[Paper Review] Universal Central Extensions of the Matrix Leibniz Superalgebras sl(m, n, A)

Naihong Hu, Dong Liu|ArXiv.org|Oct 19, 2006

Algebraic structures and combinatorial models5 references4 citations

TL;DR

This paper determines the universal central extension of the matrix Leibniz superalgebra 𝔰𝔩(m,n,𝒜) in the category of Leibniz superalgebras, showing it is isomorphic to the Steinberg Leibniz superalgebra 𝔰𝔱𝔩(m,n,𝒜) with kernel isomorphic to the image of the Connes operator 𝒟 on Hochschild homology, under mild assumptions: m+n ≥ 3, with characteristic constraints for small m+n. The result extends earlier Lie superalgebra results to the Leibniz setting using Hochschild and cyclic homology.

ABSTRACT

The universal central extensions and their extension kernels of the matrix Lie superalgebra sl(m, n, A), the Steinberg Lie superalgebra st(m, n, A) in category {\bf SLeib} of Leibniz superalgebras are determined under a weak assumption (compared with \cite{MP}) using the first Hochschild homology and the first cyclic homology group.

Motivation & Objective

• Address the gap in understanding universal central extensions for Leibniz superalgebras, particularly for matrix algebras 𝔰𝔩(m,n,𝒜), extending prior Lie superalgebra results.
• Overcome the restrictive assumption m+n ≥ 5 in earlier work by weakening it to m+n ≥ 3 with characteristic conditions.
• Establish the universal central extension of 𝔰𝔩(m,n,𝒜) in the category SLeib of Leibniz superalgebras using homological algebra tools.
• Characterize the kernel of the universal central extension as the image of the Connes operator 𝒟 on Hochschild homology.
• Extend the theory of universal central extensions from Lie superalgebras to Leibniz superalgebras, unifying homological invariants.

Proposed method

• Use the first Hochschild homology group HH₁(𝒜) as the kernel of the universal central extension in the Leibniz superalgebra setting.
• Apply the Connes operator 𝒟 to define a map on tensor powers of 𝒜, whose image captures the kernel of the extension.
• Construct the Steinberg Leibniz superalgebra 𝔰𝔱𝔩(m,n,𝒜) as the universal central extension of 𝔰𝔩(m,n,𝒜) in SLeib.
• Utilize the universal property of the quotient map π: 𝔰𝔱𝔩(m,n,𝒜) → 𝔰𝔱(m,n,𝒜) and the Lie superalgebra structure of the quotient.
• Establish a commutative diagram linking Hochschild homology, cyclic homology, and the central extension, showing Ker(π) ≅ Im(𝒟).
• Apply the adjunction between the Leibniz superalgebra and Lie superalgebra functors, using L → L_SLie to relate structures.

Experimental results

Research questions

• RQ1 What is the universal central extension of the matrix Leibniz superalgebra 𝔰𝔩(m,n,𝒜) in the category SLeib of Leibniz superalgebras?
• RQ2 How does the kernel of this universal central extension relate to Hochschild and cyclic homology groups of the associative algebra 𝒜?
• RQ3 Can the assumption m+n ≥ 5 from prior Lie superalgebra results be weakened to m+n ≥ 3 in the Leibniz superalgebra setting?
• RQ4 What is the role of the Connes operator 𝒟 in determining the kernel of the universal central extension in SLeib?
• RQ5 How does the universal central extension in SLeib relate to the classical universal central extension in SLie?

Key findings

• The universal central extension of 𝔰𝔩(m,n,𝒜) in the category SLeib is the Steinberg Leibniz superalgebra 𝔰𝔱𝔩(m,n,𝒜), valid for m+n ≥ 3 with characteristic constraints.
• The kernel of this universal central extension is isomorphic to the image of the Connes operator 𝒟 acting on the Hochschild homology group HH₁(𝒜).
• For m+n = 3, the result holds if char(K) ≠ 3; for m+n = 4, it holds if char(K) ≠ 2.
• The kernel is isomorphic to Im(𝒟) ⊆ HH₁(𝒜), which is a proper subgroup of HH₁(𝒜) in general, distinguishing it from the Lie superalgebra case.
• The universal central extension in SLeib is perfect, and the extension is universal due to the perfectness of the target algebra.
• HL₂(𝔰𝔱(m,n,𝒜)) ≅ Im(𝒟), establishing a homological characterization of the second Leibniz cohomology group.