Clifford and Weyl algebras in symmetric tensor categories

2026/07/18 by Pavel Etingof
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Abstract

Let \mathcal C be a symmetric tensor category over an algebraically closed field \mathbf k of characteristic ≠ 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.

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