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Works connected to Fyodor G. Malikov

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F. G. Malikov, B. L. Feigin, and D. B. Fuks: “Singular vectors in Verma modules over Kac–Moody algebras,” Funktsional. Anal. i Prilozhen. 20 : 2 (1986), pp. 25–37. English translation of Russian original published in Funct. Anal. Appl. 20:2 (1986). MR 847136 article

Boris L. Feigin	Related
Fyodor G. Malikov	Related

F. G. Malikov, B. L. Feigin, and D. B. Fuks: “Singular vectors in Verma modules over Kac–Moody algebras,” Funct. Anal. Appl. 20 : 2 (April 1986), pp. 103–113. English translation of Russian original published in Funktsional. Anal. i Prilozhen. 20:2 (1986). Zbl 0616.17010 article

The basic results of this paper are explicit formulas for singular vectors in reducible Verma modules over Kac–Moody algebras. As is known, Verma modules \( M(\lambda) \) over a Kac–Moody algebra \[ \mathfrak{g} = N_{-} \oplus H \oplus N_{+} \] are parametrized by linear functionals \( \lambda \) on its Cartan subalgebra \( H \). Reducible modules \( M(\lambda) \) correspond to functionals \( \lambda \) lying in the union of a countable number of hyperplanes in \( H^* \); these hyperplanes are enumerated by pairs \( (\alpha,n) \), where \( \alpha \) is a positive root of the algebra \( \mathfrak{g} \) and \( n \) is a positive integer (Kac–Kazhdan theorem [1979]). If \( \lambda \) lies in exactly one such hyperplane and the root to which, this hyperplane corresponds is real, then the module \( M(\lambda) \) contains a unique (up to proportionality) singular vector (in different terminology a zero-vector or vector of highest weight), not proportional to a vacuum vector. We give an explicit formula for this vector which however has an unusual form: to the vacuum vector oneapplies monomials in the elements of the algebra \( N_{-} \) containing these elements to nonintegral and even nonreal powers. We start by explaining in Sec. 2 how to reduce such monomials to traditional form. Then we begin to derive the formulas for singular vectors. In Sec. 3 the case of the algebra \( \mathfrak{sl}(2)\hat{} \), the simplest of the infinite-dimensional Kac–Moody algebras, is analyzed in detail; this section also contains important information about singular vectors in reducible Verma modules over \( \mathfrak{sl}(2)\hat{} \) corresponding to imaginary roots. In Sec. 4 the results of Sec. 3 are extended to the case of an arbitrary Kac–Moody algebra, where in the case of real roots one gets exhaustive info mation. In the concluding Sec. 5 we turn to the consideration of a special case, this time the case of the finite-dimensional Lie algebra \( \mathfrak{sl}(n) \); in this case we are able to reduce the answer to especially convenient form. We note that Verma modules over finite-dimensional simple Lie algebras are studied among other things in the recent paper of Zhelobenko [1986]; this paper also contains some expressions for singular vectors in Verma modules, which differ in form from those we find. The connection between our results and the results of [1986] is still unclear.

Boris L. Feigin	Related
Fyodor G. Malikov	Related