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Jordan algebras and weight modules

Olivier Mathieu, Michael Lau

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Olivier Mathieu, Michael Lau. Jordan algebras and weight modules. 2023. hal-04317040

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Jordan algebras and weight modules

Michael Lau¹* and Olivier Mathieu²

¹Université Laval

Département de mathématiques et de statistique

Québec, QC, Canada G1V 0A6

Michael.Lau@mat.ulaval.ca

²Institut Camille Jordan

UMR 5028 du CNRS, Université Claude Bernard Lyon 1

SUSTech International Center for Mathematics, Shenzhen, China

69622 Villeurbanne Cedex, France

mathieu@math.univ-lyon1.fr

**Abstract:** We consider bounded weight modules for the universal central extension $sl_2(J)$ of the Tits-Kantor-Koecher algebra of a unital Jordan algebra $J$. Universal objects called Weyl modules are introduced and studied, and a combinatorial dominance criterion is given for analogues of highest weights.

Specializing $J$ to the free Jordan algebra $J(r)$ of rank $r$, the category $C^{fin}$ of finite-dimensional $\mathbb{Z}$-graded $sl_2(J)$-modules shares many properties with the representation theory of algebraic groups. Using a deep result of Zelmanov, we show that this subcategory admits Weyl modules. By analogy, we conjecture that $C^{fin}$ is a highest weight category. The resulting homological properties would then imply cohomological vanishing results previously conjectured as a way of determining graded dimensions of free Jordan algebras.

**Keywords:** weight modules, Weyl modules, Tits-Kantor-Koecher construction, Jordan algebras, highest weight categories

<footnote>*Corresponding author</footnote>
**MSC2010:** primary 17B10; secondary 17B60, 17C05, 17C50

**Statements and Declarations:** The authors report that they have no competing interests to declare. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

# 1 Introduction

Let $J$ be a unital Jordan algebra over an algebraically closed field $k$ of characteristic 0. Tits defined a Lie algebra structure on the space $(sl_2(k) \otimes_k J) \oplus \text{Inn } J$, where $\text{Inn } J$ is the Lie algebra of inner derivations [15]. This construction was later generalized by Kantor [9] and Koecher [12] and is now called the *Tits-Kantor-Koecher algebra* and denoted by $TKK(J)$. The Lie algebra $TKK(J)$ is perfect and admits a universal central extension $sl_2(J)$ described [2]. See also [1, 13].

With the exception of $r = 1$ and $r = 2$, the structure of the free Jordan algebras $J(r)$ is unknown. However, it was proved in [10] that its structure is determined by the $sl_2(k)$-invariants of $H_*(sl_2(J(r)))$. The following conjecture was provided.

**Conjecture A [10, Conj. 3].** $H_n(sl_2(J(r)))^{sl_2(k)} = 0$, for all $n > 0$.

Verification of this conjecture would give a recursive method to compute the dimensions of the graded components of $J(r)$. In the present paper, we will interpret this conjecture in terms of representation theory.

Let $\{e, f, h\}$ be the standard basis of $sl_2(k)$, and let $h(a) = h \otimes a \in sl_2(J)$ for all $a \in J$. For any $sl_2(J)$-module $M$ and integer $j$, let $M_j$ be the weight space $M_j = \{m \in M : h(1).m = jm\}$. The module $M$ is said to be *bounded of level n* if

$$M = \bigoplus_{-n \le j \le n} M_j$$

for some nonnegative integer $n$ with $M_n \ne 0$.

A vector space $V$ endowed with a linear map $\rho: J \to \text{End}(V)$ is called a *J-space* if it satisfies

$$(J1) \quad [\rho(a), \rho(a^2)] = 0,$$

$$(J2) \quad [[\rho(a), \rho(b)], \rho(c)] = 4\rho(\partial_{a,b} c),$$
for all $a, b, c \in J$, where $\partial_{a,b}(c) = a(cb) - (ac)b$. When $\rho(1)$ acts by multiplication by $n$, $V$ is called a $J$-space of level $n$. For any bounded $sl_2(J)$-module $M$ of level $n$, the weight space $M_n$ has a structure of $J$-space of level $n$, where $\rho(a)$ is the action of $h(a)$. The $J$-space $V$ is said to be dominant of level $n$ if $V = M_n$ for some bounded $sl_2(J)$-module $M$ of level $n$.

Our first main result characterizes dominant $J$-spaces of level $n$. For any partition $\sigma = (\sigma_1, \sigma_2, \dots, \sigma_m)$ of $n+1$, we write $|C_\sigma|$ for the cardinality of the corresponding conjugacy class in $S_{n+1}$ and sgn($\sigma$) for its signature. We write $\rho_\sigma(a)$ for the expression $\rho(a^{\sigma_1})\rho(a^{\sigma_2})\cdots\rho(a^{\sigma_m})$ for all $a \in J$. The following result appears as Theorem 2.6.

**Theorem B.** Let $(V, \rho)$ be a $J$-space of level $n$. Then $V$ is dominant if and only if it satisfies the following condition

$$ \sum_{\sigma \vdash n+1} \operatorname{sgn}(\sigma) |C_{\sigma}| \rho_{\sigma}(a) = 0. $$

For any dominant $J$-space of level $n$, the Weyl module $\Delta(M)$ is the bounded $sl_2(J)$-module of level $n$ defined by the following universal property:

$$ \operatorname{Hom}_{sl_2(J)}(\Delta(V), M) = \operatorname{Hom}_J(V, M_n), $$

for all bounded $sl_2(J)$-modules $M$ of level $n$, where homomorphisms of $J$-spaces are defined to be linear maps commuting with the action of $J$.

Assume now that $J = \bigoplus_{n \ge 0} J_n$ is a finitely generated $\mathbb{Z}_+$-graded unital Jordan algebra with $J_0 = k1$. The grading on $J$ clearly induces a grading on the Lie algebra $sl_2(J)$. We show that the category $\mathcal{C}^{\text{fin}}$ of finite-dimensional $\mathbb{Z}$-graded $sl_2(J)$-modules admits a Weyl module for each dominant $J$-space.

**Theorem C.** For any $\mathbb{Z}$-graded finite-dimensional dominant $J$-space $V$ of level $n$, the Weyl module $\Delta(V)$ is finite-dimensional.

Theorem C is nontrivial and uses a deep result of Zelmanov on nil Jordan algebras. It appears as Theorem 3.1 in the paper, and shows that the category $\mathcal{C}^{\text{fin}}$ shares many properties with categories of representations of reductive algebraic groups in positive characteristic. This leads to the following conjecture.
**Conjecture D.** The category $\mathcal{C}^{fin}$ is a highest weight category, in the sense of Cline, Parshall, and Scott.

A proof of Conjecture D would also settle Conjecture A of [10].

**Theorem E.** Let $r \ge 1$. If $\mathcal{C}^{fin}$ is a highest weight category, then Conjecture A holds.

Theorem E appears below as Theorem 3.2.

# 2 Bounded weight modules

Let $J$ be a unital Jordan algebra over an algebraically closed field $k$ of characteristic zero. All vector spaces, algebras, and tensor products will be taken over $k$. For every $a, b \in J$, let $L_a: J \to J$ be the multiplication operator $L_a(b) = ab$. Write $\partial_{a,b} = [L_a, L_b]$, and let $\kappa$ be one half the Killing form on $\mathfrak{sl}_2(k)$. We write $\mathrm{Inn} J$ for the set $\{\partial_{a,b} : a, b \in J\}$ of inner derivations of $J$. The element $x \otimes a$ in the vector space $\mathfrak{sl}_2(k) \otimes J$ will be denoted by $x(a)$. We fix a standard basis $\{h, e, f\}$ of $\mathfrak{sl}_2(k)$ with $[h, e] = 2e$, $[h, f] = -2f$, and $[e, f] = h$.

## 2.1 Tits construction

In his 1962 paper, Tits defined a Lie algebra structure on the space

$$TKK(J) := \mathfrak{sl}_2(k) \otimes J \oplus \mathrm{Inn} J,$$

with Lie bracket

$$1. \text{ (T1)} \quad [x(a), y(b)] = [x, y](ab) + \kappa(x, y)\partial_{a,b}$$

$$2. \text{ (T2)} \quad [\partial, x(a)] = x(\partial a),$$

where $x(a) = x \otimes a$ for any $x, y \in \mathfrak{sl}_2$, $\partial \in \mathrm{Inn} J$, and $a, b \in J$. This construction was later generalized to Jordan pairs and triple systems by Kantor and Koecher, and $TKK(J)$ is known as the *Tits-Kantor-Koecher (TKK) algebra*.
## 2.2 Tits-Allison-Gao construction

The TKK algebra is perfect, that is, $TKK(J) = [TKK(J), TKK(J)]$, so $TKK(J)$ admits a universal central extension, which we denote by $sl_2(J)$. This Lie algebra was nicely described in the 1996 paper of Allison and Gao [2] in the context of universal coverings of the Steinberg unitary Lie algebras $stu_n(J)$ for $n \ge 3$. The case where $n = 3$ corresponds to $TKK(J)$. See also [1] for equivalent formulas written in terms of $sl_2(k)$.

As a vector space,

$$sl_2(J) = (sl_2(K) \otimes J) \oplus \{J, J\},$$

where $\{J, J\} = (\wedge^2 J)/S$ and $S = \text{Span}\{a \wedge a^2 \mid a \in J\}$. For any $a, b \in J$, we write $\{a, b\}$ for the image of $a \wedge b$ in $\{J, J\}$. The bracket on $sl_2(J)$ is given by

$$(R1) \quad [x(a), y(b)] = [x, y](ab) + \kappa(x, y)\{a, b\}$$

$$(R2) \quad [\{a, b\}, x(c)] = x(\partial_{a,b} c)$$

$$(R3) \quad [\{a, b\}, \{c, d\}] = \{\partial_{a,b} c, d\} + \{c, \partial_{a,b} d\}.$$

for all $a, b, c, d \in J$ and $x, y \in sl_2(k)$. It is a bit tricky to show that (R3) is skew-symmetric [2].

There is an obvious Lie algebra epimorphism $sl_2(J) \to TKK(J)$ which is the identity on $sl_2 \otimes J$ and sends the symbol $\{a, b\}$ to $\partial_{a,b}$. When the Jordan algebra $J$ is associative, we have $\{J, J\} = HC_1(J)$, and the construction specializes to results of Kassel and Loday[11].<fnref n="1" />

## 2.3 The short grading of $sl_2(J)$

The Lie algebra $\mathfrak{G} := sl_2(J)$ decomposes with respect to the adjoint action of $\frac{1}{2}h(1)$ as $\mathfrak{G} = \mathfrak{G}_{-1} \oplus \mathfrak{G}_0 \oplus \mathfrak{G}_1$, where

$$\mathfrak{G}_{-1} = f \otimes J,$$

$$\mathfrak{G}_0 = h \otimes J \oplus \{J, J\}, \text{ and}$$

$$\mathfrak{G}_1 = e \otimes J.$$

This decomposition is a root grading in the sense of Berman-Moody [3], and is called the short grading of $\mathfrak{G}$. In fact, every root-graded Lie algebra

<footnote n="1">¹The notation in [11] differs slightly from the modern conventions-Kassel-Loday write $HC_2$ for what is now denoted as $HC_1$.</footnote>
of type $A_1$ has a universal central extension isomorphic to $sl_2(J)$ for some unital Jordan algebra $J$. See [13] or [1] for details.

## 2.4 Bounded modules

For any $sl_2(J)$-module $V$ and any $k \in \mathbb{Z}$, let

$$V_k = \{v \in V : h(1)v = k v\}.$$

An $sl_2(J)$-module $V$ is said to be a *bounded weight module of level $\ell$* if $V = \oplus_{-\ell \le m \le \ell} V_m$, with $V_\ell \ne 0$.

## 2.5 J-spaces

Recall that a vector space $M$ endowed with a linear map $\rho : J \to \text{End}(M)$ is called a *J-space* if $\rho$ satisfies

$$(J1) \quad [\rho(a), \rho(a^2)] = 0,$$

$$(J2) \quad [[\rho(a), \rho(b)], \rho(c)] = 4\rho(\partial_{a,b} c).$$

A J-space is said to be of level $n$ if $\rho(h(1)) = n$ id.

**Lemma 2.1.** Let $(M, \tilde{\rho})$ be a representation of the Lie algebra $\mathfrak{G}_0$. Then the map $\rho : a \mapsto \tilde{\rho}(h(a))$ determines a J-space structure on $M$.

Conversely, if $M$ is a J-space, then there is a unique $\mathfrak{G}_0$-module structure $(M, \tilde{\rho})$ such that $\tilde{\rho}(h(a)) = \rho(a)$ for any $a \in J$.

**Proof.** Let $(M, \tilde{\rho})$ be a $\mathfrak{G}_0$-module. For $a \in J$, set $\rho(a) = \tilde{\rho}(a)$. Since $[h(a), h(a^2)] = 4\{a, a^2\} = 0$, it follows that $[\rho(a), \rho(a^2)] = 0$, proving (J1).

Let $a, b, c \in J$. We have $[[h(a), h(b)], h(c)] = 4h(\partial_{a,b} c)$, and therefore $[[\rho(a), \rho(b)], \rho(c)] = 4\rho(\partial_{a,b} c)$, proving (J2).

Conversely, assume that $M$ is a J-space. It is clear that $\mathfrak{G}_0$ is generated by the vector space $h \otimes J$ and defined by the relations

$$(H1) \quad [h(a), h(a^2)] = 0, \text{ and}$$

$$(H2) \quad [[h(a), h(b)], h(c)] = 4h(\partial_{a,b} c),$$

for any $a, b, c \in J$. Therefore there is a unique structure $\tilde{\rho}$ of $\mathfrak{G}_0$-module on $M$ such that $\tilde{\rho}(h(a)) = \rho(a)$ □
A linear map $\sigma : J \to \mathrm{End}_k(M)$ is a *Jordan birepresentation* (and $M$ is a *Jordan bimodule*) if the semidirect product $(a, m)(b, n) = (ab, bm + an)$ gives a Jordan algebra structure to the vector space direct sum $J \oplus M$. This condition is equivalent to the conditions

$$ (\mathrm{M1}) \quad [\sigma(a), \sigma(a^2)] = 0, \text{ and} $$

$$ (\mathrm{M2}) \quad \sigma(a^2b) + 2\sigma(a)\sigma(b)\sigma(a) = 2\sigma(ab)\sigma(a) + \sigma(a^2)\sigma(b), $$

for all $a, b \in J$. It follows from [8, II.9(47')] that (M1) and (M2) imply that $\rho = 2\sigma$ satisfies (J1) and (J2). The converse is not true, however. If $(M, \rho)$ is a $J$-space of level $n$, then setting $a = b = 1 \in J$ in (M2), we see that the only possible values for $n = \rho(1)$ are 0, 1 or 2 (Peirce decomposition), if we wish to induce a $J$-bimodule structure on $M$ with action $\sigma = \frac{1}{2}\rho$. But as we will see in Example 2.10, there exist $J$-spaces of any level.

## 2.6 Dominant $J$-spaces

Any $J$-space $M$ of level $n$ can be induced to a generalized Verma module

$$ V(M) = \mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{g}_0 \oplus \mathfrak{g}_1)} M, $$

where $\mathfrak{g}_1$ acts as zero on $M$. The main result of this section will be an analysis of which $J$-spaces $M$ of level $n$ determine $\mathfrak{g}$-modules $V(M)$ with bounded quotients $V(M)/X$, such that the composition of natural maps

$$ M \hookrightarrow V(M) \to V(M)/X $$

is an injection of $\mathfrak{g}_0 \oplus \mathfrak{g}_1$-modules. Such a quotient $V(M)/X$ will be called a *bounded $M$-quotient* of $V(M)$ of level $n$, and in this case, $M$ is said to be *dominant*. Note that if $V(M)/X$ is a bounded $M$-quotient of level $n$, then $n$ is a nonnegative integer and the weight $n$ subspace $(V(M)/X)_n$ is precisely $M$.

**Proposition 2.2.** Let $M$ be a $J$-space of nonnegative integer level $n$, with $\mathfrak{g}_0$-action given by $\rho : J \to \mathrm{End}_k(M)$. Then the generalized Verma module $V(M)$ has a bounded $M$-quotient if and only if

$$ e(1)^{n+1}f(a)^{n+1}m = 0, $$

for all $a \in J$ and $m \in M$.
**Proof.** If $V(M)$ has a nonzero bounded $M$-quotient $V(M)/X$, then $M = (V(M)/X)_n = V(M)_n$, so $X_n = 0$. By $sl_2$-theory, we see that $(V(M)/X)_m = 0$ for all $m < -n$, so $X_m = V(M)_m$ for all $m < -n$. In particular, $f(a)^{n+1}m \in V(M)_{-n-2} = X_{-n-2} \subseteq X$ for all $m \in M$, so $e(1)^{n+1}f(a)^{n+1}m \in X$ for all $a \in J$ and $m \in M$. But then $e(1)^{n+1}f(a)^{n+1}m \in V(M)_n \cap X = 0$.

Conversely, suppose that $e(1)^{n+1}f(a)^{n+1}m = 0$ for all $a \in J$ and $m \in M$. Let $Z \subset V(M)$ be a $\mathfrak{G}$-submodule which is maximal with respect to the property that $Z_n = 0$. Linearisation of the relation $e(1)^{n+1}f(a)^{n+1}m = 0$ gives

$$e(1)^{n+1}f(b_1) \cdots f(b_{n+1})m = 0,$$

for all $b_1, \dots, b_{n+1} \in J$ and $m \in M$. Then for any $a_1, \dots, a_{n+1} \in J$, we have

$$h(a_1) \dots h(a_{n+1})e(1)^{n+1}f(b_1) \cdots f(b_{n+1})m = 0,$$

from which it follows that

$$e(a_1) \cdots e(a_{n+1})f(b_1) \cdots f(b_{n+1})m = 0,$$

for all $a_i, b_j \in J$ and $m \in M$. In particular, this shows that $f(b_1) \cdots f(b_{n+1})m \in Z$, and $V(M)_k \subseteq Z$ for all $k \le -n-2$. Therefore, $V(M)_k = Z_k$ for all $k < -n$, and $V(M)/Z$ is a bounded $M$-quotient. $\square$

We now introduce some notation. Let $\sigma = (\sigma_1, \dots, \sigma_m)$ be a partition of $n+1$, that is,

$$\sigma_1 \ge \dots \ge \sigma_m \ge 1, \text{ for some } m \ge 1, \text{ where } \sigma_1 + \dots + \sigma_m = n+1.$$

We write $|C_\sigma|$ for the cardinality of the conjugacy class $C_\sigma$ of permutations in the symmetric group $S_{n+1}$ with cycle structure $\sigma$. The sign of these permutations will be denoted by $\operatorname{sgn}(\sigma)$, and we write $\rho_\sigma(a)$ for the expression $\rho(a^{\sigma_1})\rho(a^{\sigma_2})\cdots\rho(a^{\sigma_m})$ for all $a \in J$ and $\sigma \vdash n+1$. It follows easily from Condition (C1) that $[\rho(a^i), \rho(a^j)] = 0$ for all $i, j$, so this product is independent of the order of the factors.

The Newton polynomials $N_\ell(x) = x_1^\ell + \dots + x_n^\ell$ in $n$ indeterminates $x_1, \dots, x_n$ generate the ring of symmetric polynomials $k[x_1, \dots, x_n]^{S_n}$. We write

$$N_\sigma(x) = N_{\sigma_1}(x)N_{\sigma_2}(x) \cdots N_{\sigma_m}(x)$$

for each partition $\sigma = (\sigma_1, \dots, \sigma_m)$ of $n+1$. The space $P_{n+1} \subset k[x_1, \dots, x_n]^{S_n}$ of symmetric polynomials of total degree $n+1$ is clearly of dimension $p(n+$
1) - 1, where $p(n + 1)$ is the number of partitions of $n + 1$. The Newton polynomials $N_1(x), \dots, N_n(x)$ are algebraically independent, so

$$ \{N_{\sigma}(x) : \sigma \vdash n + 1 \text{ such that } \sigma \neq (1, 1, \dots, 1)\} $$

is a basis for $P_{n+1}$, and the set

$$ \{N_{\sigma}(x) : \sigma \vdash n + 1\} $$

has exactly one linear dependence relation, up to scalar multiple. We include an amusing representation-theoretic argument below, that we have not seen elsewhere in the literature.

**Proposition 2.3.** Up to scalar multiple, the unique linear dependence relation on the set $\{N_{\sigma}(x) : \sigma \vdash n + 1\}$ is $\sum_{\sigma \vdash n+1} \operatorname{sgn}(\sigma)|C_{\sigma}|N_{\sigma}(x) = 0$.

**Proof.** By the Frobenius character formula,

$$ N_{\sigma}(x) = \sum_{\lambda} \chi_{\lambda}(\sigma) S_{\lambda}, $$

where the sum is taken over all partitions $\lambda \neq (1, 1, \dots, 1)$ of $n + 1$. Here $\chi_{\lambda}(\sigma)$ is the character (evaluated at any permutation of cycle structure $\sigma$) of the Specht module associated with $\lambda$, and $S_{\lambda}$ is the Schur polynomial associated to $\lambda$. See [6] for details.

Since $\operatorname{sgn}$ is the character of the sign representation, the Specht module associated to $(1, \dots, 1)$, we see that

$$ \begin{aligned} \sum_{\sigma \vdash n+1} \operatorname{sgn}(\sigma) |C_{\sigma}| N_{\sigma}(x) &= \sum_{\sigma \in S_{n+1}} \operatorname{sgn}(\sigma) N_{\sigma}(x) \\ &= \sum_{\sigma \in S_{n+1}} \chi_{(1, \dots, 1)}(\sigma) \sum_{\lambda \neq (1, \dots, 1)} \chi_{\lambda}(\sigma) S_{\lambda} \\ &= \sum_{\lambda \neq (1, \dots, 1)} S_{\lambda} \sum_{\sigma \in S_{n+1}} \chi_{(1, \dots, 1)}(\sigma) \chi_{\lambda}(\sigma), \end{aligned} $$

and the inner product

$$ (\chi_{(1, \dots, 1)}, \chi_{\lambda}) = \sum_{\sigma \in S_{n+1}} \chi_{(1, \dots, 1)}(\sigma) \chi_{\lambda}(\sigma) $$

is 0 whenever $\lambda \neq (1, \dots, 1)$, by the orthogonality relations. $\square$
**Remark 2.4.** Proposition 2.3 can also be proved with a more standard combinatorial argument: Let $V$ be a vector space of dimension $n$, with basis $\{e_1, \dots, e_n\}$. Let $h: V \to V$ be defined by $h(e_i) = x_i e_i$, and let $h^{\otimes (n+1)}: V^{\otimes (n+1)} \to V^{\otimes (n+1)}$ be the induced endomorphism. For $\sigma \in S_{n+1}$, we have

$$ \mathrm{Tr}(h^{\otimes (n+1)} \circ \sigma) = N_{\sigma}(x). $$

Since $\sum_{\sigma \in S_{n+1}} \mathrm{sgn}(\sigma)\sigma$ acts as zero on $V^{\otimes (n+1)}$, the dependence relation follows.

The following well-known formula, originally due to Garland [7] and reinterpreted by Chari and Pressley [4], will be used to prove boundedness conditions.

**Lemma 2.5.** Let $p: \mathcal{U}(\mathfrak{G}) \to \mathcal{U}(\mathfrak{G}_{-1})\mathcal{U}(\mathfrak{G}_0)$ be the projection relative to the (vector space) decomposition $\mathcal{U}(\mathfrak{G}) = \mathcal{U}(\mathfrak{G})\mathfrak{G}_1 \oplus \mathcal{U}(\mathfrak{G}_{-1})\mathcal{U}(\mathfrak{G}_0)$. Then $p(e(1)^r f(a)^{n+1})$ is the coefficient of $u^{n+1}$ in the generating function

$$ \frac{(-1)^r r!(n+1)!}{(n+1-r)!} \left( \sum_{s=1}^{\infty} f(a^s)u^s \right)^{n+1-r} \exp \left( -\sum_{t=1}^{\infty} \frac{h(a^t)}{t} u^t \right). $$

□

**Theorem 2.6.** Let $(M, \rho)$ be a J-space of level $n$. Then the following conditions are equivalent:

(1) $M$ is dominant.

(2) $e(1)^{n+1}f(a)^{n+1}m = 0$ for all $a \in J$ and $m \in M$.

(3) $\sum_{\sigma \vdash n+1} \mathrm{sgn}(\sigma)|C_{\sigma}|\rho_{\sigma}(a) = 0$.

**Proof.** By Proposition 2.2, Conditions (1) and (2) are equivalent, so we need only prove that (2) and (3) are equivalent.

Since $e(1)^{n+1}f(a)^{n+1}$ is homogeneous of degree 0 with respect to the grading induced by $\mathrm{ad}(h \otimes 1)$, we see that its action on any highest weight vector $m$ is given by the action of its projection $p$ on the subspace $\mathcal{U}(\mathfrak{G}_0)$ with respect to the decomposition $\mathcal{U}(\mathfrak{G})_0 = \mathcal{U}(\mathfrak{G}_0) \oplus (\mathcal{U}(\mathfrak{G})\mathfrak{G}_1 \cap \mathcal{U}(\mathfrak{G})_0)$ of the space $\mathcal{U}(\mathfrak{G})_0$ of degree 0 elements of $\mathcal{U}(\mathfrak{G})$.
By Lemma 2.5, $p(e(1)^{n+1}f(a)^{n+1})$ is the coefficient of $u^{n+1}$ in the generating series

$$(-1)^{n+1}(n+1)!(n+1)!\exp\left(-\sum_{k=1}^{\infty} \frac{h(a^k)}{k} u^k\right).$$

Computing directly, the coefficient of $u^{n+1}$ in $\exp\left(-\sum_{k=1}^{\infty} \frac{h(a^k)}{k} u^k\right)$ is

$$\sum_{\sigma \vdash n+1} (-1)^{r_{\sigma}} \frac{h_{\sigma}(a)}{(\prod_{i=1}^{r_{\sigma}} \sigma_i) (\prod_{j=1}^{m_{\sigma}} a_j!)}$$

where $\sigma_i$ is the length of the $i$th row of the Young frame $T_\sigma$ associated to the partition $\sigma$, $r_\sigma$ is the number of rows of $T_\sigma$, $m_\sigma$ is the number of columns of $T_\sigma$, $a_j$ is the number of rows of length $j$ in $T_\sigma$, and $h_\sigma(a) = h(a^{\sigma_1})h(a^{\sigma_2})\cdots h(a^{\sigma_{r_\sigma}})$. If $\text{odd}(r_\sigma)$ (respectively, $\text{even}(r_\sigma)$) is the number of odd-length (respectively, even-length) rows of $T_\sigma$, we see that $(-1)^{n+1} = (-1)^{\text{odd}(r_\sigma)}$, so

$$(-1)^{n+1}(-1)^{r_{\sigma}} = (-1)^{n+1}(-1)^{\text{odd}(r_{\sigma})}(-1)^{\text{even}(r_{\sigma})} = (-1)^{\text{even}(r_{\sigma})} = \text{sgn}(\sigma).$$

By elementary counting arguments,

$$|C_{\sigma}| = \frac{(n+1)!}{(\prod_{i=1}^{r_{\sigma}} \sigma_i) (\prod_{j=1}^{m_{\sigma}} a_j!)}.$$

See [14, Proposition 1.1.1], for instance. The projection of $e(1)^{n+1}f(a)^{n+1}$ on $\mathcal{U}(\mathfrak{G}_0)$ is thus $(n+1)! \sum_{\sigma \vdash n+1} \text{sgn}(\sigma)|C_{\sigma}|h_{\sigma}(a)$, so Conditions (2) and (3) are equivalent. $\square$

**Example 2.7.** By Theorem 2.6, any dominant $J$-space $M$ of level 0 is trivial, in the sense that $\rho: J \to \text{End}_k(M)$ is the zero map and $M$, equipped with the trivial $\mathfrak{G}$-action, is the unique bounded $M$-quotient of $V(M)$.

**Example 2.8.** Dominant $J$-spaces $(M, \rho)$ of level 1 satisfy $\rho(a^2) = \rho(a)^2$ for all $a \in J$, so

$$\rho(ab) = \frac{1}{2}(\rho(a)\rho(b) + \rho(b)\rho(a)), \qquad (2.9)$$
for all $a, b \in J$ by linearization. Dominant $J$-spaces $(M, \rho)$ of level 1 are thus precisely associative specializations, Jordan algebra homomorphisms $\rho$ from $J$ to special Jordan algebras of linear operators on a vector space $M$.

**Example 2.10.** For levels higher than 2, dominant $J$-spaces are never Jordan bimodules. See the discussion after Lemma 2.1 for details. Many such $J$-spaces exist. For example, it follows immediately from Proposition 2.3 and Theorem 2.6, that the map

$$ \rho: k[t] \to \mathrm{End}_k(k[x_1, \dots, x_n]^{S_n}) \qquad (2.11) \\ t^\ell \mapsto N_\ell(x) $$

defines a dominant $J$-space of level $n$ for the (associative) Jordan algebra $J = k[t]$, an example we will consider in more detail in Section 3.

# 3 Weyl modules and highest weight categories

Let $n$ be a nonnegative integer. The categories $\mathcal{C}^b(M)$ of bounded weight modules attached to bounded $J$-spaces $M$ of level $n$ admit universal objects $\Delta(M)$, called *Weyl modules*. Every bounded $M$-quotient of level $n$ is a homomorphic image of $\Delta(M)$, and it is clear that

$$ \Delta(M) = V(M) / \mathcal{U}(\mathfrak{g}) \sum_{\ell < -n} V(M)_{\ell} = \bigoplus_{\ell=0}^{n} \Delta(M)_{n-2\ell}, $$

where $\Delta(M)_{n-2\ell}$ is the vector subspace of weight $n-2\ell$. Identifying

$$ \mathfrak{g}_{-1} = \{f \otimes a : a \in J\} \subset \mathfrak{g} = (\mathfrak{sl}_2(k) \otimes J) \oplus \{J, J\} $$

with $J$, the weight space $V(M)_{n-2\ell}$ identifies with the vector space $S^\ell J \otimes M$ for $\ell = 0, \dots, n$.

## 3.1 Weyl modules for finite dimensional dominant $J$-spaces

Let $J = \bigoplus_{\ell=0}^\infty J_\ell$ be a finitely generated $\mathbb{Z}_+$-graded unital Jordan algebra with $J_0 = k1$. Let $M = \bigoplus_{\ell=0}^\infty M_\ell$ be a $\mathbb{Z}$-graded dominant $J$-space of level $n$. We now prove one of our main results, that the category $\mathcal{C}^{\mathrm{fin}}$ of finite-dimensional $\mathbb{Z}$-graded $\mathfrak{sl}_2(J)$-modules contains its Weyl modules.
**Theorem 3.1.** Let $M$ be a $\mathbb{Z}$-graded dominant $J$-space of level $n$, for a $\mathbb{Z}_+$-graded and finitely generated Jordan algebra $J$ with $J_0 = k1$. Then the Weyl module $\Delta(M)$ is finite dimensional if and only if $M$ is finite dimensional.

**Proof.** If $\Delta(M)$ is finite dimensional, then $M \subseteq \Delta(M)$ is clearly also finite dimensional. Conversely, assume that $M$ is finite dimensional. Up to a possible shift in grading, we may assume that $M$ is $\mathbb{Z}_+$-graded. Let $N$ be the largest nonnegative integer for which the graded component $M_N$ is nonzero. Let $a \in J$ be a homogeneous element with $\deg a > N$, and let $v \in M$. By Lemma 2.5, $e(1)^n f(a)^{n+1}v$ is the coefficient of $u^{n+1}$ in the formal series

$$(-1)^n n!(n+1)! \sum_{s=1}^{\infty} f(a^s) u^s \exp \left( - \sum_{t=1}^{\infty} \frac{h(a^t)}{t} u^t \right) v.$$

By degree considerations, $h(a^t)v = 0$ for all $t \ge 1$, so

$$e(1)^n f(a)^{n+1} v = (-1)^n n!(n+1)! f(a^{n+1}) v,$$

and $f(a^{n+1})v = 0$ as an element of $\Delta(M)$.

In particular, $f(b^{(N+1)(n+1)}) M = 0$ for all $b$ in the (non-unital) Jordan subalgebra $J^+ = \bigoplus_{\ell=1}^{\infty} J_\ell \subset J$. Let $I = \{x \in J^+ : f(x)M = 0\}$. For all $x \in J^+$, $y \in I$, and $m \in M$,

$$\begin{aligned} 0 &= h(x)f(y)m \\ &= f(y)h(x)m - 2f(xy)m \\ &= -2f(xy)m \end{aligned}$$

since $h(x)M \subseteq M$ and $f(y)M = 0$. Therefore, $xy \in I$ and $I$ is an ideal of $J^+$.

Since $b^{(N+1)(n+1)} \in I$ for all $b \in J^+$, the Jordan algebra $J^+/I$ is nil of bounded index, hence locally nilpotent by a result of Zelmanov [16]. But $J$, and thus $J^+/I$, is finitely generated, so $J^+/I$ is nilpotent and there exists $N' > 0$ such that every product (in any association) of $N'$ elements of $J^+$ is in $I$. The (finitely many) generators of $J$ may be chosen to be homogeneous and of positive degree at most $r$ for some $r > 0$. In particular, $J_s \subseteq I$ for all $s \ge rN'$. That is, $f(a)M = 0$ for all $a \in J$ with $\deg a \ge rN'$.

The weight space $\Delta(M)_{n-2\ell}$ is spanned by monomials of the form

$$f(a_1) \cdots f(a_\ell)w$$
with $a_1, \dots, a_\ell \in J$ and $w \in M$. Since the $f(a_i)$ commute with each other, the set

$$ \{f(a_1) \cdots f(a_\ell)w : w \in M \text{ and } a_1, \dots, a_\ell \in J \text{ with } \deg a_i < rN' \text{ for all } i\} $$

already spans $\Delta(M)_{n-2\ell}$. As $M$ and $J_i$ are finite dimensional for all $i$, it now follows that $\dim \Delta(M)_{n-2\ell} < \infty$ for all $\ell$, and the Weyl module $\Delta(M) = \bigoplus_{\ell=0}^n \Delta(M)_{n-2\ell}$ is also finite dimensional. $\square$

## 3.2 Highest weight categories and character formulas for free Jordan algebras

Cline, Parshall, and Scott [5] introduced the notion of *highest weight category* as a unifying theme in representation theory, modelled after highest weight representations of semisimple algebraic groups and their Lie algebras. Their definition requires labelling simple objects by a poset $\Lambda$, and the existence of enough injectives, as well as costandard objects labelled by the same index set as the simples and satisfying various axioms. Given the similarities between the category $C^{fin}$ of finite-dimensional $\mathbb{Z}$-graded $\mathfrak{sl}_2(J(r))$-modules and the representation theory of reductive algebraic groups in positive characteristic, we conjecture that $C^{fin}$ is a highest weight category, with the Weyl modules and their duals (twisted by the Cartan involution) as the standard and costandard objects, respectively. In a highest weight category, the higher ext-groups $\mathrm{Ext}^i(\Delta(\lambda), \nabla(\mu)) = 0$ for all $i > 0$ and $\lambda, \mu \in \Lambda$. If $C^{fin}$ is indeed a highest weight category as conjectured above, the vanishing of higher ext-groups would, in fact, settle the main conjecture of [10] and thus describe the graded dimensions of the free Jordan algebras $J(r)$.

**Theorem 3.2.** If $C^{fin}$ is a highest weight category with Weyl modules and their duals as its standard and costandard objects, then $H_i(\mathfrak{sl}_2(J(r)))$ contains no nonzero trivial $\mathfrak{sl}_2(k)$-modules for $i > 0$.

**Proof.** Let $J$ be the free unital Jordan algebra $J(r)$ on $r$ generators, and suppose that $C^{fin}$ is a highest weight category as in the hypotheses of the theorem. As noted above, in a highest weight category, $\mathrm{Ext}^i(\Delta(\lambda), \nabla(\mu))$ vanishes for all $i > 0$ and indices $\lambda, \mu$ of simples, where $\Delta(\lambda)$ and $\nabla(\mu)$ are the corresponding standard and costandard objects. In $C^{fin}$, the Weyl and
dual Weyl modules corresponding to the trivial 1-dimensional sl₂(J)-module k are themselves 1-dimensional, so

$$ \mathrm{Ext}^i_{\mathrm{Cfin}}(k, k) = 0 \quad \text{for all } i > 0. $$

But $\mathrm{Ext}^i_{\mathrm{Cfin}}(k, k) = H^i(\mathfrak{sl}_2(J))$, and the cohomology ring

$$ H^*(\mathfrak{sl}_2(J)) = H^*(\mathfrak{sl}_2(k)) \otimes H^*(\mathfrak{sl}_2(J), \mathfrak{sl}_2(k)). $$

As $H^0(\mathfrak{sl}_2(J)) = H^0(\mathfrak{sl}_2(k)) = k$, we see that $H^*(\mathrm{sl}_2(J)) = k$ and the relative cohomology $H^i(\mathfrak{sl}_2(J), \mathfrak{sl}_2(k)) = 0$, for all $i > 0$. The result now follows from the universal coefficient theorem and the interpretation of the relative cohomology as the $\mathfrak{sl}_2(k)$-invariants in $H^i(\mathfrak{sl}_2(J))$. $\square$

### 3.3 Example: Weyl modules for free Jordan algebras of rank 1

For any Jordan algebra $J$ with unit 1 and $n \in \mathbb{Z}_+$, let

$$ T(J) = k1 \oplus J \oplus (J^{\otimes 2}) \oplus (J^{\otimes 3}) \oplus \dots $$

be its tensor algebra, and let $I \subseteq T(J)$ be the two-sided ideal generated by the relations

$$ 1 - n1, \qquad (3.3) $$

$$ a \otimes a^2 - a^2 \otimes a, \qquad (3.4) $$

$$ a \otimes b \otimes c + c \otimes b \otimes a - b \otimes a \otimes c - c \otimes a \otimes b + b(ac) - a(bc), \qquad (3.5) $$

$$ \sum_{\sigma \vdash n+1} \operatorname{sgn}(\sigma) |C_{\sigma}| T_{\sigma}(a), \qquad (3.6) $$

for all $a, b, c \in J$, where $T_{\sigma}(a) = a^{\sigma_1} \otimes a^{\sigma_2} \otimes \dots \otimes a^{\sigma_m}$ for all partitions $\sigma = (\sigma_1, \dots, \sigma_m) \vdash n + 1$. The associative algebra $U_n(J) = T(J)/I$ is called the *universal J-space envelope of level n*. There is a unique associative algebra homomorphism $\tilde{\rho}: T(J) \to \mathrm{End}_k(M)$ extending the action $\rho: J \to \mathrm{End}_k(M)$ of any J-space $(M, \rho)$ of level n, and in light of Lemma 2.1 and Theorem 2.6, the map $\tilde{\rho}$ descends to the quotient $U_n(J)$. By construction, dominant J-spaces of level n and left $U_n(J)$-modules are equivalent
notions, and a J-space $(M, \rho)$ of level $n$ is said to be free of rank $r$ if $(M, \tilde{\rho})$ is a free $\mathcal{U}_n(J)$-module.

Let $F = \mathcal{U}_n(J)$ be the universal J-space envelope of level $n$ for a unital Jordan algebra $J$. If $J$ is finitely generated as a Jordan algebra, then $F$ is finitely generated as an associative algebra, by Relation (3.6). For example, if $J = k[t]$ is the free Jordan algebra of rank 1, then $F = k[x_1, \dots, x_n]^{S_n}$ is the algebra of symmetric polynomials, where $t^\ell$ corresponds to the Newton polynomial $N_\ell(x) = x_1^\ell + \dots + x_n^\ell \in F$. If $J$ is free of rank $m$, then $\mathcal{U}_0(J) = k$ and $\mathcal{U}_1(J)$ is the quotient of the free associative algebra in $m$ generators by the ideal generated by the relation $a \otimes b \otimes c + c \otimes b \otimes a = b \otimes a \otimes c + c \otimes a \otimes b$ for all $a, b, c \in J$.

Let $L$ be the two-dimensional simple $\mathfrak{sl}_2(k)$-module. The Jordan algebra $J = k[t]$ is commutative and associative, and it is easy to see that $\{J, J\} = 0$ and the TKK algebra $\mathfrak{G} = \mathfrak{sl}_2(J) = \mathfrak{sl}_2(k) \otimes J$ is centrally closed. The space $L[t] = L \otimes k[t]$ is obviously a $\mathfrak{G}$-module, where

$$ (x \otimes p(t)).(v \otimes q(t)) = xv \otimes p(t)q(t), $$

for all $x \in \mathfrak{sl}_2(k)$, $v \in L$, and $p(t), q(t) \in J$. This gives a $\mathfrak{G}$-module structure on the space $S^n(L[t]) \subset T(L[t])$ of homogeneous symmetric tensors of degree $n$.

**Proposition 3.7.** Let $F = k[x_1, \dots, x_n]^{S_n}$ be the rank 1 free $\mathcal{U}_n(k[t])$-module. Then the Weyl module $\Delta_n(F)$ is isomorphic to $S^n(L[t])$.

**Proof.** Let $v \in L$ be a nonzero vector of weight 1 with respect to the action of $h \in \mathfrak{sl}_2(k)$. There is a natural injection $\iota: F \to S^n(L[t])$, with

$$ \iota: \sum_{\sigma \in S_n} x_{\sigma(1)}^{a_{\sigma(1)}} \cdots x_{\sigma(n)}^{a_{\sigma(n)}} \mapsto \sum_{\sigma \in S_n} (v \otimes t^{a_{\sigma(1)}}) \otimes \cdots \otimes (v \otimes t^{a_{\sigma(n)}}). $$

This maps extends uniquely to a $\mathfrak{G}$-module epimorphism

$$ \begin{gathered} V(F) \to S^n(L[t]) \\ u.p \mapsto u.\iota(p), \quad \text{for all } u \in U(\mathfrak{G}) \text{ and } p \in F, \end{gathered} $$

with kernel $\sum_{\ell<-n} V(F)_\ell$. $\square$

**Remark 3.8.** In fact, for every prime Jordan algebra $J$ and every $n \ge 2$, there is a dominant J-space of level $n$, on which the Lie algebra $\mathfrak{G}_0(J) =$
(h ⊗ J) ⊕ {J, J} acts faithfully. If J is special, then there is a faithful as-
sociative specialization ρ : J → End$_{k}$(M), and M is a J-space of level 1.
The faithfulness of the extension ˜ρ : G₀ → End$_{k}$(M) on {J, J} follows im-
mediately from the assumption that J is prime. We can then take the n-fold
tensor product of M to obtain a faithful J-space of level n.

If $J$ is the Albert algebra $\mathbb{A}$, then we can construct a faithful $\mathfrak{G}_0$-module of
level $n$ as a tensor product of copies of the level 2 and level 3 representations
of the Albert algebra, obtained from representations of the exceptional Lie
algebra $E_6$, viewed as the subalgebra $(kh \otimes \mathbb{A}) \oplus \{\mathbb{A}, \mathbb{A}\}$ of the Lie algebra
$sl_2(\mathbb{A})$.

This observation is clearly not true for arbitrary (non-prime) Jordan alge-
bras. For example, the Lie algebra $\mathfrak{G}_0(k[t, t^{-1}])$ has a nontrivial centre that
acts as 0 on all bounded modules.

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