Instruct: Identify the main and secondary category of Arxiv papers based on the titles
Query: Purity, algebraic compactness, direct sum decompositions, and
  representation type
15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
31
The categories of this paper are: ao-sci
12
The categories of this paper are: nucl-th
7
The categories of this paper are: hep-th
29
The categories of this paper are: funct-an
23
The categories of this paper are: solv-int
0
The categories of this paper are: hep-ph
4
The categories of this paper are: cond-mat