Cartan-type algebras

From the text (translated from the Russian): "We define a new class of associative algebras (CAA) that are said to be contragredient by analogy with contragredient Lie algebras (CLA), introduced by V. Kac in 1968 [{it Infinite-dimensional Lie algebras}, second edition, Cambridge Univ. Press, Cambridge, 1985; [msn] MR0823672 (87c:17023) [/msn]]. par "The category of CLA is embedded into CAA by means of the classical functor U (germ g mapsto U(germ g), where germ g is a Lie algebra). The category CAA also contains various extensions and deformations of the algebra U(germ g), including quantum algebras introduced by Drinfelʹd and Jimbo in 1985, Yangians, Weyl algebras, their various superanalogues, etc. par "The naturality of the axiomatics presented is based on the possibility of constructing a unified theory of representations in the category CAA, especially in some subcategory CTA (a Cartan-type algebra). The category CTA is a natural domain for the development of extremal (projective) methods presented by us in 1983 [Dokl. Akad. Nauk SSSR {bf 273} (1983), no.~4, 785--788; [msn] MR0728272 (85f:17004) [/msn]], originally in the framework of the category CLA. The original (narrower) version of the definition of CAA was presented in a later paper of ours [J. Group Theory Phys. {bf 1} (1993), no. 1, 198--223; per bibl.], where some problems of a cohomological character were considered that are related to the theory of representations in the category CAA."