Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Generalized Hirota Equations and Representation Theory. I. The case of
  $SL(2)$ and $SL_q(2)$"   This paper begins investigation of the concept of ``generalized
$\tau$-function'', defined as a generating function of all the matrix elements
of a group element $g \in G$ in a given highest-weight representation of a
universal enveloping algebra ${\cal G}$. In the generic situation, the
time-variables correspond to the elements of maximal nilpotent subalgebras
rather than Cartanian elements. Moreover, in the case of quantum groups such
$\tau$-``functions'' are not $c$-numbers but take their values in
non-commutative algebras (of functions on the quantum group $G$). Despite all
these differences from the particular case of conventional $\tau$-functions of
integrable (KP and Toda lattice) hierarchies (which arise when $G$ is a
Kac-Moody (1-loop) algebra of level $k=1$), these generic $\tau$-functions also
satisfy bilinear Hirota-like equations, which can be deduced from manipulations
with intertwining operators. The most important applications of the formalism
should be to $k>1$ Kac-Moody and multi-loop algebras, but this paper contains
only illustrative calculations for the simplest case of ordinary (0-loop)
algebra $SL(2)$ and its quantum counterpart $SL_q(2)$, as well as for the
system of fundamental representations of $SL(n)$.

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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn
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The categories of this paper are: atom-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: stat
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The categories of this paper are: math
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The categories of this paper are: q-bio
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The categories of this paper are: math-ph
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The categories of this paper are: q-alg
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The categories of this paper are: cs
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The categories of this paper are: astro-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: gr-qc
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The categories of this paper are: funct-an