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1. Universal Tensor Categories Generated by Dual Pairs

   https://doi.org/10.1007/s10485-021-09640-2  

   Chirvasitu, Alexandru ; Penkov, Ivan ( October 2021 , Applied Categorical Structures)

   Abstract Let $$V_*\otimes V\rightarrow {\mathbb {C}}$$ V ∗ ⊗ V → C be a non-degenerate pairing of countable-dimensional complex vector spaces V and $$V_*$$ V ∗ . The Mackey Lie algebra $${\mathfrak {g}}=\mathfrak {gl}^M(V,V_*)$$ g = gl M ( V , V ∗ ) corresponding to this pairing consists of all endomorphisms $$\varphi $$ φ of V for which the space $$V_*$$ V ∗ is stable under the dual endomorphism $$\varphi ^*: V^*\rightarrow V^*$$ φ ∗ : V ∗ → V ∗ . We study the tensor Grothendieck category $${\mathbb {T}}$$ T generated by the $${\mathfrak {g}}$$ g -modules V , $$V_*$$ V ∗ and their algebraic duals $$V^*$$ V ∗ and $$V^*_*$$ V ∗ ∗ . The category $${{\mathbb {T}}}$$ T is an analogue of categories considered in prior literature, the main difference being that the trivial module $${\mathbb {C}}$$ C is no longer injective in $${\mathbb {T}}$$ T . We describe the injective hull I of $${\mathbb {C}}$$ C in $${\mathbb {T}}$$ T , and show that the category $${\mathbb {T}}$$ T is Koszul. In addition, we prove that I is endowed with a natural structure of commutative algebra. We then define another category $$_I{\mathbb {T}}$$ I T of objects in $${\mathbb {T}}$$ T which are free as I -modules. Our main result is that the category $${}_I{\mathbb {T}}$$ I T is also Koszul, and moreover that $${}_I{\mathbb {T}}$$ I T is universal among abelian $${\mathbb {C}}$$ C -linear tensor categories generated by two objects X , Y with fixed subobjects $$X'\hookrightarrow X$$ X ′ ↪ X , $$Y'\hookrightarrow Y$$ Y ′ ↪ Y and a pairing $$X\otimes Y\rightarrow {\mathbf{1 }}$$ X ⊗ Y → 1 where 1 is the monoidal unit. We conclude the paper by discussing the orthogonal and symplectic analogues of the categories $${\mathbb {T}}$$ T and $${}_I{\mathbb {T}}$$ I T . 

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2. Twisting of affine algebraic groups, II

   https://doi.org/10.1093/imrn/rnaa381  

   Gelaki, Shlomo ( February 2021 , International Mathematics Research Notices)

   We use [11] to study the algebra structure of twisted cotriangular Hopf algebras ${}_J\mathcal{O}(G)_{J}$, where $J$ is a Hopf $2$-cocycle for a connected nilpotent algebraic group $G$ over $\mathbb{C}$. In particular, we show that ${}_J\mathcal{O}(G)_{J}$ is an affine Noetherian domain with Gelfand–Kirillov dimension $\dim (G)$, and that if $G$ is unipotent and $J$ is supported on $G$, then ${}_J\mathcal{O}(G)_{J}\cong U({\mathfrak{g}})$ as algebras, where ${\mathfrak{g}}={\textrm{Lie}}(G)$. We also determine the finite dimensional irreducible representations of ${}_J\mathcal{O}(G)_{J}$, by analyzing twisted function algebras on $(H,H)$-double cosets of the support $H\subset G$ of $J$. Finally, we work out several examples to illustrate our results.

    

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3. Cohomology of finite tensor categories: Duality and Drinfeld centers

   https://doi.org/10.1090/tran/8548  

   Negron, Cris ; Plavnik, Julia ( March 2022 , Transactions of the American Mathematical Society)

   We consider the finite generation property for cohomology of a finite tensor category C \mathscr {C} , which requires that the self-extension algebra of the unit \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},\mathbf {1}) is a finitely generated algebra and that, for each object V V in C \mathscr {C} , the graded extension group \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on C \mathscr {C} . For example, the stated result holds when C \mathscr {C} is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0 0 , we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes. 

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4. New lower bounds for matrix multiplication and

   https://doi.org/10.1017/fmp.2023.14  

   Conner, Austin ; Harper, Alicia ; Landsberg, J.M. ( January 2023 , Forum of Mathematics, Pi)

   Let$M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }\in \mathbb C^{\mathbf {u}\mathbf {v}}{\mathord { \otimes } } \mathbb C^{\mathbf {v}\mathbf {w}}{\mathord { \otimes } } \mathbb C^{\mathbf {w}\mathbf {u}}$denote the matrix multiplication tensor (and write$M_{\langle \mathbf {n} \rangle }=M_{\langle \mathbf {n},\mathbf {n},\mathbf {n}\rangle }$), and let$\operatorname {det}_3\in (\mathbb C^9)^{{\mathord { \otimes } } 3}$denote the determinant polynomial considered as a tensor. For a tensorT, let$\underline {\mathbf {R}}(T)$denote its border rank. We (i) give the first hand-checkable algebraic proof that$\underline {\mathbf {R}}(M_{\langle 2\rangle })=7$, (ii) prove$\underline {\mathbf {R}}(M_{\langle 223\rangle })=10$and$\underline {\mathbf {R}}(M_{\langle 233\rangle })=14$, where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was$M_{\langle 2\rangle }$, (iii) prove$\underline {\mathbf {R}}( M_{\langle 3\rangle })\geq 17$, (iv) prove$\underline {\mathbf {R}}(\operatorname {det}_3)=17$, improving the previous lower bound of$12$, (v) prove$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.32\mathbf {n}$for all$\mathbf {n}\geq 25$, where previously only$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1$was known, as well as lower bounds for$4\leq \mathbf {n}\leq 25$, and (vi) prove$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.6\mathbf {n}$for all$\mathbf {n} \ge 18$, where previously only$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+2$was known. The last two results are significant for two reasons: (i) they are essentially the first nontrivial lower bounds for tensors in an “unbalanced” ambient space and (ii) they demonstrate that the methods we use (border apolarity) may be applied to sequences of tensors.

   The methods used to obtain the results are new and “nonnatural” in the sense of Razborov and Rudich, in that the results are obtained via an algorithm that cannot be effectively applied to generic tensors. We utilize a new technique, calledborder apolaritydeveloped by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensorTand an integerr, in a finite number of steps, either outputs that there is no border rankrdecomposition forTor produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable whenThas a large symmetry group, in which case it outputs potential decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory.

    

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5. Stability in the homology of Deligne–Mumford compactifications

   https://doi.org/10.1112/S0010437X21007582  

   Tosteson, Philip ( December 2021 , Compositio Mathematica)

   Abstract Using the theory of ${\mathbf {FS}} {^\mathrm {op}}$ modules, we study the asymptotic behavior of the homology of ${\overline {\mathcal {M}}_{g,n}}$ , the Deligne–Mumford compactification of the moduli space of curves, for $n\gg 0$ . An ${\mathbf {FS}} {^\mathrm {op}}$ module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via copies that glue on marked projective lines, we give the homology of ${\overline {\mathcal {M}}_{g,n}}$ the structure of an ${\mathbf {FS}} {^\mathrm {op}}$ module and bound its degree of generation. As a consequence, we prove that the generating function $\sum _{n} \dim (H_i({\overline {\mathcal {M}}_{g,n}})) t^n$ is rational, and its denominator has roots in the set $\{1, 1/2, \ldots, 1/p(g,i)\},$ where $p(g,i)$ is a polynomial of order $O(g^2 i^2)$ . We also obtain restrictions on the decomposition of the homology of ${\overline {\mathcal {M}}_{g,n}}$ into irreducible $\mathbf {S}_n$ representations. 

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