Representations of the BRST algebra and unsolvable algebraic problems

Abstract

Complete solution of the decomposition problem for finite-dimensional representations π of the BRST algebra 𝒜 is presented. As shown earlier by the authors [Rev. Math. Phys. 5, 191 (1993)], 𝒜 coincides with the Lie superalgebra l(1,1). It is proved that an arbitrary π either has the decomposition into a direct sum of irreducible and/or indecomposable representations (IR and IDR, respectively) or has the set of indecomposable subrepresentations that does not admit any classification. All the series of IDR are explicitly described, and all the unclassifiable cases are reduced to definite unsolvable algebraic problems. The absence of classification is established by means of a special computer method, which opens a new possibility to search for unsolvable algebraic problems.