Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Kernel function and quantum algebras   We introduce an analogue $K_n(x,z;q,t)$ of the Cauchy-type kernel function
for the Macdonald polynomials, being constructed in the tensor product of the
ring of symmetric functions and the commutative algebra $\mathcal{A}$ over the
degenerate $\mathbb{C} \mathbb{P}^1$. We show that a certain restriction of
$K_n(x,z;q,t)$ with respect to the variable $z$ is neatly described by the
tableau sum formula of Macdonald polynomials. Next, we demonstrate that the
integer level representation of the Ding-Iohara quantum algebra naturally
produces the currents of the deformed $\mathcal{W}$ algebra. Then we remark
that the $K_n(x,z;q,t)$ emerges in the highest-to-highest correlation function
of the deformed $\mathcal{W}$ algebra.

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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-th
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The categories of this paper are: eess
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The categories of this paper are: adap-org
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The categories of this paper are: math-ph
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The categories of this paper are: hep-th
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The categories of this paper are: astro-ph
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The categories of this paper are: physics
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The categories of this paper are: alg-geom
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The categories of this paper are: dg-ga
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The categories of this paper are: q-alg
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The categories of this paper are: cmp-lg
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The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol