Speciality of Kantor doubles of Jordan brackets: sufficiency of Shestakov's identity (4)

The Kantor double $\operatorname{Kan}(\Gamma,\{,\})$ of a Jordan bracket — a superanticommutative bracket on a unital associative commutative superalgebra $\Gamma$ for which the double is a Jordan superalgebra — is one of the main sources of Jordan superalgebras, including the superalgebras attached to superconformal algebras. Martínez, Shestakov and Zelmanov (2001) proved that every Jordan bracket embeds into a Poisson bracket on a Laurent extension, so every Kantor double is $i$-special, and settled speciality only for restricted classes of brackets. Shestakov (2009) reduced the general question to a precise conjecture: with $D(a) = \{a,1\}$ and the corrected bracket $\langle a,b\rangle = \{a,b\} - D(a)b + aD(b)$, a special Jordan bracket must satisfy the linear identity $\langle\langle a,b\rangle,c\rangle = -D(a)\langle b,c\rangle + (-1)^{|a||b|}D(b)\langle a,c\rangle$, and this identity is conjectured to be sufficient for the Kantor double to be special. The problem is to prove or refute this conjecture — equivalently, to give a complete speciality criterion for Kantor doubles of Jordan brackets, valid for arbitrary $\Gamma$ without finite-generation or nilpotence assumptions. The conjecture is open: positive subclasses (vector brackets, Poisson brackets with $\{\{\Gamma,\Gamma\},\Gamma\} = 0$, and the Novikov-Poisson class, i.e. brackets built from Novikov-Poisson algebras, which combine a commutative associative product with a compatible second product) are settled, exceptional doubles are known, but since all doubles are $i$-special the boundary cannot be detected by identities.

Background

Jordan algebras are commutative algebras satisfying the identity $(x^2 y) x = x^2 (y x)$; they arose as algebras of quantum observables with the product $a \cdot b = \frac{1}{2}(ab+ba)$. The graded counterpart is defined via the Grassmann envelope: given the infinite exterior (Grassmann) algebra $G = G_0 + G_1$ on odd generators, a Jordan superalgebra is a $\mathbb{Z}_2$-graded algebra $J = J_0 + J_1$ whose envelope $G_0 \otimes J_0 + G_1 \otimes J_1$ satisfies the ordinary Jordan identities. Every associative superalgebra $A$ yields a Jordan superalgebra $A^{(+)}$ on the same space with the symmetrized product $a \cdot b = \frac{1}{2}(ab + (-1)^{|a||b|}ba)$, where $|a| \in \{0,1\}$ is the parity. A Jordan superalgebra is special if it embeds into some $A^{(+)}$, exceptional otherwise, and $i$-special if it is a homomorphic image of a special one — equivalently, if it satisfies every $s$-identity, i.e. every identity holding in all $A^{(+)}$.

The Kantor double construction is one of the main sources of Jordan superalgebras. Let $\Gamma = \Gamma_0 + \Gamma_1$ be a unital associative commutative superalgebra over a field $F$ of characteristic $\neq 2$, and let $\{,\} : \Gamma \times \Gamma \to \Gamma$ be a superanticommutative bilinear bracket with $\{\Gamma_i, \Gamma_j\} \subseteq \Gamma_{i+j}$ (superanticommutative means $\{a,b\} = -(-1)^{|a||b|}\{b,a\}$ for homogeneous $a, b$). On the vector space $\operatorname{Kan}(\Gamma, \{,\}) = \Gamma + \Gamma x$, where $x$ is a formal symbol and the grading is $\operatorname{Kan}(\Gamma,\{,\})_0 = \Gamma_0 + \Gamma_1 x$ and $\operatorname{Kan}(\Gamma,\{,\})_1 = \Gamma_1 + \Gamma_0 x$, Kantor (1992) defined the multiplication

$$a \cdot b = ab, \quad a \cdot bx = (ab)x, \quad (bx) \cdot a = (-1)^{|a|}(ba)x, \quad ax \cdot bx = (-1)^{|b|}\{a,b\},$$

for homogeneous $a, b \in \Gamma_0 \cup \Gamma_1$. The bracket is a Jordan bracket when this double is a Jordan superalgebra; Kantor proved that every Poisson bracket — one with $D(a) = \{a,1\} = 0$ for all $a$, in which case $\Gamma$ is a Lie superalgebra under $\{,\}$ — is Jordan. For a Jordan bracket, $D : a \mapsto \{a,1\}$ is a derivation of $\Gamma$. (In characteristic 3 the Jordan-bracket condition involves one additional identity on the bracket, which follows from the others in every characteristic $\neq 2, 3$; see Shestakov 2009.) Simple Kantor doubles account for several classes in the classification of simple Jordan superalgebras, and Kac–Martínez–Zelmanov (2001) showed that the Jordan superalgebras attached to the superconformal algebras of mathematical physics — the infinite-dimensional simple $\mathbb{Z}$-graded Lie superalgebras acting on chiral fields in two-dimensional conformal field theory — are Kantor doubles or are embeddable into Kantor doubles (the Cheng–Kac series, an infinite-dimensional family $J_{CK}(A,d)$ of simple Jordan superalgebras that are free modules of rank 8 over an algebra of functions, attached to the Cheng–Kac superconformal algebra).

Which doubles are special? Three classical results delimit the picture. McCrimmon (1992) proved that every vector bracket, $\{a,b\} = D(a)b - aD(b)$, is special. Shestakov (1993) proved that Poisson doubles are $i$-special, and special exactly when $\{\{\Gamma,\Gamma\},\Gamma\} = 0$; in particular the double of the classical Poisson bracket is exceptional. Martínez–Shestakov–Zelmanov (2001) proved that every Jordan bracket embeds into a Poisson bracket on a Laurent extension of $\Gamma$ (a localization adjoining the inverse of an element), hence every Kantor double is $i$-special, and that the Cheng–Kac superalgebras are special; their speciality criteria for more general brackets required auxiliary hypotheses on $\Gamma$ (such as finite generation and absence of nilpotent ideals).

The decisive difficulty is that the special/exceptional boundary for doubles is invisible to identity-theoretic methods: every Kantor double is $i$-special, so every $s$-identity holds in it, and exceptionality can never be witnessed by a failing identity — only structural arguments apply. Shestakov (2009) introduced the corrected bracket

$$\langle a, b \rangle = \{a, b\} - D(a)b + aD(b),$$

which equals $\{,\}$ for Poisson brackets and is trivial for vector brackets, and proved that speciality of the double forces the linear identity

$$\langle\langle a,b\rangle,c\rangle = -D(a)\langle b,c\rangle + (-1)^{|a||b|}\,D(b)\langle a,c\rangle \tag{4}$$

on $\langle,\rangle$ (Theorem 2.4 of that paper). Identity (4) generalizes both classical criteria: it is trivial on vector brackets and coincides with the condition $\{\{\Gamma,\Gamma\},\Gamma\} = 0$ on Poisson brackets. Shestakov conjectured that (4) is also sufficient for speciality.

Problem Statement

Let $F$ be a field of characteristic $\neq 2$, let $\Gamma = \Gamma_0 + \Gamma_1$ be a unital associative commutative superalgebra over $F$, and let $\{,\}$ be a Jordan bracket on $\Gamma$: a superanticommutative bracket, $\{a,b\} = -(-1)^{|a||b|}\{b,a\}$ and $\{\Gamma_i, \Gamma_j\} \subseteq \Gamma_{i+j}$, for which the Kantor double $\operatorname{Kan}(\Gamma,\{,\}) = \Gamma + \Gamma x$ is a Jordan superalgebra. The double carries the multiplication

$$a \cdot b = ab, \quad a \cdot bx = (ab)x, \quad (bx) \cdot a = (-1)^{|a|}(ba)x, \quad ax \cdot bx = (-1)^{|b|}\{a,b\},$$

for homogeneous $a, b \in \Gamma_0 \cup \Gamma_1$, and the grading $\operatorname{Kan}(\Gamma,\{,\})_0 = \Gamma_0 + \Gamma_1 x$, $\operatorname{Kan}(\Gamma,\{,\})_1 = \Gamma_1 + \Gamma_0 x$. Set $D(a) = \{a,1\}$ (a derivation of $\Gamma$) and $\langle a,b\rangle = \{a,b\} - D(a)b + aD(b)$.

Shestakov's speciality conjecture (2009). If the bracket $\langle,\rangle$ satisfies identity (4),

$$\langle\langle a,b\rangle,c\rangle = -D(a)\langle b,c\rangle + (-1)^{|a||b|}\,D(b)\langle a,c\rangle$$

for all homogeneous $a, b, c \in \Gamma$, then the Kantor double $\operatorname{Kan}(\Gamma,\{,\})$ is special, i.e. embeddable into $A^{(+)}$ for some associative superalgebra $A$.

Prove this conjecture, or refute it by exhibiting a Jordan bracket $(\Gamma,\{,\})$ whose associated bracket $\langle,\rangle$ satisfies identity (4) while $\operatorname{Kan}(\Gamma,\{,\})$ is not special. Equivalently: determine whether identity (4), known to be necessary for speciality, is a complete criterion for the speciality of Kantor doubles of Jordan brackets. The answer must be valid for arbitrary $\Gamma$ — no finite-generation, nilpotence-freeness, or auxiliary generation conditions on $\Gamma$ or $\Gamma_1$ may be assumed — and over arbitrary ground fields of characteristic $\neq 2$ (in characteristic 3 the definition of a Jordan bracket includes one additional identity, redundant in every other characteristic; the conjecture is understood with these standard conventions). Note that $i$-speciality is not at issue: all Kantor doubles are $i$-special (Martínez–Shestakov–Zelmanov 2001), so the target is exactly the special/exceptional boundary.

References

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