Central extensions and outer derivations of Borcherds-Kac-Moody superalgebras: the super analog of Borcherds's 1991 central-extension theorems

In 'Central extensions of generalized Kac-Moody algebras' (J. Algebra 140 (1991) 330-335), Borcherds presented the universal central extension (UCE) of a generalized Kac-Moody algebra by generators $e_i, f_i, h_{ij}$ with $[e_i,f_j]=h_{ij}$, proved that $h_{ij}$ is nonzero only when two columns of the defining matrix are equal, that the algebra is perfect and its own UCE when the matrix has no zero columns, and noted two directions he did not develop: (i) Theorem 1 should extend to generalized Kac-Moody superalgebras, with the involution replaced by an order-4 element and the extra hypothesis that there are no odd real simple roots; (ii) equal columns yield outer derivations generating an $O_n(\mathbb{R})$ action that does not commute with the Cartan subalgebra. In the subsequent literature (standard references, moonshine and physics applications), characterization theorems, character formulas and examples for Borcherds-Kac-Moody superalgebras exist, but the super central-extension structure theory and the derivation classification do not. The problem of constructing them remains open.

Background

A generalized Kac-Moody algebra (also called a Borcherds, GKM or BKM algebra) is the Lie algebra associated to a real symmetric matrix $A=(a_{ij})_{i,j\in I}$ with $a_{ij}\le 0$ for $i\ne j$ and $2a_{ij}/a_{ii}\in\mathbb{Z}$ whenever $a_{ii}>0$; unlike an ordinary Kac-Moody matrix it may have $a_{ii}\le 0$, whose simple roots are called imaginary (Borcherds 1988). It is generated by Chevalley generators $e_i, f_i$ and a Cartan part, with the usual Serre relations only for real simple roots ($a_{ii}>0$) and commuting relations for imaginary pairs with $a_{ij}=0$. A Lie algebra $G$ is perfect if $[G,G]=G$; every perfect Lie algebra has a universal central extension (UCE), and $G$ is its own UCE exactly when its second homology vanishes. Borcherds (1991) proved: a $\mathbb{Z}$-graded Lie algebra with an involution $\omega$ acting as $-1$ on the degree-0 part and an invariant bilinear form that is almost positive definite ($(a,-\omega(a))>0$ on each nonzero graded piece) is, when the nonzero graded pieces are finite dimensional, precisely a central extension with a presentation by generators $e_i, f_i, h_{ij}$ and relations $[e_i,f_j]=h_{ij}$, $[h_{ij},e_k]=\delta_i^j a_{ik}e_k$, the Serre relations for real roots and the commuting relations above (Theorem 1); $h_{ij}\ne 0$ only when the $i$th and $j$th columns of $A$ are equal, and for $i\ne j$ such $h_{ij}$ is central (Theorem 2); the $h_{ij}$ with equal columns form a basis of the h-subalgebra, and if $A$ has no zero columns then the algebra is perfect and its own UCE (Theorem 2); the presentation without Serre relations is its own UCE (Theorem 3); the $h_{ij}$ are linearly independent even when $A$ has zero columns, via lowest-weight modules (Theorem 4). The monster Lie algebra illustrates the dimensions: the UCE Cartan is one-dimensional per real simple root plus $24^2$ dimensions for each positive integer. For ordinary Kac-Moody algebras no two columns can be equal, so they are their own UCE.

A Borcherds-Kac-Moody (BKM) superalgebra is the $\mathbb{Z}_2$-graded analog: the simple generators $e_i, f_i$ carry a parity, the invariant form is supersymmetric, roots may occur with negative multiplicity (superroots), and Serre relations become super-Serre relations (Borcherds 1992; Ray 2006). The 1991 paper remarks that Theorem 1 should generalize to superalgebras after the 'usual' changes, replacing the involution by an element of order 4 whose square is $1$ on even and $-1$ on odd elements, and adding the hypothesis that there are no odd real simple roots; many finite-dimensional simple superalgebras satisfy it. Ray (2000) instead characterized a class of graded Lie superalgebras from root-system data, and Ray (2001) observed that the almost-positive-definite condition itself has little meaning for superalgebras (only the classical Lie superalgebra families $A(1,n)$, $B(0,n)$ and $B^{(1)}(0,n)$ among those that are not ordinary Lie algebras satisfy it), so a super analog must be rebuilt on different hypotheses. Modern working definitions (Harrison-Paquette-Volpato 2019; Palmkvist 2012; Cederwall-Palmkvist 2023) use only generators $e_i, f_i, h_i$ with $[e_i,f_j]=\delta_{ij}h_j$ and never the $h_{ij}$. The general categorical theory of UCE of Lie superalgebras (perfect objects, existence) is standard (Neher 2003).

A derivation of a Lie (super)algebra $G$ is a linear map $d$ with $d([x,y])=[dx,y]+[x,dy]$ (with the super sign); inner derivations have the form $\operatorname{ad} x$, and $\operatorname{Der}(G)/\operatorname{ad}G$ measures outer derivations. Borcherds (1991, remark after Theorem 4) noted that equal columns of $A$ produce outer derivations $h$ with $[h,e_i]=e_j$, $[h,e_j]=-e_i$, $[h,f_i]=f_j$, $[h,f_j]=-f_i$ (zero on the other Chevalley generators), that $n$ equal columns generate an $O_n(\mathbb{R})$ action, and that these derivations do not always commute with the Cartan subalgebra (in the universal central extension, $[h,h_{ii}]=h_{ij}+h_{ji}\ne 0$). The general framework for GKM algebras (Ray 2001) quotients out part of the center and adjoins commuting outer derivations; the non-commuting $O_n(\mathbb{R})$ family lies outside that framework.

Problem Statement

Part I (super analog of the central extension theorems). Let $A=(a_{ij})_{i,j\in I}$ be a real symmetric matrix with $a_{ij}\le 0$ for $i\ne j$ and $2a_{ij}/a_{ii}\in\mathbb{Z}$ whenever $a_{ii}>0$, and let $G(A)$ be the associated Borcherds-Kac-Moody (BKM) superalgebra in the sense of Borcherds 1992 and Ray 2006: the $\mathbb{Z}_2$-graded Lie superalgebra over $\mathbb{R}$ (or $\mathbb{C}$, consistently) generated by an even Cartan subalgebra and by Chevalley generators $e_i, f_i$ of assigned parities, with the supersymmetric form and super-Serre relations for real simple roots and the commuting relations for imaginary pairs with $a_{ij}=0$. Establish the super analog of the central extension theory of Borcherds 1991:

(1) Presentation and characterization theorem. Prove the super analog of the assertion stated (without proof) in the 1991 remark: a $\mathbb{Z}$-graded Lie superalgebra whose nonzero graded pieces are finite dimensional, with an element of order 4 whose square is $1$ on the even and $-1$ on the odd part, carrying an invariant supersymmetric form that is almost positive definite on the graded pieces (with the super sign conventions), is a central extension of a BKM superalgebra with no odd real simple roots, presented by generators $e_i, f_i, h_{ij}$ and relations $[e_i,f_j]=h_{ij}$, $[h_{ij},e_k]=\delta_i^j a_{ik}e_k$ (with the super signs), the super-Serre relations, and $[e_i,e_j]=0=[f_i,f_j]$ when $a_{ii}\le 0$, $a_{jj}\le 0$, $a_{ij}=0$. Since, as the Background records, the non-super almost-positive-definite condition has little meaning for superalgebras as it stands (Ray 2001), part of the task is to give the super sign conventions and the correct super version of that hypothesis, and to determine whether it together with the 'no odd real simple roots' hypothesis is exactly what is needed: prove what fails, and prove the corrected characterization theorem, when odd real simple roots are allowed or when the literal transcription of the almost-positive-definite hypothesis degenerates.

(2) Structure of the center. Prove the super analogs of Theorems 2 and 4: $h_{ij}=0$ unless the $i$th and $j$th columns of $A$ are equal; the $h_{ij}$ with equal columns form a basis of the h-subalgebra and are linearly independent (including when $A$ has zero columns); with the parity bookkeeping explicit - the Cartan subalgebra is even, so the statement must say where an $h_{ij}$ of odd parity (arising from equal columns of differing parity) lives and how the h-subalgebra is graded.

(3) Perfectness and self-UCE. Prove that when $A$ has no zero columns, $G(A)$ is perfect and its own universal central extension, or else identify the precise super obstruction and prove the corrected statement. The general theory of universal central extensions of Lie superalgebras (Neher 2003) guarantees the UCE exists for perfect $G(A)$; what is required is its explicit identification, as the 1991 paper achieves in the non-super case.

Part II (outer derivations). The source constructs, for a GKM algebra, the outer derivations attached to a set of $n$ equal columns of $A$, notes that they generate an $O_n(\mathbb{R})$ action, and notes that they do not always commute with the Cartan subalgebra, but does not develop this observation further. Develop it: for a GKM algebra $G(A)$, and as the super extension of the same question for a BKM superalgebra, classify the outer derivations: determine whether every derivation is a sum of an inner derivation, a standard commuting (Cartan-type) derivation (a derivation induced by a linear map of the Cartan subalgebra that extends to $G(A)$), and a member of the $O_n(\mathbb{R})$ family attached to a set of $n$ equal columns of $A$; in the super case also account for the odd derivations. Determine the structural role of the $O_n(\mathbb{R})$ family noted in the source: these derivations do not commute with the Cartan subalgebra (in the universal central extension, $[h,h_{ii}]=h_{ij}+h_{ji}\ne 0$), so they are not covered by the standard semidirect general setup of Ray 2001; determine how they act on the center of the universal central extension and how the derivation algebra combines with the central extension structure of Part I.

The two parts form one objective: the enveloping structure theory of Borcherds-Kac-Moody (super)algebras - universal central extensions plus derivations. Part I(1) proves the super direction that the source paper explicitly states; Parts I(2)-(3) and Part II formulate the corresponding super completions of the source's remaining central-extension theorems and of its undeveloped derivation observation.

References

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