Indecomposable finite-dimensional unital bimodules over the Jordan superalgebra $D(0)$, the unitization of the Kaplansky superalgebra $K_3$

The 4-dimensional Jordan superalgebras $D(t)$ form a one-parameter family in Kac's classification of simple finite-dimensional Jordan superalgebras. Martínez and Zelmanov classified the indecomposable finite-dimensional bimodules over $D(t)$ for $t \neq -1, 0, 1$, and Kashuba and Serganova later resolved the two degenerate simple values: for $t = 1$ the category of unital bimodules is semisimple with simple objects $\Lambda^a(V)$, and for $t = -1$ ($\cong M(1,1)^{(+)}$) they gave a full block-by-block description, with some blocks of wild representation type. The remaining excluded value $t = 0$ is different in nature: $D(0) = F{\cdot}1 \oplus K_3$ is the unitization of the Kaplansky superalgebra $K_3$ and hence not simple, and its bimodule category is not semisimple (the regular module is a non-split extension of the trivial module by the ideal $K_3$). For $t = 0$ only the irreducible modules are classified (Trushina, in characteristic 0 and in characteristic $p \neq 2$); no indecomposable-level or block-level description exists in the literature. This problem asks for a classification of the indecomposable finite-dimensional unital $D(0)$-bimodules over an algebraically closed field of characteristic 0, equivalently a block-level description of the category including the representation type of each block, together with the analogous determination for one-sided (special) $D(0)$-bimodules.

Background

A Jordan superalgebra is a $\mathbb{Z}_2$-graded algebra $J = J_{\bar 0} \oplus J_{\bar 1}$ over a field of characteristic $\neq 2$ whose multiplication is supercommutative, $a \circ b = (-1)^{|a||b|} b \circ a$ for homogeneous $a, b$, and satisfies the Jordan superidentity (the graded linearization of $(a^2 \circ b) \circ a = a^2 \circ (b \circ a)$). The basic example is the symmetrization $A^{(+)}$ of an associative superalgebra $A$, with product $a \circ b = \frac{1}{2}(ab + (-1)^{|a||b|} ba)$. Over an algebraically closed field of characteristic 0, the simple finite-dimensional Jordan superalgebras were classified by Kac and completed in later work; besides the Hermitian series $M_{m,n}^{(+)}$, $Q(n)^{(+)}$, $\operatorname{Josp}(m,2n)$, $JP(n)$ and the Kantor series, the list contains the superalgebra $J(V,f)$ of a nondegenerate supersymmetric bilinear form $f$ on a superspace $V$ (obtained by adjoining a unit $1$ to $V$: $1 \circ v = v$ and $v \circ w = f(v,w){\cdot}1$ for $v, w \in V$), a one-parameter family of 4-dimensional superalgebras $D(t)$, the 3-dimensional non-unital Kaplansky superalgebra $K_3$, and the exceptional 10-dimensional $K_{10}$.

For a parameter $t$, the superalgebra $D(t)$ has even part $\langle e_1, e_2\rangle$ and odd part $\langle x, y\rangle$, with $e_i^2 = e_i$, $e_1 \circ e_2 = 0$, $e_1 \circ x = e_2 \circ x = x/2$ (and similarly for $y$), $x \circ y = -y \circ x = e_1 + t e_2$, and unit $1 = e_1 + e_2$; it is simple exactly for $t \neq 0$. Two parameter values are isomorphism degenerations: $D(-1) \cong M(1,1)(F)^{(+)}$, the symmetrized $2 \times 2$ matrix superalgebra, and $D(1) \cong J(V,f)$ with $V = \langle e_1 - e_2\rangle_{\bar 0} \oplus \langle x, y\rangle_{\bar 1}$. At $t = 0$ the superalgebra is not simple: $D(0) = F{\cdot}1 \oplus K_3$ is the unitization (unital hull) of the Kaplansky superalgebra $K_3 = \langle e_1, x, y\rangle$ (where $x \circ y = e_1$), a 3-dimensional simple non-unital superalgebra that occurs as an ideal of $D(0)$; $D(0)$ is also not a superform algebra, since $x \circ y$ has a nonzero component along $e_1 - e_2$.

A Jordan bimodule over $J$ is a superspace $M$ with left and right actions of $J$ such that the split null extension $J \oplus M$ (the multiplication of $J$ extended by $M \circ M = 0$ and the given actions) is a Jordan superalgebra. When $J$ is unital, every Jordan bimodule decomposes into a direct sum of a unital bimodule (the identity acts as the identity operator) and a one-sided (or special) bimodule, characterized by $\{J, M, J\} = 0$ where $\{a, m, b\} = (a m) b + a (m b) - (-1)^{|a||m|} m (a \circ b)$ is the Jordan triple product; on a one-sided bimodule the identity acts as $\frac{1}{2}\operatorname{id}$ (the degenerate alternative, the identity acting as $0$, forces the whole action of $J$ to be trivial). For an even idempotent $e \in J$, every bimodule carries a Peirce decomposition $M = M_0(e) \oplus M_{1/2}(e) \oplus M_1(e)$ according to the eigenvalues $0, \frac{1}{2}, 1$ of the action of $e$. Two Lie-theoretic tools underlie the known classifications: the Tits–Kantor–Koecher (TKK) construction, which assigns to $J$ a short $\mathbb{Z}$-graded Lie superalgebra $\operatorname{Lie}(J)$ (grading concentrated in degrees $-1, 0, 1$) and relates unital Jordan bimodules to short-graded modules over $\operatorname{Lie}(J)$ and its universal central extension, and the universal associative enveloping algebra, which realizes one-sided bimodules as modules over an associative algebra. By these methods, Martínez and Zelmanov classified the indecomposable finite-dimensional bimodules over $D(t)$ for all $t \neq -1, 0, 1$, and the categories of unital and one-sided bimodules have since been described at the block level for the degenerate simple values $t = \pm 1$ (see Kashuba–Serganova [10]); the value $t = 0$ is the one not covered by any of these works.

Problem Statement

Let $F$ be an algebraically closed field of characteristic 0, and let $D(0)$ be the Jordan superalgebra at the parameter value $t = 0$ of the family $D(t)$: even part $\langle e_1, e_2 \rangle$, odd part $\langle x, y \rangle$, $e_i \circ e_i = e_i$, $e_1 \circ e_2 = 0$, $e_i \circ x = e_i \circ y = x/2$ (respectively $y/2$), $x \circ y = -y \circ x = e_1$, unit $1 = e_1 + e_2$; equivalently $D(0) = F{\cdot}1 \oplus K_3$ is the unitization of the Kaplansky superalgebra $K_3 = \langle e_1, x, y \rangle$. Let $D(0)\text{-mod}_1$ denote the category of finite-dimensional unital Jordan bimodules over $D(0)$ (superspaces $M$ on which $1$ acts as the identity and such that the split null extension $D(0) \oplus M$, with $M \circ M = 0$, is a Jordan superalgebra), and $D(0)\text{-mod}_{1/2}$ the category of finite-dimensional one-sided (special) bimodules (those on which the identity acts as $\frac{1}{2}\operatorname{id}$ and the Jordan triple product $\{a, m, b\} = (a m) b + a (m b) - (-1)^{|a||m|} m (a \circ b)$ vanishes identically for homogeneous $a, b \in D(0)$ and $m \in M$). Determine the structure of these categories, as follows.

1. Determine the block structure of $D(0)\text{-mod}_1$: for each block (connected component under nontrivial extensions), decide whether it is equivalent to the category of finite-dimensional modules over an explicitly given associative algebra, and determine the representation type of each block (finite: only finitely many indecomposable objects up to isomorphism; tame: indecomposables of each dimension occur in finitely many one-parameter families; wild: the classification problem contains the classification of finite-dimensional modules over a free associative algebra on two generators, so no explicit list of all indecomposables can exist).

2. Classify the indecomposable finite-dimensional unital $D(0)$-bimodules explicitly and completely; if some block has wild representation type, prove this and give the block-level description of item 1 as the definitive answer. A classification of $D(0)\text{-mod}_1$ recovers the irreducible unital $D(0)$-modules classified by Trushina and the regular module $\operatorname{Reg}(D(0))$ (the bimodule defined on $D(0)$ itself by its own multiplication) as a non-split extension of the trivial module (the one-dimensional module on which $1$ acts as the identity and $K_3$ as $0$) by the ideal $K_3$.

3. Determine whether nonzero finite-dimensional one-sided $D(0)$-bimodules exist; if so, classify their indecomposables explicitly and completely or describe the blocks of $D(0)\text{-mod}_{1/2}$ in the same sense as item 1.

The characteristic-0 case is the target; modular variants are not required.

References

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