engleski

Whittaker modules for the affine Lie algebra A_1 ^(1)

We prove the irreducibility of the universal non-degenerate Whittaker modules for the affine Lie algebra $\widehat{; ; ; ; ; sl_2}; ; ; ; ; $ of type $A_1^{; ; ; ; ; (1)}; ; ; ; ; $ with noncritical level. These modules can become simple Whittaker modules over $\widetilde{; ; ; ; ; sl_2}; ; ; ; ; =\widehat{; ; ; ; ; sl_2}; ; ; ; ; + C d $ with the same Whittaker function and central charge. We have to modulo a central character for $sl_2$ to obtain simple degenerate Whittaker $\widehat{; ; ; ; ; sl_2}; ; ; ; ; $-modules with noncritical level. In the case of critical level the universal Whittaker module is reducible. We prove that the quotient of universal Whittaker $\widehat{; ; ; ; ; sl_2}; ; ; ; ; $--module by a submodule generated by a scalar action of central elements of the vertex algebra $V_{; ; ; ; ; -2}; ; ; ; ; (sl_2)$ is simple as $\widehat{; ; ; ; ; \sl_2}; ; ; ; ; $--module. We also explicitly describe the simple quotients of universal Whittaker modules at the critical level for $\widetilde{; ; ; ; ; \sl_2}; ; ; ; ; $. Quite surprisingly, with the same Whittaker function some simple degenerate $\widetilde{; ; ; ; ; sl_2}; ; ; ; ; $ Whittaker modules at the critical level have semisimple action of $d$ and others have free action of $d$. At last, by using vertex algebraic techniques we present a Wakimoto type construction of a family of simple generalized Whittaker modules for $\widehat{; ; ; ; ; sl_2}; ; ; ; ; $ at the critical level. This family includes all classical Whittaker modules at critical level. We also have Wakimoto type realization for degenerate Whittaker modules for $\widehat{; ; ; ; ; sl_2}; ; ; ; ; $ at noncritical level.

Wakimoto modules ; Whittaker modules ; affine Lie algebra ; vertex algebra ; critical level

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