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Principal \(\widehat{{\mathfrak{sl}}_3}\) subspaces and quantum Toda Hamiltonian. (English) Zbl 1206.17012

Miwa, Tetsuji (ed.) et al., Algebraic analysis and around in honor of Professor Masaki Kashiwara’s 60th birthday. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-51-8/hbk). Advanced Studies in Pure Mathematics 54, 109-166 (2009).
Let \(\mathfrak n=\mathbb C e_{21} \oplus \mathbb C e_{32},\mathbb C e_{31}\) be the nilpotent subalgebra of the complex Lie algebra \(\mathfrak {sl}_3\) and let \(\hat {\mathfrak n}=\mathfrak n \oplus \mathbb C[t,t^{-1}]\) be the correspondent current algebra.
In the paper the authors study a class of \(\hat{\mathfrak n}\)-modules the simplest example of which is the principal subspace \(V^k\) of the level \(k\) vacuum representation of the affine Lie algebra \(\mathfrak {sl}_3\), namely, let \(M^k\) be level \(k\) vacuum representation of the affine Lie algebra \({sl}_3\). For a highest weight vector \(v^k \in M^k\) we have \(V^k=U(\hat {\mathfrak n})v^k\).
The following fermionic formula for the character of \(V^k\) was obtained in a series of papers. The authors propose a new approach to computation of the character formulas by combinatorial method. They obtain a bosonic formula for the character of \(\hat {\mathfrak n}\)-module \(M\) in three steps:
(1) find the set of extremal vectors of \(M\);
(2) find the contribution of each vector;
(3) prove that the sum of all contributions equals to the character of \(M\).
Structural functions in the bosonic formula is closely related to the Whittaker vectors in Verma modules of the quantum groups.
The structure of the paper is the following: sec. 2 contains the recalling of the bosonic and fermionic formulas for the simplest case of \({sl}_2\); in sections 3, 4, 5 the recurrent upper and lower estimates for the characters of the families of \(\hat {\mathfrak n}\)-modules are derived; sections 6, 7 are devoted to the investigation of the inductive structure of the bosonic formula with respect to the rank of the algebra and to the discussion of one special case; and in sec. 8 the connection of the given formulas with the Whittaker vectors in Verma modules of the quantum groups is considered.
For the entire collection see [Zbl 1160.32002].

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B65 Infinite-dimensional Lie (super)algebras
81R12 Groups and algebras in quantum theory and relations with integrable systems
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