AI 中文总结
研究特征≠2的代数闭域上对称张量范畴\(\mathcal C\)中带（斜）对称双线性形式对象的克利福德代数和外尔代数，建立其单性和阿祖马亚性质，计算Verlinde范畴中的此类代数，引入辛维特群并给出特定条件下的表示。
AI 中文摘要
设\(\mathcal C\)是特征\(\neq2\)的代数闭域\(\mathbf k\)上的对称张量范畴。我们研究\(\mathcal C\)中具有（斜）对称双线性形式的对象的克利福德代数和外尔代数。当形式非退化时，在适当假设下建立此类代数的单性和阿祖马亚性质。还计算了 Verlinde 范畴\({\rm Ver}_p\)中的克利福德代数和外尔代数，并用于证明若\(\mathcal C\)是弗罗贝尼乌斯正合的，则\(\mathcal C\)中具有有限对称代数的辛对象的外尔代数是阿祖马亚的。据此引入辛维特群\(\mathcal S\mathcal W(\mathcal C)\)，并在特定条件下用\(G\)的正交表示的第二施蒂费尔 - 惠特尼类来表示它。
英文摘要
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 $G$.
Comments22 pages, latex
