×

The tensor category of linear maps and Leibniz algebras. (English) Zbl 0909.18003

[Note: Some diagrams below cannot be displayed correctly in the web version. Please use the PDF version for correct display.]
Let \(K\) be a fixed field and let \({\mathcal L} {\mathcal M}\) be the category whose objects are \(K\)-linear maps \(f:V\to W\) of \(K\)-vector spaces, denoted by \((V,W)\), and whose morphisms \((\alpha, \overline\alpha): (V,W)\to (V',W')\) are such that the diagram \[ \begin{tikzcd} V \ar[r,"\alpha"]\ar[d,,"f" '] & V^\prime\\ W \ar[r, "\overline{\alpha}" '] & W^\prime \ar[u, "f^\prime" ']\end{tikzcd} \] is commutative. Equipped with a suitable tensor product \(\otimes\), the category \({\mathcal L} {\mathcal M}\) becomes a symmetric tensor category, i.e., a \(K\)-linear symmetric strict monoidal category with symmetry isomorphism \(\tau\). A Lie algebra in \({\mathcal L} {\mathcal M}\) is an object \((V,W)\) of \({\mathcal L} {\mathcal M}\) equipped with a morphism \(\mu: (V,W)\otimes (V,W)\to (V,W)\) satisfying \[ \mu\circ \tau=-\mu\quad \text{and} \quad\mu (1\otimes\mu) -\mu(\mu \otimes 1)+ \mu(\mu \otimes 1)(1\otimes \tau)=0. \] A bialgebra in \({\mathcal L} {\mathcal M}\) is an object \((M,H)\) of \({\mathcal L} {\mathcal M}\), where \(H\) is a bialgebra and \(M\) is an \(H\)-bimodule and an \(H\)-bicomodule such that the maps \(\Delta_1: M\to M\otimes H\) and \(\Delta_2:M\to H\otimes M\) defining the two comodule structures are \(H\)-bimodule maps. If \(H\) is a Hopf algebra, then \((M,H)\) is called a Hopf algebra in \({\mathcal L} {\mathcal M}\), which is cocommutative if so is \(H\).
In this paper, the authors show that a Lie algebra in \({\mathcal L} {\mathcal M}\) is equivalent to a \(K\)-linear map \(f:M\to {\mathfrak g}\), where \({\mathfrak g}\) is a Lie algebra, \(M\) is a (right) \({\mathfrak g}\)-module, and \(f\) is \({\mathfrak g}\)-equivariant. In particular, if \(M\) is a Leibniz algebra \({\mathfrak h}\), i.e., a \(K\)-vector space equipped with a \(K\)-bilinear map \([\cdot,\cdot]: {\mathfrak h}\times {\mathfrak h}\to {\mathfrak h}\) such that \[ \bigl[x,[y,z]\bigr]-\bigl[[x,y],z\bigr] +\bigl[[x,z],y\bigr]=0 \quad \text{for all }x,y,z \text{ in } {\mathfrak h}, \] and \({\mathfrak g}\) is the quotient \({\mathfrak h}_{\text{Lie}}\) of \({\mathfrak h}\) by the 2-sided ideal generated by the elements \([x,x]\) for all \(x\) in \({\mathfrak h}\), then the surjective map \({\mathfrak h}\to {\mathfrak h}_{\text{Lie}}\) is a Lie algebra in \({\mathcal L} {\mathcal M}\) and is the image of \({\mathfrak h}\) by a functor (Leib)\(\to\)(Lie in \({\mathcal L} {\mathcal M})\) of the category of Leibniz algebras into the category of Lie algebras in \({\mathcal L} {\mathcal M}\). By means of this functor, the authors show that there exists some factorization of the functor \(UL\) of (Leib) into the category (As) of associative unital algebra [J.-L. Loday and T. Pirashvili, Math. Ann. 296, No. 1, 139-158 (1993; Zbl 0821.17022)] through the category (Hopf in \({\mathcal L} {\mathcal M}\)) of cocommutative Hopf algebras in \({\mathcal L} {\mathcal M}\). They also prove PoincarĂ©-Birkhoff-Witt and Milnor-Moore type theorems in \({\mathcal L} {\mathcal M}\).

MSC:

18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
17A32 Leibniz algebras
17B35 Universal enveloping (super)algebras
18B99 Special categories

Citations:

Zbl 0821.17022
PDFBibTeX XMLCite
Full Text: EuDML EMIS