Clifford and Weyl algebras in symmetric tensor categories

Abstract: Let $\mathcal C$ be a symmetric tensor category over an algebraically closed field $\mathbf k$ of characteristic $\ne 2$. We study Clifford and Weyl algebras of objects of $\mathcal C$ with a (skew-)symmetric bilinear form. When the form is non-degenerate, we establish simplicity and the Azumaya property for such algebras under suitable assumptions. We also compute Clifford and Weyl algebras in the Verlinde category ${\rm Ver}_p$ and use them to prove that if $\mathcal C$ is Frobenius exact then the Weyl algebra of a symplectic object of $\mathcal C$ with finite symmetric algebra is Azumaya. Using this, we introduce the symplectic Witt group $\mathcal S\mathcal W(\mathcal C)$, the subgroup of the Brauer group ${\rm Br}(\mathcal C)$ consisting of Morita classes of such Azumaya algebras, and when $\mathcal C={\rm Rep}(G)\boxtimes{\rm sVec}$ for a finite group $G$ of order coprime to ${\rm char}(\mathbf k)$, express $\mathcal S\mathcal W(\mathcal C)$ in terms of second Stiefel-Whitney classes of orthogonal representations of $G$.

• The paper constructs Clifford and Weyl algebras internally to symmetric tensor categories and proves PBW theorems using contraction operators and finite Moyal–Weyl products.
• It establishes simplicity and the Azumaya property under finite-length or characteristic-dependent symmetric-power conditions, then classifies key examples in the Verlinde category.
• The paper defines a symplectic Witt group within the Brauer group and relates its classes for representation categories to second Stiefel–Whitney classes, with surjectivity for abelian groups but failures for some metacyclic groups.

Overview

This paper develops a theory of Clifford and Weyl algebras internal to an arbitrary symmetric tensor category $\mathcal{C}$ over an algebraically closed field $k$ of characteristic $\ne 2$. The author constructs these algebras as quotients of enveloping algebras of Heisenberg (super)algebras, proves a PBW theorem, establishes simplicity and the Azumaya property under suitable hypotheses, computes explicit examples in the Verlinde category $\mathrm{Ver}_p$, and uses the resulting Azumaya algebras to define a symplectic Witt group $\mathcal{SW}(\mathcal{C})$ inside the Brauer group. The final main result identifies this group for $$\mathcal{C} = \mathrm{Rep}(G) \boxtimes \mathrm{sVec}$$ with data built from second Stiefel–Whitney classes of orthogonal representations of $G$.

Definitions and basic structure

For an object $V \in \mathcal{C}$ with a symmetric or skew-symmetric bilinear form $B$, the object $V \oplus \mathbf{1}$ carries the structure of a Lie superalgebra in $k$0: the Heisenberg algebra when $k$1 is skew-symmetric (purely even), and the Heisenberg superalgebra when $k$2 is symmetric ($k$3 even, $k$4 odd). The associated algebra is

$k$5

called the Weyl algebra in the skew case and the Clifford algebra in the symmetric case; for $k$6 this recovers the classical definitions (with the normalization $k$7).

Two structural facts organize the theory. First, twisting by a super-line $k$8 exchanges symmetry types: there is a natural isomorphism $k$9, so in any category containing a super-line the Weyl and Clifford theories are equivalent. Second, for objects $\ne 2$0 carrying forms of the same type one has $\ne 2$1, with the super-tensor product in the Clifford case.

Contraction operators and the PBW theorem

The proof machinery rests on bicontraction maps $\ne 2$2 built from the coproduct components of the Hopf algebra $\ne 2$3 and the form $\ne 2$4. Iterated contractions satisfy $\ne 2$5; consequently, if $\ne 2$6, then $\ne 2$7, a fact that governs all positive-characteristic phenomena in the paper.

The PBW theorem states that the natural surjection $\ne 2$8 (respectively $\ne 2$9) is an isomorphism, so $\mathrm{Ver}_p$0 is a filtered quantization of the Poisson algebra $\mathrm{Ver}_p$1 (or super-Poisson algebra $\mathrm{Ver}_p$2). The proof reduces to Weyl algebras via the super-line trick and defines an explicit Moyal–Weyl product on $\mathrm{Ver}_p$3 by $\mathrm{Ver}_p$4 with $\mathrm{Ver}_p$5; associativity follows from the identity $\mathrm{Ver}_p$6. In characteristic zero this recovers $\mathrm{Ver}_p$7; in characteristic $\mathrm{Ver}_p$8 the sum terminates because $\mathrm{Ver}_p$9, which is what makes the formula valid without division issues beyond the factors $\mathcal{SW}(\mathcal{C})$0 (permissible since $\mathcal{SW}(\mathcal{C})$1). A shorter alternative proof via the schematic Heisenberg group $\mathcal{SW}(\mathcal{C})$2 and known PBW results for group schemes in symmetric tensor categories is also given.

Simplicity and the Azumaya property

The central simplicity result concerns $\mathcal{SW}(\mathcal{C})$3 for a simple (ind-)algebra $\mathcal{SW}(\mathcal{C})$4 in $\mathcal{SW}(\mathcal{C})$5. Two cases are established:

• Skew case: if $\mathcal{SW}(\mathcal{C})$6 whenever $\mathcal{SW}(\mathcal{C})$7, then $\mathcal{SW}(\mathcal{C})$8 is simple.
• Hyperbolic case: for $\mathcal{SW}(\mathcal{C})$9 with the canonical pairing, if $$\mathcal{C} = \mathrm{Rep}(G) \boxtimes \mathrm{sVec}$$0, then $$\mathcal{C} = \mathrm{Rep}(G) \boxtimes \mathrm{sVec}$$1 is simple.

Both proofs are filtration arguments: commutators with generators produce contractions that strictly lower degree, and injectivity of the relevant maps $$\mathcal{C} = \mathrm{Rep}(G) \boxtimes \mathrm{sVec}$$2 below degree $$\mathcal{C} = \mathrm{Rep}(G) \boxtimes \mathrm{sVec}$$3 forces any nonzero ideal to meet $$\mathcal{C} = \mathrm{Rep}(G) \boxtimes \mathrm{sVec}$$4, hence to be everything. A corollary extends this to orthogonal direct sums of such summands.

The main application is to Frobenius exact categories. If $$\mathcal{C} = \mathrm{Rep}(G) \boxtimes \mathrm{sVec}$$5 is Frobenius exact and $$\mathcal{C} = \mathrm{Rep}(G) \boxtimes \mathrm{sVec}$$6 has finite length, then:

1. $$\mathcal{C} = \mathrm{Rep}(G) \boxtimes \mathrm{sVec}$$7 is simple for symplectic $$\mathcal{C} = \mathrm{Rep}(G) \boxtimes \mathrm{sVec}$$8;
2. the natural map $$\mathcal{C} = \mathrm{Rep}(G) \boxtimes \mathrm{sVec}$$9 is an isomorphism;
3. $G$0 is a central Azumaya algebra.

The key input is a lemma showing that a category tensor-generated by $G$1 with $G$2 (or $G$3) of finite length has moderate growth; in characteristic zero it is then super-Tannakian by Deligne's theorem, while in characteristic $G$4 it admits a fiber functor to $G$5. The positive-characteristic argument analyzes the decomposition of $G$6 into simples $G$7: since $G$8 for $G$9 and $V \in \mathcal{C}$0 (finite length of $V \in \mathcal{C}$1), the situation decomposes into copies of $V \in \mathcal{C}$2 (even $V \in \mathcal{C}$3) and hyperbolic planes $V \in \mathcal{C}$4 (odd $V \in \mathcal{C}$5, reducing to the previously established cases. The Azumaya property follows from the observation that $V \in \mathcal{C}$6 is isometric to $V \in \mathcal{C}$7 hyperbolic, so the left-right action map identifies with $V \in \mathcal{C}$8.

Top symmetric powers and queer algebras

For a symplectic object $V \in \mathcal{C}$9 with $B$0 of finite length in a Frobenius exact category, writing $B$1 for the top degree with $B$2, the object $B$3 is invertible, and natural duality isomorphisms $B$4 hold, together with $B$5. Moreover $B$6, so $B$7 is a super-line precisely when $B$8 is odd. This is proved by reducing along fiber functors to $B$9 and $V \oplus \mathbf{1}$0, using that $V \oplus \mathbf{1}$1 is Frobenius via the Verlinde fiber functor.

The paper also classifies simple algebras in $V \oplus \mathbf{1}$2: they are exactly $V \oplus \mathbf{1}$3 and the queer algebras $V \oplus \mathbf{1}$4 for nonzero $V \oplus \mathbf{1}$5. This follows from the classification of indecomposable exact module categories over $V \oplus \mathbf{1}$6 (corresponding to the ADET graphs $V \oplus \mathbf{1}$7 and $V \oplus \mathbf{1}$8) plus standard reconstruction. A concrete identification connects this to classical representation theory: for $V \oplus \mathbf{1}$9, the Weyl algebra $k$00 is isomorphic to $k$01, where $k$02 is the image of the spin representation of $k$03 in $k$04 — obtained by transporting the classical isomorphism $k$05 through the Verlinde fiber functor.

The symplectic Witt group

For Frobenius exact $k$06, the classes $k$07 of symplectic objects with $k$08 of finite length, modulo those with $k$09, form an elementary abelian 2-group $k$10, the symplectic Witt group. Since $k$11, the Azumaya property gives $k$12, so every class is 2-torsion, and the assignment $k$13 defines an inclusion

$k$14

the 2-torsion subgroup of the Brauer group. Injectivity holds because Brauer-triviality of $k$15 reconstructs $k$16 as $k$17 for some $k$18.

Computation for Rep(G) ⊠ sVec

Let $k$19 be finite of order coprime to $k$20 and $k$21. By Carnovale's theorem, $k$22, where $k$23 denotes cohomology with the modified multiplication $k$24. For an orthogonal representation $k$25 of $k$26, stabilizing to $k$27 yields a class $k$28 as the pullback of the Clifford-group extension. The main computation gives

$k$29

so $k$30, where $k$31 is generated by the $k$32. The proof identifies $k$33 with the classical Clifford superalgebra $k$34, computes its Picard restriction to $k$35 as $k$36, and constructs explicit twisted-adjoint lifts $k$37 realizing the multiplier $k$38. A subsequent remark refines this: under the Künneth decomposition, $k$39 versus $k$40, so the raw pullback requires correction by the mixed cocycle $k$41 in odd dimension.

Stiefel–Whitney classes over ℂ

Over $k$42, writing $k$43 for the Stiefel–Whitney classes of an orthogonal representation, one has $k$44 and $k$45. The Whitney sum formula shows the modified multiplication on $k$46 corresponds to $k$47, and the key vanishing $k$48 holds since $k$49 is a coboundary. Defining the Stiefel–Whitney subgroup

$k$50

one obtains $k$51 as sets, where $k$52, and this is a subgroup for ordinary multiplication as well. Notably, $k$53 need not be a subgroup for $k$54 — for $k$55 the class $k$56 of two coordinate characters is nonzero.

The resulting surjectivity criterion is sharp:

Statement	Condition / example
$k$57 surjective	iff $k$58
$k$59 surjective	whenever $k$60 is abelian
$k$61 not always surjective	type-1 split metacyclic groups

For abelian $k$62, $k$63, so cup products of characters exhaust the 2-torsion. For non-surjectivity, the metacyclic examples of Gunarwardena–Kahn–Thomas supply a degree-two class outside the span of second Stiefel–Whitney classes; the argument uses that $k$64 consists of Kummer classes of complex characters, themselves realizable as $k$65 of oriented real two-planes, so no correction can rescue the class.

Limitations and open questions

Several hypotheses are essential and their removal is not addressed. Simplicity and the Azumaya property require $k$66 (or finite length of $k$67); the behavior of $k$68 when symmetric powers grow indefinitely, e.g., for negligible objects, is not treated. The symplectic Witt group is defined only within Frobenius exact categories, and the explicit Stiefel–Whitney computation requires $k$69 with $k$70; the non-semisimple or non-super-Tannakian analogues are open. The classification of simple algebras is specific to $k$71 and relies on the known module-category classification there. Finally, the failure of surjectivity of $k$72 for general finite groups leaves open a precise characterization of which 2-torsion Brauer classes arise from symplectic objects beyond the abelian and metacyclic cases considered here.

Conclusion

The paper supplies a categorical framework in which Clifford and Weyl algebras behave as in classical algebra — PBW, simplicity, Azumaya — provided symmetric powers vanish above the characteristic, and demonstrates that Frobenius exactity converts finite-length hypotheses into verifiable fiber-functor computations in $k$73. The resulting symplectic Witt group links the Brauer 2-torsion of super-representation categories to Stiefel–Whitney theory, with a complete answer for abelian groups and a concrete obstruction class for the general case.