[Paper Review] Universal central extensions of $\mathfrak{sl}(m, n, A)$ of small rank over associative superalgebras

Xabier García‐Martínez, M. Ladra|arXiv (Cornell University)|May 15, 2014

Advanced Topics in Algebra10 references3 citations

TL;DR

This paper completes the classification of universal central extensions and second homology groups for the Lie superalgebras $\mathfrak{sl}(m,n,A)$ and $\mathfrak{st}(m,n,A)$ over associative superalgebras $A$, resolving the remaining cases for small ranks ($m+n=3,4$) where $\mathfrak{sl}(2,1,A)$, $\mathfrak{sl}(3,1,A)$, and $\mathfrak{sl}(2,2,A)$ were previously open. Using super 2-cocycles and homomorphism constructions, it establishes that $\mathfrak{st}(m,n,A)$ is the universal central extension of $\mathfrak{sl}(m,n,A)$ in these cases, with explicit descriptions of $\operatorname{H}_2(\mathfrak{sl}(m,n,A))$ and $\operatorname{H}_2(\mathfrak{st}(m,n,A))$ in terms of cyclic homology and quotients of $A$ by ideals including $mA + [A,A]$. The results unify and complete prior work for all $m+n \geq 3$. The key contribution is a full characterization of the second homology for all such superalgebras.

ABSTRACT

We complete the solution of the problem of finding the universal central extension of the matrix superalgebras $\mathfrak{sl}(m, n, A)$ where $A$ is an associative superalgebra and computing $H_2\big(\mathfrak{sl}(m, n, A)\big)$. The Steinberg Lie superalgebra $\mathfrak{st}(m, n, A)$ has a very important role and we will also find out $H_2\big(\mathfrak{st}(m, n, A)\big)$. In Chen and Sun (arXiv:1311.7079, 2013) it is solved the problem where $m+n \geq 5$ and in Chen and Guay (Algebr. Represent. Theory, 2013) it is solved when $n=0$, so here we work out the three remaining cases $\mathfrak{sl}(2, 1, A), \mathfrak{sl}(3, 1, A)$ and $\mathfrak{sl}(2, 2, A)$.

Motivation & Objective

• To complete the classification of universal central extensions for $\mathfrak{sl}(m,n,A)$ and $\mathfrak{st}(m,n,A)$ over associative superalgebras when $m+n=3,4$, which were previously unresolved.
• To compute $\operatorname{H}_2(\mathfrak{sl}(m,n,A))$ and $\operatorname{H}_2(\mathfrak{st}(m,n,A))$ for all $m+n \geq 3$, completing the characterization of second homology in the superalgebra setting.
• To establish that $\mathfrak{st}(m,n,A)$ is the universal central extension of $\mathfrak{sl}(m,n,A)$ in the three remaining cases: $\mathfrak{sl}(2,1,A)$, $\mathfrak{sl}(3,1,A)$, and $\mathfrak{sl}(2,2,A)$.
• To unify results from prior works on $\mathfrak{sl}(m,n,A)$ for $m+n \geq 5$ and $n=0$, extending them to a complete framework for all $m+n \geq 3$.

Proposed method

• Construction of a Lie superalgebra homomorphism $\rho: \mathfrak{st}(m,n,A)^\sharp \to \widetilde{\mathfrak{st}}(m,n,A)$ via a super 2-cocycle to verify universality of the central extension.
• Use of the Steinberg superalgebra $\mathfrak{st}(m,n,A)$ as a universal central extension, with generators $F_{ij}(a)$ satisfying specific relations derived from the supertrace and bracket identities.
• Verification of the cocycle conditions (C1)–(C3) on the map $v_{ijkl}(a)$, including antisymmetry, graded Jacobi identity, and invariance under superalgebra commutators.
• Explicit computation of the kernel of the universal central extension via the second homology group, using the isomorphism $\ker(\rho) \cong \operatorname{H}_2(\mathfrak{sl}(m,n,A))$.
• Application of the parity change functor $\Pi$ and quotient algebras $A_m = A / (mA + [A,A])$ to describe the structure of $\operatorname{H}_2(\mathfrak{st}(m,n,A))$.
• Combination of results from [3], [4], and this work to produce a complete classification of $\operatorname{H}_2(\mathfrak{sl}(m,n,A))$ and $\operatorname{H}_2(\mathfrak{st}(m,n,A))$ for all $m+n \geq 3$.

Experimental results

Research questions

• RQ1 What is the universal central extension of $\mathfrak{sl}(2,1,A)$ for an associative superalgebra $A$?
• RQ2 What is the structure of $\operatorname{H}_2(\mathfrak{sl}(3,1,A))$ and $\operatorname{H}_2(\mathfrak{st}(3,1,A))$?
• RQ3 How does the second homology of $\mathfrak{sl}(2,2,A)$ and $\mathfrak{st}(2,2,A)$ depend on the superalgebra $A$?
• RQ4 Can the universal central extension of $\mathfrak{sl}(m,n,A)$ be fully characterized for all $m+n \geq 3$?
• RQ5 What is the precise relationship between $\operatorname{H}_2(\mathfrak{sl}(m,n,A))$ and $\operatorname{H}_2(\mathfrak{st}(m,n,A))$ in the small-rank cases?

Key findings

• The universal central extension of $\mathfrak{sl}(2,1,A)$ is $\mathfrak{st}(2,1,A)$, and $\operatorname{H}_2(\mathfrak{st}(2,1,A)) = 0$.
• For $\mathfrak{sl}(3,1,A)$, the universal central extension is $\mathfrak{st}(3,1,A)$, and $\operatorname{H}_2(\mathfrak{st}(3,1,A)) = \Pi(A_2)^6$, where $A_2 = A / (2A + [A,A])$.
• For $\mathfrak{sl}(2,2,A)$, the universal central extension is $\mathfrak{st}(2,2,A)$, and $\operatorname{H}_2(\mathfrak{st}(2,2,A)) = A_2^4 \oplus A_0^2$, with $A_2 = A / (2A + [A,A])$ and $A_0 = A / (0A + [A,A])$.
• The second homology of $\mathfrak{sl}(m,n,A)$ is $\operatorname{HC}_1(A)$ for $m+n \geq 5$ or $m=2,n=1$, and $\operatorname{HC}_1(A) \oplus \operatorname{H}_2(\mathfrak{st}(m,n,A))$ in the remaining cases.
• The full classification of $\operatorname{H}_2(\mathfrak{sl}(m,n,A))$ and $\operatorname{H}_2(\mathfrak{st}(m,n,A))$ for all $m+n \geq 3$ is now complete, with explicit formulas in terms of cyclic homology and quotient algebras.
• The results unify and extend prior work on $\mathfrak{sl}(m,n,A)$ for $m+n \geq 5$ and $n=0$, completing the homological characterization of these superalgebras.