Actions of Monoidal Categories and Representations of Cartan Type Lie Algebras

Abstract

Using crossed homomorphisms, we show that the category of weak representations (respectively admissible representations) of Lie–Rinehart algebras (respectively Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, the corresponding bifunctor of monoidal categories is established to give new weak representations (respectively admissible representations) of Lie–Rinehart algebras (respectively Leibniz pairs). This generalises and unifies various existing constructions of representations of many Lie algebras by using this new bifunctor. We construct some crossed homomorphisms in different situations and use our actions of monoidal categories to recover some known constructions of representations of various Lie algebras and to obtain new representations for generalised Witt algebras and their Lie subalgebras. The cohomology theory of crossed homomorphisms between Lie algebras is introduced and used to study linear deformations of crossed homomorphisms.

3-post-Lie algebras and relative Rota-Baxter operators of nonzero weight on 3-Lie algebras

“…(x * y) • z = x * y, z, w = z, w, x * y = 0, (31) x * (yz) = {xy, z, w} = 0, (32) x * z, w, t = { z, w, t , x, y} = 0, (33) where operations Here the notation (•, •, •) stands for the associator with respect to the operation * , i.e., for all x, y, z ∈ A,…”

“…Moreover, they construct a controlling algebra that characterizes deformations of relative Rota-Baxter operators on Lie algebras, on 3-Lie algebras, and on Leibniz algebras respectively [35,36,37]. Recently, Pei and his colleagues established crossed homomorphisms on Lie algebras via the same methods, and generalized constructions of many kinds of Lie algebras by using bifunctors [31]. Besides, the first author and the corresponding author investigated cohomology and linear deformations of LieYRep pairs and explored several properties of relative Rota-Baxter-Nijenhuis structures on LieYRep pairs in [44], and cohomology and deformations of relative Rota-Baxter operators on Lie-Yamaguti algebras in [45].…”

Cohomology and deformations of crossed homomorphisms between Lie-Yamaguti algebras

“…Moreover, they construct a controlling algebra that characterizes deformations of relative Rota-Baxter operators (also called O-operators) on Lie algebras, on 3-Lie algebras, and on Leibniz algebras respectively [29,30,31]. Recently, Pei and his colleagues established crossed homomorphisms on Lie algebras via the same methods, and generalized constructions of many kinds of Lie algebras by using bifunctors [25]. Besides, the first two authors investigated cohomology and linear deformations of LieYRep pairs and explored several properties of relative Rota-Baxter-Nijenhuis structures on LieYRep pairs in [37], and cohomology and deformations of relative Rota-Baxter operators on Lie-Yamaguti algebras in [38].…”

“…An example of crossed homomorphisms is a differential operator of weight 1, and a flat connection 1-form of a trivial principle bundle is also a crossed homomorphism. In [25], authors showed that the category of weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp.…”

Koszul duality, minimal model and $L_\infty$-structure for differential algebras with weight

“…Studies of cohomology and homotopy for algebra structures with linear operators have become very active recently. The structures include differential operators and Rota-Baxter operators on associative and Lie algebras [21,36,51,52]. Due to the complexity of the algebraic structures, the L ∞ -algebras in these studies are obtained by direct constructions without using the language of operads.…”