Speciality of finite torsion-free Lie conformal (super)algebras via Gröbner–Shirshov bases with arbitrary locality functions

Whether every finite torsion-free Lie conformal (super)algebra is special, i.e., embeds into an associative conformal algebra, is a foundational open problem of conformal algebra theory — the analogue of the Poincaré–Birkhoff–Witt theorem — conjectured affirmatively by Roitman around 2000. Kolesnikov (2018) reduced it to a combinatorial criterion: for a Lie conformal algebra $L$ that is free of finite rank over $\mathbb{k}[\partial]$ with basis $X$, $L$ is special if and only if some locality function $N:X\times X\to\mathbb{Z}_+$ makes the Gröbner–Shirshov basis (GSB) of the universal associative conformal envelope $U(L;X,N)$ free of nonzero $\mathbb{k}[\partial]$-linear combinations of generators. The problem asks for a resolution of the speciality question in the general case. Whether the GSB completion of the envelope relations terminates with a finite GSB, which is what makes the criterion effective, is a closely related open issue but not part of the required resolution.

Background

Conformal algebras were introduced by Kac in the 1990s as the algebraic skeleton of vertex operator algebras from two-dimensional conformal field theory: a conformal algebra is a vector space $C$ over a field $\mathbb{k}$ of characteristic zero equipped with a derivation $\partial$ and bilinear products $(n)$, $n\in\mathbb{Z}_+$, satisfying sesquilinearity ($\partial a\,(n)\,b=-n\,a\,(n-1)\,b$, $a\,(n)\,\partial b=\partial(a\,(n)\,b)+n\,a\,(n-1)\,b$) and locality ($a\,(n)\,b=0$ for all sufficiently large $n$). A Lie conformal algebra is one whose $\lambda$-bracket $[a(\lambda)b]=\sum_{n\ge 0}a\,(n)\,b\,\lambda^n$ is skew-symmetric and satisfies the conformal Jacobi identity; the basic finite examples are the current algebra $\mathrm{Cur}\,\mathfrak{g}$ over a Lie algebra $\mathfrak{g}$ and the Virasoro conformal algebra $\mathrm{Vir}$ with $[v(\lambda)v]=\partial v+2\lambda v$. An associative conformal algebra satisfies the conformal associativity identity, and every associative conformal algebra $A$ becomes a Lie conformal algebra $A^{(-)}$ under the conformal commutator $[x(\lambda)y]=x(\lambda)y-\{y(\lambda)x\}$, where $\{a(n)b\}=\sum_{s\ge 0}(-1)^{n+s}\partial^{(s)}(a\,(n+s)\,b)$ and $\partial^{(s)}$ denotes divided powers, i.e., $h^{(n)}=h^n/n!$ for a formal variable $h$ (so $\lambda^{(n)}=\lambda^n/n!$ as well). By analogy with the Poincaré–Birkhoff–Witt (PBW) theorem and with special Jordan algebras, a Lie conformal algebra $L$ is called special if it embeds into some associative conformal algebra so that its bracket is the conformal commutator. Roitman constructed the universal enveloping conformal algebra $U(L)$ and showed that the naive conformal PBW statement fails: a finitely generated Lie conformal algebra embeddable into an associative one must have at most linear growth of its locality function, while free Lie conformal algebras have quadratic growth and are therefore non-special. He also proved that uniformly bounded locality implies embeddability, conjectured that every Lie conformal algebra of finite type (finite rank over $\mathbb{k}[\partial]$) is embeddable, and posed the companion conjecture that every torsion-free finite type Lie conformal algebra has a faithful finite-rank conformal module (a conformal Ado theorem); for finite type algebras the locality function has merely linear, not bounded, growth, so his embedding theorem does not decide their speciality. On the computational side, Bokut, Fong, and Ke developed a Composition–Diamond lemma (the noncommutative Gröbner-basis theory now called Gröbner–Shirshov bases, GSB) for associative conformal algebras whose generators have a bounded locality function: a GSB is a confluent system of defining relations relative to which the irreducible (normal) words form a linear basis. Kolesnikov (2018) removed the boundedness restriction: for an arbitrary locality function $N:X\times X\to\mathbb{Z}_+$ he realized the free associative conformal algebra $\mathrm{Conf}(X,N)$ as a module over an ordinary associative algebra generated by symbols $\partial$, $L^a_n$ and $R^a_n$ (left and right conformal multiplication operators; for a Lie conformal algebra $L$ with basis $X$ the corresponding operator algebra is denoted $A(L;X)$) and proved that its GSB consists of relations (7), (8), (9), so that GSB theory for conformal algebras follows from the CD-lemma for modules over associative algebras and existing computer algebra systems can in principle run the computations. For a Lie conformal algebra $L$ with $\mathbb{k}[\partial]$-basis $X$ and bracket $[a(\lambda)b]=\sum_c f^{a,b}_c(\partial,\lambda)c$, he then proved the criterion: $L$ is special if and only if there exists $N$ such that the GSB completion of the ideal generated by the commutator relations $a\,(n)\,b-\{b\,(n)\,a\}-g^{a,b}_n$ in $\mathrm{Conf}(X,N)$ contains no nonzero $\mathbb{k}[\partial]$-linear combination of elements of $X$; the corresponding quotient is the universal envelope $U(L;X,N)$. He carried out two finite computations — the Heisenberg–Virasoro algebra $\mathrm{Vir}\ltimes\mathrm{Cur}\,\mathbb{k}$ (confirming speciality, with basis $\partial^s(L^v_0)^k a$) and $\mathrm{Vir}$ at $N(v,v)=3$ — but the paper gives no general criterion for when this completion terminates, and both Roitman's conjectures above and the termination question remain undetermined for arbitrary finite torsion-free $L$.

Problem Statement

Let $\mathbb{k}$ be a field of characteristic zero and let $L$ be a Lie conformal algebra that is torsion-free and finite, i.e., free of finite rank as a $\mathbb{k}[\partial]$-module (the $\partial$-torsion of any conformal algebra lies in its annihilator, so this is the natural hypothesis), with $\mathbb{k}[\partial]$-basis $X$ and structure constants $[a(\lambda)b]=\sum_{c\in X}f^{a,b}_c(\partial,\lambda)c$. For a locality function $N:X\times X\to\mathbb{Z}_+$, let $U(L;X,N)$ be the universal associative conformal envelope of $L$ relative to $N$: the quotient of the free associative conformal algebra $\mathrm{Conf}(X,N)$ by the ideal generated by $a\,(n)\,b-\{b\,(n)\,a\}-g^{a,b}_n$ for $a,b\in X$, $n\in\mathbb{Z}_+$, where $g^{a,b}_n$ is the coefficient of $\lambda^{(n)}$ in the bracket. The question below covers Lie conformal superalgebras exactly as in the source: in the $\mathbb{Z}_2$-graded case all defining identities are read with the standard Koszul sign conventions, and the operator relations of the $A(L;X)$-module acquire the parity factors $(-1)^{p(a)p(b)}$ of Remark 3 of [1] (for instance, the commutator relations become $L^a_n b-(-1)^{p(a)p(b)}R^a_n b-g^{a,b}_n$), where $p(a)\in\mathbb{Z}_2$ denotes the parity of $a\in X$. The problem is the following question: Is every finite torsion-free Lie conformal (super)algebra special, i.e., does there always exist a locality function $N$ such that the canonical map $L\to U(L;X,N)$ is injective? Equivalently, by the criterion of [1], does there exist $N$ such that the completed Gröbner–Shirshov basis of the ideal generated by the envelope relations in $\mathrm{Conf}(X,N)$, computed as the GSB of the corresponding $A(L;X)$-module generated by $X$, contains no nonzero $\mathbb{k}[\partial]$-linear combination of elements of $X$?

References

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