Representation theory of W-algebras

By T. Arakawa. 2005

172 citations recorded by the source corpus.

We study the representation theory of the $\mathcal{W}$ -algebra $\mathcal{W}_k(\bar{\mathfrak{g}})$ associated with a simple Lie algebra $\bar{\mathfrak{g}}$ at level k. We show that the “-” reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k∈ℂ. Moreover, we show that the character of each irreducible highest weight representation of $\mathcal{W}_k(\bar{\mathfrak{g}})$ is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra $\mathfrak{g}$ of $\bar{\mathfrak{g}}$ . As a consequence we complete (for the “-” reduction) the proof of the conjecture of E. Frenkel, V. Kac and M. Wakimoto on the existence and the construction of the modular invariant representations of $\mathcal{W}$ -algebras.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2606d26e899e8784f86f6c5cc4bc228119b33a09

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In Algebra and Number Theory

1. A. Sole
2. Vyacheslav Futorny
3. S. Ovsienko
4. V. Kac
5. Alexander I. Molev
6. I. Losev
7. A. Premet
8. Pavel S. Kolesnikov

In Geometry and Topology

1. A. Sole
2. Vyacheslav Futorny
3. S. Ovsienko
4. V. Kac
5. Alexander I. Molev
6. I. Losev
7. A. Premet
8. Pavel S. Kolesnikov

In Mathematical Physics

1. S. Ovsienko
2. V. Kac
3. Alexander I. Molev
4. Vyacheslav Futorny
5. A. Sole
6. I. Losev

Citations in the corpus

• The Gelfand-Kirillov conjecture and Gelfand-Tsetlin modules for finite W-algebras (2010, background): “[36] and Arakawa [3] have shown that the two definitions of finite W -algebras are equivalent.”
• Finite vs affine W-algebras (2005, result): “ reduction definition of finite W-algebras adopted in the paper and the definition via the Whittaker models which goes back to [Ko] (see [P1, GG]) are equivalent. This result was independently proved in [A2]. In fact in [A2] the result is stated only in the case of principal nilpotent f, cf. [KS], but ”
• Rationality of W-algebras: principal nilpotent cases (2012, unknown): “) with coefficient in A(L), which is identical to A(V)† in Losev’s notation [Los], see Section 3. In the case that f = fprin the classification problem is relatively simple since A(Wk(g,fprin)) ∼= Z(g) ([A2]), where Z(g) is the center of the universal enveloping algebra U(g) of g, and hence, A(Wk(g))) ”