The tensor product of representations of \(U_q(\mathfrak{sl}_2)\) via quivers (Q1400954)

The main goal of the present paper is to obtain a geometric realization of the tensor product of a finite number of finite-dimensional irreducible representations of \(U_q(\mathfrak{sl}_2)\) using quiver varieties. A quiver variety corresponding to the tensor product of finitely many integrable highest weight modules of a Kac-Moody algebra with a simply laced root system was defined by \textit{H. Nakajima} [Invent. Math. 146, No. 2, 399--449 (2001; Zbl 1023.17008)] and \textit{A. Malkin} [Duke Math. J. 116, No. 3, 477--524 (2003; Zbl 1048.20029)]. The paper under review considers the simplest case \(\mathfrak{sl}_2\) and recovers the complete structure of \(U_q(\mathfrak{sl}_2)\) via the tensor product variety and not just its crystal structure as in the more general cases. In the paper under review the tensor product variety \(\mathfrak{T}(\mathbf{d})\) is defined over the finite field \(\mathbb{F}_{q^2}\) with \(q^2\) elements rather than over the complex numbers. The author finds three spaces of invariant functions on \(\mathfrak{T}(\mathbf{d})\) which all are isomorphic to the tensor product \(V_{d_1}\otimes\cdots\otimes V_{d_k}\) of finite-dimensional simple \(U_q(\mathfrak{sl}_2)\)-modules. The natural bases of theses spaces correspond to the elementary basis \(\mathcal{B}_e\), Lusztig's canonical basis \(\mathcal{B}_c\), and a basis \(\mathcal{B}_s\) compatible with the decomposition of \(V_{d_1}\otimes\cdots\otimes V_{d_k}\) into a direct sum of simple modules, respectively. The bases \(\mathcal{B}_c\) and \(\mathcal{B}_s\) are characterized by their relation to the irreducible components of \(\mathfrak{T}(\mathbf{d})\). Here an irreducible component of \(\mathfrak{T}(\mathbf{d})\) defined over \(\mathbb{F}_{q^2}\) consists of the \(\mathbb{F}_{q^2}\)-rational points of the irreducible component of the corresponding variety defined over the algebraic closure of \(\mathbb{F}_{q^2}\). Elements of the bases \(\mathcal{B}_c\) and \(\mathcal{B}_s\) are equal to a non-zero constant on the set of dense points of an irreducible component of \(\mathfrak{T}(\mathbf{d})\) with supports contained in distinct irreducible components. However, the elements of \(\mathcal{B}_s\) have disjoint supports unlike the elements of \(\mathcal{B}_c\). The paper contains also a geometric description of the space of intertwiners \(\mathrm{Hom}_{U_q(\mathfrak{sl}_2)}(V_{d_1}\otimes\cdots\otimes V_{d_k},V_\lambda)\) and the natural basis of this space is again characterized by its relation to the irreducible components of \(\mathfrak{T}(\mathbf{d})\). An important tool used in this paper is the Penrose-Kauffman graphical calculus of intertwiners for \(U_q(\mathfrak{sl}_2)\) expanded by \textit{I. B. Frenkel} and \textit{M. G. Khovanov} [Duke Math. J. 87, No. 3, 409--480 (1997; Zbl 0883.17013)] in order to prove various results for Lusztig's (dual) canonical basis.