## 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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