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Register a new userMotzkin Algebras and the $A_n$ Tensor Categories of Bimodules
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We discuss the structure of the Motzkin algebra $M_k(D)$ by introducing a sequence of idempotents and the basic construction. We show that $cup_{kgeq 1}M_k(D)$ admits a factor trace if and only if $Din {2cos(pi/n)+1|ngeq 3}cup [3,infty)$ and higher commutants of these factors depend on $D$. Then a family of irreducible bimodules over the factors are constructed. A tensor category with $A_n$ fusion rule is obtained from these bimodules.
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We prove that every rigid C*-bicategory with finite-dimensional centers (finitely decomposable horizontal units) can be realized as Connes bimodules over finite direct sums of II$_1$ factors. In particular, we realize every multitensor C*-category as bimodules over a finite direct sum of II$_1$ factors.
A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces (i.e., is the representation category of an affine proalgebraic supergroup) if and only if it has moderate growth (i.e., the lengths of tensor powers of an object grow at most exponentially). In this paper we prove a characteristic p version of this theorem. Namely we show that a pre-Tannakian category over an algebraically closed field of characteristic p>0 admits a fiber functor into the Verlinde category Ver_p (i.e., is the representation category of an affine group scheme in Ver_p) if and only if it has moderate growth and is Frobenius exact. This implies that Frobenius exact pre-Tannakian categories of moderate growth admit a well-behaved notion of Frobenius-Perron dimension. It follows that any semisimple pre-Tannakian category of moderate growth has a fiber functor to Ver_p (so in particular Delignes theorem holds on the nose for semisimple pre-Tannakian categories in characteristics 2,3). This settles a conjecture of the third author from 2015. In particular, this result applies to semisimplifications of categories of modular representations of finite groups (or, more generally, affine group schemes), which gives new applications to classical modular representation theory. For example, it allows us to characterize, for a modular representation V, the possible growth rates of the number of indecomposable summands in V^{otimes n} of dimension prime to p.
We show that the equivariant hypertoric convolution algebras introduced by Braden-Licata-Proudfoot-Webster are affine quasi hereditary in the sense of Kleshchev and compute the Ext groups between standard modules. Together with the main result of arXiv:2009.03981, this implies a number of new homological results about the bordered Floer algebras of Ozsvath-Szabo, including the existence of standard modules over these algebras. We prove that the Ext groups between standard modules are isomorphic to the homology of a variant of the Lipshitz-Ozsvath-Thurston bordered strands dg algebras.
We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field $bf k$. If ${rm char}({bf k})=p>0$, we use this method to construct generalizations ${rm Ver}_{p^n}$, ${rm Ver}_{p^n}^+$ of the incompressible abelian symmetric tensor categories defined in arXiv:1807.05549 for $p=2$ and by Gelfand-Kazhdan and Georgiev-Mathieu for $n=1$. Namely, ${rm Ver}_{p^n}$ is the abelian envelope of the quotient of the category of tilting modules for $SL_2(bf k)$ by the $n$-th Steinberg module, and ${rm Ver}_{p^n}^+$ is its subcategory generated by $PGL_2(bf k)$-modules. We show that ${rm Ver}_{p^n}$ are reductions to characteristic $p$ of Verlinde braided tensor categories in characteristic zero, which explains the notation. We study the structure of these categories in detail, and in particular show that they categorify the real cyclotomic rings $mathbb{Z}[2cos(2pi/p^n)]$, and that ${rm Ver}_{p^n}$ embeds into ${rm Ver}_{p^{n+1}}$. We conjecture that every symmetric tensor category of moderate growth over $bf k$ admits a fiber functor to the union ${rm Ver}_{p^infty}$ of the nested sequence ${rm Ver}_{p}subset {rm Ver}_{p^2}subsetcdots$. This would provide an analog of Delignes theorem in characteristic zero and a generalization of the result of arXiv:1503.01492, which shows that this conjecture holds for fusion categories, and then moreover the fiber functor lands in ${rm Ver}_p$.
We consider algebras in a modular tensor category C. If the trace pairing of an algebra A in C is non-degenerate we associate to A a commutative algebra Z(A), called the full centre, in a doubled version of the category C. We prove that two simple algebras with non-degenerate trace pairing are Morita-equivalent if and only if their full centres are isomorphic as algebras. This result has an interesting interpretation in two-dimensional rational conformal field theory; it implies that there cannot be several incompatible sets of boundary conditions for a given bulk theory.
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