Oct 13, 2013 at 1:42
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1$\begingroup$ What is the source of definitions and notation on which the question is based? There are somewhat different approaches in the literature to the Jacobson-Witt algebras, but for instance where does the claim $\dim W(2) = 3$ over $\mathbb{F}_2$ come from? $\endgroup$– Jim HumphreysCommented Oct 13, 2013 at 14:29
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$\begingroup$ The main source is a paper titled: " Some new simple Lie algebras in characteristic 2" published by Professor B.Eick. $\endgroup$– user118746Commented Oct 13, 2013 at 14:48
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3$\begingroup$ After the recent edit this question is even more unclear... $\endgroup$– Salvatore SicilianoCommented Nov 10, 2013 at 17:01
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2 Answers
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I think there is only a problem of notation here. The Jacobson-Witt algebra $W(m; \underline{n})$ is known to be simple of dimension $mp^{\vert n\vert}$ (where $\vert n \vert=n_1+n_2 \cdots + n_m$) except when $m = 1$ and the ground field has characteristic $p=2$. In the latter case the derived subalgebra $W(1; \underline{n})^{(1)}$ is simple of dimension $2^{\vert n \vert} − 1$, provided $\vert n \vert >1$. So I think that $W(1; \underline{2})^{(1)}$ is the Lie algebra the user is really referring to. Up to isomorphisms, this is the only $3$-dimensional simple Lie algebra over $GF(2)$. It has a basis $\{a,b,c \}$ such that $[a,b]=c$, $[b,c]=a$, $[c,a]=b$.
answered Oct 13, 2013 at 22:24
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$\begingroup$ this is the exact answer I was kooking for, in addition to that I would like to know how the dimension of W(1;1) and W(2;1) be computed over GF(2)? thank you in advance for your time. $\endgroup$ Commented Oct 14, 2013 at 20:37
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1$\begingroup$ A standard reference is "H. Strade - R. Farnsteiner: Modular Lie Algebras and Their Representation, Dekker, New York, 1988." You are referred to Chapter 4 of this book. $\endgroup$ Commented Oct 15, 2013 at 13:18
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$\begingroup$ I have already studied this book, but I am a little confused about somethings, for instance according theorem from mentioned book (page 149) if m=1 and p=2 then W(m;n) is not simple, whereas W(2;1) over GF(2) is 16-dimensional simple Lie algebra!this matter has been stated in paper titled "Some new simple Lie algebras in characteristic 2" by B. Eick. $\endgroup$ Commented Oct 15, 2013 at 19:49
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1$\begingroup$ W(2;(1,1)) has indeed dimension 8 over GF(2) and it is isomorphic to sl(3, GF(2)), so it is not clear to me what you are really asking... Maybe, you could find the paper "Simple Lie algebras of low dimension over GF(2)" by Vaughan-Lee useful for your purpose. As a final suggestion, if the problem is due to notation you may need to contact the paper's author directly. $\endgroup$ Commented Oct 15, 2013 at 21:34
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Since it not exactly clear, which Lie algebra is meant, let me note that the Lie algebra $W(1;\underline{2})$ is $4$-dimensional over $\Bbb F_2$, see the article by Helmut Strade Lie algebras of small dimension. Its commutator subalgebra $W(1;\underline{2})^{(1)}$ is the unique simple Lie algebra of dimension $3$ over $\Bbb F_2$ mentioned above.
answered Oct 9, 2019 at 9:38