On the toroidal Leibniz algebras
Let \(\dot {\mathfrak{g}}\) be a finite-dimensional simple Lie algebra over the complex numbers \({\mathbb {C}}\), and let \[ A = {\mathbb {C}}[t_1, \;t_1^{-1}, \;t_2, \;t_2^{-1}, \;\ldots , \;t_{\nu}, \;t_{\nu}^{-1}] . \] Then \(\mathfrak{g} = \dot{\mathfrak{g}} \otimes A\) is a Lie algebra with bracket \[ [x \otimes f_1, \;y \otimes f_2] = [x, \;y ] \otimes f_1 f_2 , \] where \(x\), \(y \in \dot{\mathfrak{g}}\) and \(f_1\), \(f_2 \in A\). The universal central extension of \(\mathfrak{g}\), namely \(U = \mathfrak{g} \oplus (\Omega_A / dA)\), is called a toroidal Lie algebra [\textit{R. V. Moody, S. Eswara Rao, T. Yomonuma}, Geom. Dedicata 35, No. 1--3, 283--307 (1990Zbl 0704.17011)], while an affine Lie algebra arises from the case \(\nu = 1\). The bracket on \(\mathfrak{g} = \dot{\mathfrak{g}} \otimes A\) also affords \(\mathfrak{g}\) with the structure of a Leibniz algebra, and \textit{Y. Gao} [J. Pure Appl. Algebra 140, No. 1, 33--56 (1999; Zbl 0944.17001)] has identified the universal central extension of \(\mathfrak{g}\) in the category of Leibniz algebras when \(A\) is more generally a commutative ring. This Leibniz extension has form: \[ \mathfrak{g}_{LT} = \dot{\mathfrak{g}} \otimes A \oplus \Omega^1_{A} \] with bracket \[ \begin{aligned} & [x \otimes f, \;y \otimes g] = [x, \;y] \otimes fg + (x, \, y)f \, dg\\ & [\Omega_A, \;\mathfrak{g}_{LT}] = 0, \end{aligned} \] where \((x, \, y)\) denotes the Killing from. The paper under review determines the universal central extension of the Leibniz algebra \(\dot{\mathfrak{g}} \otimes D\), where \(D\) is a commutative, unital dialgebra. Recall [\textit{J.-L. Loday, A. Frabetti, F. Chapoton, F. Goichot}, Dialgebras and related operads. Lecture Notes in Mathematics. 1763. Berlin: Springer (2001; Zbl 0970.00010)] that a dialgebra \(D\) over a field \(k\) is a \(k\)-vector space with two \(k\)-linear operations \(\dashv, \;\vdash : D \otimes D \to D\), satisfying: \[ \begin{aligned} & a \dashv (b \dashv c) = (a \dashv b) \dashv c = a \dashv (b \vdash c) \\ & (a \vdash b) \dashv c = a \vdash (b \dashv c) \\ & (a \vdash b) \vdash c = a \vdash (b \vdash c) = (a \dashv b) \vdash c, \end{aligned} \] while \(D\) is commutative if \(a \dashv b = b \vdash a\). The Leibniz bracket on \(\dot{\mathfrak{g}} \otimes D\) is given by \[ [x \otimes a, \;y \otimes b] = [x, \;y] \otimes (a \dashv b). \] The main result of the paper is that the universal (Leibniz) central extension of \(\dot{\mathfrak{g}} \otimes D\) is the Leibniz algebra \(\dot{\mathfrak{g}} \otimes D \oplus \Omega^1_D\) with bracket \[ [x \otimes a, \;y \otimes b] = [x,\;y] \otimes (a \dashv b) + (x, \, y) (a \dashv db). \] Also, the derivations of \(\mathfrak{g}_{LT}\) are identified as \[ {\text{Inn}}(\dot{\mathfrak{g}}) \otimes A \oplus {\text{Der}}(A), \] where Inn denotes inner derivations. An additional result proven is that for a perfect Leibniz algebra \(L\) with universal central extension \(\hat{L}\), every automorphism \(\theta\) of \(L\) admits a unique extension \(\hat{\theta}\) to an automorphism of \(\hat{L}\).
Leibniz algebras, universal central extensions
scientific article; zbMATH DE number 5278758
scientific article