Noncommutatively graded algebras (Q1980729)

This paper introduces the notion of noncommutatively graded algebra and obtains results generalizing well-known facts on graded algebras. If \(R\) denotes a unital, commutative and associative ring, and \(G\) is a group, an \(R\)-algebra \(A\) is said to be \emph{noncommutatively \(G\)-graded} if it decomposes as a direct sum of \(R\)-submodules \(A=\oplus_{g\in G}A_g\) such that \(A_gA_h\subseteq A_{gh}+A_{hg}\), for any \(g,h\in G\). A motivation for such definition is next. If \(A\) is an associative \(G\)-graded algebra, and \([x,y]=xy-yx\) and \(x\circ y=xy+yx\) denote the commutator and anticommutator respectively, then the induced Lie and Jordan algebras \(A^-=(A,[\ ,\ ])\) and \(A^+=(A,\circ)\) are clearly noncommutatively graded algebras. But observe that they are not \(G\)-graded algebras if \(A\) was not a commutative or anticommutative algebra. A more general example is the following. For any fixed \(\lambda,\mu\colon G\times G\to R\) and \(A\) a \(G\)-graded algebra, then the new algebra \((A,\bullet)\), where the product of homogeneous elements is given by \(a_g\bullet b_h=\lambda(g,h) a_gb_h+\mu(h,g) b_ha_g\), is a noncommutatively graded algebra. The algebras \(A^-\) and \(A^+\) arise from the constant maps \(\lambda=\mp\mu= 1\) and an associative starting algebra \(A\). The following step is to extend standard results to the new category \(G\)-NCGA of noncommutatively \(G\)-graded algebras. For instance, if \(A\) is unital, then the identity element \(1\in A_e\), for \(e \) the neutral element in \(G\), and a right invertible homogeneous element of degree \(g\) in \(A\) has some right inverse of degree \(g^{-1}\); direct and inverse limits exist in the category \(G\)-NCGA; the \emph{opposite} gives an automorphism of the category; and some functors between the categories of \(G\)-NCGA and \(H\)-NCGA are provided, fixed a group monomorphism or epimorphism between two groups \(G\) and \(H\) (adjointness studied too). Besides, a graded tensor algebra of a \(G\)-graded module is built. In the case of a noncommutatively \(G\)-graded Lie algebra, a universal (noncommutatively) \(G\)-graded enveloping algebra is constructed when the original Lie algebra is a free module over \(R\) possesing a homogeneous basis. (This happens, for instance, if \(R\) is a field.) In such case, a graded analogue of the Poincaré-Birkhoff-Witt theorem follows easily. For the entire collection see [Zbl 1467.16001].