Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Nichols (braided) Lie algebras   We prove {\rm (i)} Nichols algebra $\mathfrak B(V)$ of vector space $V$ is
finite-dimensional if and only if Nichols braided Lie algebra $\mathfrak L(V)$
is finite-dimensional; {\rm (ii)} If the rank of connected $V$ is $2$ and
$\mathfrak B(V)$ is an arithmetic root system, then $\mathfrak B(V) = F \oplus
\mathfrak L(V);$ and {\rm (iii)} if $\Delta (\mathfrak B(V))$ is an arithmetic
root system and there does not exist any $m$-infinity element with $p_{uu}
\not= 1$ for any $u \in D(V)$, then $\dim (\mathfrak B(V) ) = \infty$ if and
only if there exists $V'$, which is twisting equivalent to $V$, such that $
\dim (\mathfrak L^ - (V')) = \infty.$ Furthermore we give an estimation of
dimensions of Nichols Lie algebras and two examples of Lie algebras which do
not have maximal solvable ideals.

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The categories of this paper are: plasm-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: cs
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The categories of this paper are: econ
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The categories of this paper are: physics
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The categories of this paper are: nucl-th
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The categories of this paper are: funct-an
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The categories of this paper are: hep-lat
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The categories of this paper are: chem-ph
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The categories of this paper are: q-fin
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The categories of this paper are: quant-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn