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The $$q$$-characters of representations of quantum affine algebras and deformations of $$W$$-algebras. (English) Zbl 0973.17015
Jing, Naihuan (ed.) et al., Recent developments in quantum affine algebras and related topics. Proceedings of the international conference on representations of affine and quantum affine algebras and their applications, North Carolina State University, Raleigh, NC, USA, May 21-24, 1998. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 248, 163-205 (1999).
Let $$\mathfrak g$$ be a simple Lie algebra, $$\widehat{\mathfrak g}$$ the corresponding non-twisted affine Kac-Moody algebra and $$U_q\widehat{\mathfrak g}$$ the corresponding quantized universal enveloping algebra. Let $$\text{Rep} U_q\widehat{\mathfrak g}$$ be the Grothendieck ring of the category of finite-dimensional representations of $$U_q\widehat{\mathfrak g}$$. By using the universal R-matrix of $$U_q\widehat{\mathfrak g}$$ and an analogue of the Harish-Chandra homomorphism, the authors construct an injective homomorphism $$\chi_q :{ \text{Rep}} U_q\widehat{\mathfrak g}\to \mathcal Y ={\mathbb Z}[Y^{\pm 1}_{i, a_i}]_{i=1,\dots,\ell; a_i\in{\mathbb C}^\times}$$, where $$Y^{\pm 1}_{i, a_i}$$ are built from the Heisenberg algebra elements in the Drinfeld realization of $$U_q\widehat{\mathfrak g}$$. The authors show that $$\chi_q$$ behaves well with respect to the restriction to $$U_q\mathfrak g$$ and to the quantum affine subalgebras of $$U_q\widehat{\mathfrak g}$$. The authors call $$\chi_q(V)$$ the $$q$$-character of representation $$V$$. This construction is motivated by the theory of deformed $$\mathcal W$$-algebras [E. Frenkel and N. Reshetikhin, Commun. Math. Phys. 178, 237-264 (1996; Zbl 0869.17014)], and the authors conjecture that the image of $$\chi_q$$ equals the intersection of certain screening operators $$S_i$$ defined on $$\mathcal Y$$. The $$q$$-characters are closely related to a character theory in [H. Knight, J. Algebra 174, 187-196 (1995; Zbl 0868.17009)].
For the entire collection see [Zbl 0932.00043].
Reviewer: M.Primc (Zagreb)

MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 17B68 Virasoro and related algebras 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations
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