Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Groups with involution, and quasigroups with cracovian representations   In groups with involution a nonassociative product of elements is defined,
which leads to the definition of a certain type of quasigroups. These
quasigroups are represented by square tables of complex numbers, with inverses,
which differ from the matrix representations of groups in the rule of
performing the product of two tables. The row-by-column product of two matrices
in representations of groups is replaced by the column-by-column product, which
is called the cracovian product, in representations of the defined type of
quasigroups. The matrices undergoing the column-by-column product are called
cracovians. The basic properties of the quasigroups connected with groups with
involution are determined while only a summary of the properties of cracovian
algebra is presented, as the basis of cracovian representation theory for the
quasigroups connected with groups with involution. Clifford groups are groups
with involution and the quasigroups connected with them are determined. The
orthogonal and pseudo-orthogonal rotation groups belong to groups with
involution. An analogy is drawn between Weyl's "hidden" symmetry group of an
object, and the quasigroup connected with the group with involution.

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The categories of this paper are: supr-con
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The categories of this paper are: adap-org
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
3
The categories of this paper are: physics
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The categories of this paper are: plasm-ph
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The categories of this paper are: ao-sci
34
The categories of this paper are: chem-ph
11
The categories of this paper are: hep-lat
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
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The categories of this paper are: cs
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The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat