Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The index of symmetry for a left-invariant metric on a solvable
  three-dimensional Lie group   We find the index of symmetry for all solvable three-dimensional Lie groups
with a left-invariant metric. When combined with the work of Reggiani on
unimodular three-dimensional Lie groups, the index of symmetry is then known
for all three-dimensional Lie groups with a left-invariant metric. Similar to
the unimodular case, for every solvable three-dimensional Lie algebra there is
at least one left-invariant metric for which the index of symmetry is positive.
Also, the index is never equal to 2.

7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
3
The categories of this paper are: physics
4
The categories of this paper are: cond-mat
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
24
The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex