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In this paper, we used the free fields of Wakimoto to construct a class of irreducible representations for the general linear Lie superalgebra $mathfrak{gl}_{m|n}(mathbb{C})$. The structures of the representations over the general linear Lie superalgebra and the special linear Lie superalgebra are studied in this paper. Then we extend the construction to the affine Kac-Moody Lie superalgebra $widehat{mathfrak{gl}_{m|n}}(mathbb{C})$ on the tensor product of a polynomial algebra and an exterior algebra with infinitely many variables involving one parameter $mu$, and we also obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter $mu$ is nonzero.
Let $mathfrak l:= mathfrak q(n)timesmathfrak q(n)$, where $mathfrak q(n)$ denotes the queer Lie superalgebra. The associative superalgebra $V$ of type $Q(n)$ has a left and right action of $mathfrak q(n)$, and hence is equipped with a canonical $math
The Capelli problem for the symmetric pairs $(mathfrak{gl}times mathfrak{gl},mathfrak{gl})$ $(mathfrak{gl},mathfrak{o})$, and $(mathfrak{gl},mathfrak{sp})$ is closely related to the theory of Jack polynomials and shifted Jack polynomials for special
We introduce and define the quantum affine $(m|n)$-superspace (or say quantum Manin superspace) $A_q^{m|n}$ and its dual object, the quantum Grassmann superalgebra $Omega_q(m|n)$. Correspondingly, a quantum Weyl algebra $mathcal W_q(2(m|n))$ of $(m|n
Let $mathfrak g = mathfrak{gl}_N(k)$, where $k$ is an algebraically closed field of characteristic $p > 0$, and $N in mathbb Z_{ge 1}$. Let $chi in mathfrak g^*$ and denote by $U_chi(mathfrak g)$ the corresponding reduced enveloping algebra. The Kac-
Generalizations of the AGT correspondence between 4D $mathcal{N}=2$ $SU(2)$ supersymmetric gauge theory on ${mathbb {C}}^2$ with $Omega$-deformation and 2D Liouville conformal field theory include a correspondence between 4D $mathcal{N}=2$ $SU(N)$ su