Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hopf-like Algebras and Extended P-B-W Theorems   Based on invariant algebras, we introduce representations$^{6-th}$ of Lie
algebras and representations$^{< 4-th>}$ of Leibniz algebras, give the extended
P-B-W Theorems in the context of the new representations of Lie algebras and
Leibniz algebras, and generalize the Hopf-algebra structure on the enveloping
algebras of Lie Algebras.

24
The categories of this paper are: dg-ga
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
22
The categories of this paper are: chao-dyn
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
11
The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
27
The categories of this paper are: comp-gas
3
The categories of this paper are: physics
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph