Kevin McGerty
Lusztig has constructed a Frobenius morphism for quantum groups at an $\ell$-th root of unity, which gives an integral lift of the Frobenius map on universal enveloping algebras in positive characteristic. Using the Hall algebra we give a simple construction of this map for the positive part of the quantum group attached to an arbitrary Cartan datum in the nondivisible case.
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The quantum Frobenius map and it splitting are shown to descend to corresponding maps for generalized $q$-Schur algebras at a root of unity. We also define analogs of $q$-Schur algebras for any affine algebra, and prove the corresponding results for them.
We prove a super Frobenius formula for the characters of the cyclotomic Hecke algebras by applying the super Schur-Weyl reciprocity between the quantum superalgebras and cyclotomic Hecke algebras, which is a super analogue of the Frobenius formula in (T. Shoji, J. Algebra 226: 2000, 818--856) and a cyclotomic analogue of the super Frobeinus formula in (H. Mitsuhashi, Linear Multilinear Algebra 58: 2010, 941--955).
Mason and Ng have given a generalization to semisimple quasi-Hopf algebras of Linchenko and Montgomery's generalization to semisimple Hopf algebras of the classical Frobenius-Schur theorem for group representations. We give a simplified proof, in particular a somewhat conceptual derivation of the appropriate form of the Frobenius-Schur indicator that indicates if and in which of two possible fashions a given simple module is self-dual.
We establish a super Frobenius formula for the characters of Iwahori-Hecke algebras. We show that the Hall-Littlewood sypersymmetric function, up to constant, generates the values of the irreducible characters of Iwahori-Hecke algebras at the elements corresponding to cycle permutations. Our formula in this article includes both the ordinary quantum case and the classical super case.
We show that the bigroupoid of separable symmetric Frobenius algebras over an algebraically closed field and the bigroupoid of finitely semi-simple Calabi-Yau categories are equivalent. To this end, we construct a trace on the category of finitely-generated representations of a symmetric, separable Frobenius algebra, given by the composite of the Frobenius form with the Hattori-Stallings trace. In order to put this result into perspective, we introduce the concept of the equivariantization of a 2-functor between bicategories endowed with a G-action. We prove that the equivalence between Froben…
For a finite dimensional semisimple Lie algebra and a root of unity, Lusztig defined an infinite dimensional quantum group of divided powers. Under certain restrictions on the order of the root of unity, he constructed a Frobenius homomorphism with finite dimensional Hopf kernel and with image the universal enveloping algebra. In this article we define and describe the Frobenius homomorphism for arbitrary roots of unity by systematically using the theory of Nichols algebras. In several new exceptional cases the Frobenius-Lusztig kernel is associated to a different Lie algebra than the initial …
The classical Frobenius-Schur indicators for finite groups are character sums defined for any representation and any integer m greater or equal to 2. In the familiar case m=2, the Frobenius-Schur indicator partitions the irreducible representations over the complex numbers into real, complex, and quaternionic representations. In recent years, several generalizations of these invariants have been introduced. Bump and Ginzburg, building on earlier work of Mackey, have defined versions of these indicators which are twisted by an automorphism of the group. In another direction, Linchenko and Montg…
Double affine Frobenius Hecke algebras
该研究定义了受超对称Frobenius超代数修饰的GLₙ、SLₙ根数据对应的双仿射Hecke代数，推导了其基定理、中心、多项式表示及SL₂(ℤ)外自同构作用，还得到量子 wreath Hecke代数的新双仿射版本。
We characterize Frobenius and separable monoidal algebra extensions $i: R\ra S$ in terms given by $R$ and $S$. For instance, under some conditions, we show that the extension is Frobenius, respectively separable, if and only if $S$ is a Frobenius, respectively separable, algebra in the category of bimodules over $R$. In the case when $R$ is separable we show that the extension is separable if and only if $S$ is a separable algebra. Similarly, in the case when $R$ is Frobenius and separable in a sovereign monoidal category we show that the extension is Frobenius if and only if $S$ is a Frobeniu…
We present characterizations of braided co-Frobenius Hopf algebras in the braided tensor category of Yetter-Drinfeld modules over a Hopf algebra extending those already known for co-Frobenius Hopf algebras.
Frobenius monoidal functors induced by Frobenius extensions of Hopf algebras
本文研究了由霍普夫代数的弗罗贝尼乌斯扩展诱导的单子函子，证明其在一般情况下为弗罗贝尼乌斯单子函子，并探讨了其在 Yetter-Drinfeld 模拟中的扩展条件。
Frobenius algebra objects in Temperley-Lieb categories at roots of unity
本文在根为单位的Temperley-Lieb范畴中定义了Frobenius结构，并展示了其Nakayama映射的阶为2，扩展了Frobenius代数的概念。
This note extends Radford's formula for the fourth power of the antipode of a finite dimensional Hopf algebra to co-Frobenius Hopf algebras and studies equivalent conditions to a Hopf algebra being involutory for finite dimensional and co-Frobenius Hopf algebras.
In this paper, we construct the quasi-Frobenius-Lusztig kernel for any simple Lie algebra $\mathfrak{g}$.
We introduce a new filtration on Hopf algebras, the standard filtration, generalizing the coradical filtration. Its zeroth term, called the Hopf coradical, is the subalgebra generated by the coradical. We give a structure theorem: any Hopf algebra with injective antipode is a deformation of the bosonization of the Hopf coradical by its diagram, a connected graded Hopf algebra in the category of Yetter-Drinfeld modules over the latter. We discuss the steps needed to classify Hopf algebras in suitable classes accordingly. For the class of co-Frobenius Hopf algebras, we prove that a Hopf algebra …
We establish relations between Frobenius parts and between flat-dominant dimensions of algebras linked by Frobenius bimodules. This is motivated by the Nakayama conjecture and an approach of Martinez-Villa to the Auslander-Reiten conjecture on stable equivalences. We show that the Frobenius parts of Frobenius extensions are again Frobenius extensions. Further, let $A$ and $B$ be finite-dimensional algebras over a field $k$, and let $\dm(_AX)$ stand for the dominant dimension of an $A$-module $X$. If $_BM_A$ is a Frobenius bimodule, then $\dm(A)\le \dm(_BM)$ and $\dm(B)\le \dm(_A\Hom_B(M, B))$.…
We investigate the transformed Hopf algebras in Hopf Galois extensions. The final goal of this paper is to introduce certain triangular Hopf algebras associated with restricted Frobenius Lie algebras over a field of characteristic $p>0$.
We study Frobenius-Schur indicators of the regular representations of finite-dimensional semisimple Hopf algebras, especially group-theoretical ones. Those of various Hopf algebras are computed explicitly. In view of our computational results, we formulate the theorem of Frobenius for semisimple Hopf algebras and give some partial results on this problem.
We consider Frobenius algebras in the monoidal category of right comodules over a Hopf algebra $H$. If $H$ is a group Hopf algebra, we study a more general Frobenius type property and uncover the structure of graded Frobenius algebras. Graded symmetric algebras are also investigated.
A dichotomy between twisted tensor products of bialgebras and Frobenius algebras
该研究为扭曲张量积定义余单位与余乘法结构，刻画其成为余代数、双代数、Frobenius代数等的条件，方法适用于辫子幺半范畴，并得到相关代数构造应用。
We study quasi-Hopf algebras and their subobjects over certain commutative rings from the point of view of Frobenius algebras. We introduce a type of Radford formula involving an anti-automorphism and the Nakayama automorphism of a Frobenius algebra, then view several results in quantum algebras from this vantage-point. In addition, separability and strong separability of quasi-Hopf algebras are studied as Frobenius algebras.
In this note we associate to each Frobenius algebra a vertex algebra, the simplest example being the Virasoro vertex algebra. This construction is analogous to the procedure which associates to a Lie algebra with an invariant bilinear form the vacuum module over the corresponding Kac-Moody Lie algebra.
We investigate when a weak Hopf algebra H is Frobenius; we show this is not always true, but it is true if the semisimple base algebra A has all its matrix blocks of the same dimension. However, if A is a semisimple algebra not having this property, there is a weak Hopf algebra H with base A which is not Frobenius (and consequently, it is not Frobenius "over" A either). We give, moreover, a categorical counterpart of the result that a Hopf algebra is a Frobenius algebra for a noncoassociative generalization of weak Hopf algebra.
A Lie version of Turaev's $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a \textit{$\frak{g}$-quasi-Frobenius Lie algebra} for $\frak{g}$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(\frak{q},β)$ together with a left $\frak{g}$-module structure which acts on $\frak{q}$ via derivations and for which $β$ is $\frak{g}$-invariant. Geometrically, $\frak{g}$-quasi-Frobenius Lie algebras are the Lie algebra structures associated to symplectic Lie grou…
A Lie Correspondence in the Setting of Hopf and Frobenius Algebras
本文在 Hopf 与 Frobenius 代数框架下引入 Hopf-Frobenius 模，构造相应的 Lie 群与 Lie 代数，并证明经典 Lie 对应是该构造的特例。
The zx-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of dagger-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of this structure, Hopf-Frobenius algebras, starting from a single Hopf algebra which is not necessarily commutative or cocommutative. We provide a few necessary and sufficient conditions for a Hopf algebra to be a Hopf-Frobenius algebra, and show that every Hopf algebra in the category of finite dimensional vector spaces is a Hopf-Frobenius algebra. In additi…
"Co-Frobenius" coalgebras were introduced as dualizations of Frobenius algebras. Recently, it was shown in \cite{I} that they admit left-right symmetric characterizations analogue to those of Frobenius algebras: a coalgebra $C$ is co-Frobenius if and only if it is isomorphic to its rational dual. We consider the more general quasi-co-Frobenius (QcF) coalgebras; in the first main result we show that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or equivalently right) rational dual $Rat(C^*)$, in the sense that certain coproduct or prod…
We construct, for q a root of unity of odd order, an embedding of the projective special linear group PSL(n) into the group of bi-Galois objects over u_q(sl(n))*, the coordinate algebra of the Frobenius-Lusztig kernel of SL(n), which is shown to be an isomorphism at n=2.
We show that the equivalence between several possible characterizations of Frobenius algebras, and of symmetric Frobenius algebras, carries over from the category of vector spaces to more general monoidal categories. For Frobenius algebras, the appropriate setting is the one of rigid monoidal categories, and for symmetric Frobenius algebras it is the one of sovereign monoidal categories. We also discuss some properties of Nakayama automorphisms.
We extend to the co-Frobenius case a result of Drinfeld and Radford related to the fourth power of the antipode of a finite dimensional (co)quasitriangular Hopf algebra.
On extended Frobenius structures
本文系统研究了域、幺半范畴和幺半函子框架下的扩展Frobenius代数，以分类非定向2维拓扑量子场论，并给出了相关例子、分类结果与一般构造。
Generalized quantum cluster structure of reduced Lê--Roger--Yang skein algebras and their Frobenius maps
本文为拟三角剖分带孔曲面的约化Lê--Roger--Yang skein代数建立广义量子簇代数结构，定义并应用Frobenius映射，确定相关代数的中心并计算其作为中心模的秩。
Frobenius Algebras and Dual Bimodules in Monoidal 2-Categories
利用Frobenius代数结构在弱幺半2-范畴中显式构造对偶双模，证明其相干性，并论证2Vect中每个特殊Frobenius代数是刚性的。
We present a unified apoach to the study of separable and Frobenius algebras. The crucial observation is thsat both cases are related to the nonlinear equation $R^{12}R^{23}=R^{23}R^{13}=R^{13}R^{12}$, called the FS-equation. Given a solution of the FS-equation satisfying certain normalizing conditions, we can construct a Frobenius algebra or a separable algebra A(R). The main result of this paper in the structure of this two fundamental types of algebras: a finitely generated projective Frobenius or separable $k$-algebra $A$ is isomorphic to such A(R). If $A$ is free as a $k$-module, then A(R…
Quasi-Frobenius algebras in finite tensor categories
本文在有限张量范畴中引入拟Frobenius代数，证明其等价于与Frobenius代数Morita等价，并指出对称Frobenius代数类在pivotal范畴中关于Morita等价封闭。
The `spider theorem' for a general Frobenius algebra $A$, classifies all maps $A^{\otimes m}\to A^{\otimes n}$ that are built from the operations and, in a graphical representation, represented by a {\it connected} diagram. Here the algebra can be noncommutative and the Frobenius form can be asymmetric. We view this theorem as reducing any connected diagram to a standard form with $j$ beads $B$, where $j$ is the number of bounded connected components of the original diagram. We study the associated F-dimension Hilbert series $\dim_x=\sum_{j=0}^\infty x^j\dim_j$, where $\dim_j=ε\circ B^j\circ 1…
Symmetric monoidal equivalences of topological quantum field theories in dimension two and Frobenius algebras
本文证明二维拓扑量子场论与交换Frobenius代数之间的范畴等价具有对称幺半结构，并由此重新推导出乘积代数对应的流形不变量等于各代数不变量相乘。
Graded Frobenius Algebras
本研究构造了编码带分次2D-TQFT的PROP，定义了分次Frobenius代数，给出其映射关系描述与相关结构实例，并全面刻画了PROP上代数悬置的符号问题。
On non-counital Frobenius algebras
本文研究有限维自内射（拟Frobenius）代数的余乘结构，证明包括有限维weak Hopf代数在内的大类此类代数存在非余单位的非零余乘映射，并推测该结构对一般自内射代数成立。
We prove that a Hopf algebra with a finite coradical filtration is co-Frobenius, i. e. there is a non-zero integral on it. As a consequence, we show that algebras of functions on quantum groups at roots of one are co-Frobenius. We also characterize co-Frobenius Hopf algebras with coradical a Hopf subalgebra. This characterization is in the framework of the lifting method due to H.-J. Schneider and the first-named author. Here is our main result. Let H be a Hopf algebra whose coradical is a Hopf subalgebra. Let gr H be the associated graded Hopf algebra and let R be the diagram of H. Then the f…
In Section 1 we introduce Frobenius coordinates in the general setting that includes Hopf subalgebras. In Sections 2 and 3 we review briefly the theories of Frobenius algebras and augmented Frobenius algebras with some new material in Section 3. In Section 4 we study the Frobenius structure of an FH-algebra H \cite{Par72} and extend two recent theorems in \cite{EG}. We obtain two Radford formulas for the antipode in H and generalize in Section 7 the results on its order in \cite{FMS}. We study the Frobenius structure on an FH-subalgebra pair in Sections 5 and 6. In Section 8 we show that the q…
The Hopf category of Frobenius algebras
本研究证明Frobenius代数间的通用测量余代数与余测量代数可分别使其范畴成为Hopf范畴与Hopf反范畴，推广了相关经典结论，还涉及有限特征下的例子及拓扑量子场论相关思考。
We define a dg-category of looped diagrams which we use to construct operations on the Hochschild complex of commutative Frobenius dg-algebras. We show that we recover the operations known for symmetric Frobenius dg-algebras constructed using Sullivan chord diagrams as well as all formal operations for commutative algebras (including Loday's lambda operations) and prove that there is a chain level version of a suspended Cactus operad inside the complex of looped diagrams. This recovers the suspended BV algebra structure on the Hochschild homology of commutative Frobenius algebras defined by Ab…
For each Frobenius algebra there is defined a skein module of surfaces embedded in a given 3-manifold and bounding a prescribed curve system in the boundary. The skein relations are local and generate the kernel of a certain natural extension of the corresponding topological quantum field theory. In particular the skein module of the 3-ball is isomorphic to the ground ring of the Frobenius algebra. We prove a presentation theorem for the skein module with generators incompressible surfaces colored by elements of a generating set of the Frobenius algebra, and with relations determined by tubing…
In this paper, we show that $\C{G}$-Frobenius algebras (for $\C{G}$ a finite groupoid) correspond to a particular class of Frobenius objects in the representation category of $D(k[\C{G}])$, where $D(k[\C{G}])$ is the Drinfeld double of the quantum groupoid $k[\C{G}]$.
We develop filtered-graded techniques for algebras in monoidal categories with the main goal of establishing a categorical version of Bongale's 1967 result: A filtered deformation of a Frobenius algebra over a field is Frobenius as well. Towards the goal, we first construct a monoidal associated graded functor, building on prior works of Ardizzoni-Menini, of Galatius et al., and of Gwillian-Pavlov. Next, we produce equivalent conditions for an algebra in a rigid monoidal category to be Frobenius in terms of the existence of categorical Frobenius form; this builds on work of Fuchs-Stigner. Thes…
Frobenius Algebras Derived from the Kauffman Bracket Skein Algebra
本文研究了特定曲面在$2N$次单位根处的skein模，并由此构造出Frobenius代数。
A cohomology theory for multiplications and comultiplications of Frobenius algebras is developed in low dimensions in analogy with Hochschild cohomology of bialgebras based on deformation theory. Concrete computations are provided for key examples. Skein theoretic constructions give rise to solutions to the Yang-Baxter equation using multiplications and comultiplications of Frobenius algebras, and 2-cocycles are used to obtain deformations of R-matrices thus obtained.
Previously, we introduced a duality transformation for Euler $G$--Frobenius algebras. Using this transformation, we prove that the simple $A,D,E$ singularities and Pham singularities of coprime powers are mirror self--dual where the mirror duality is implemented by orbifolding with respect to the symmetry group generated by the grading operator and dualizing. We furthermore calculate orbifolds and duals to other $G$--Frobenius algebras which relate different $G$--Frobenius algebras for singularities. In particular, using orbifolding and the duality transformation we provide a mirror pairs for …
The Frobenius-Perron dimension for an abelian category was recently introduced. We apply this theory to the category of representations of the finite-dimensional radical squared zero algebras associated to certain modified ADE graphs. In particular, we take an ADE quiver with arrows in a certain orientation and an arbitrary number of loops at each vertex. We show that the Frobenius-Perron dimension of this category is equal to the maximum number of loops at a vertex. Along the way, we introduce a result which can be applied in general to calculate the Frobenius-Perron dimension of a radical sq…
We discuss algebraic and representation theoretic structures in braided tensor categories C which obey certain finiteness conditions. Much interesting structure of such a category is encoded in a Hopf algebra H in C. In particular, the Hopf algebra H gives rise to representations of the modular group SL(2,Z) on various morphism spaces. We also explain how every symmetric special Frobenius algebra in a semisimple modular category provides additional structure related to these representations.
An important breakthrough in understanding the geometry of Schubert varieties was the introduction of the notion of Frobenius split varieties and the result that the flag varieties G/P are Frobenius split. The aim of this article is to give in this case a complete and self contained representation theoretic approach to this method. The geometric Frobenius method in (char k=p>0) will here be replaced by Lusztig's Frobenius maps for quantum groups at roots of unity (which exist not only for primes but any odd integer \ell >1).
We explicitly show that symmetric Frobenius structures on a finite-dimensional, semi-simple algebra stand in bijection to homotopy fixed points of the trivial SO(2)-action on the bicategory of finite-dimensional, semi-simple algebras, bimodules and intertwiners. The results are motivated by the 2-dimensional Cobordism Hypothesis for oriented manifolds, and can hence be interpreted in the realm of Topological Quantum Field Theory.
In the present paper, we study the relationship between deformation quantizations and Frobenius-projective structures defined on an algebraic curve in positive characteristic. A Frobenius-projective structure is an analogue of a complex projective structure on a Riemann surface, which was introduced by Y. Hoshi. Such an additional structure has some equivalent objects, e.g., a dormant $\mathrm{PGL}_2$-oper and a projective connection having a full set of solutions. The main result of the present paper provides a canonical construction of a Frobenius-constant quantization on the cotangent space…
Algebraic Phase Theory III: Structural Quantum Codes over Frobenius Rings
本文通过弗罗贝尼乌斯环的代数结构，揭示了量子稳定子码的代数起源，展示了量子化作为代数相诱导而非解析完成的过程。
Relative Poisson bialgebras and Frobenius Jacobi algebras
本文通过研究相对泊松代数的Manin三元组和相对泊松双代数，引入相对泊松Yang-Baxter方程及其反对称解的构造方法，并由此给出弗罗贝尼乌斯-雅可比代数的新构造途径。
This paper addresses a Hom-associative algebra built as a direct sum of a given Hom-associative algebra $(\mathcal{A}, \cdot, α)$ and its dual $(\mathcal{A}^{\ast}, \circ, α^{\ast}),$ endowed with a non-degenerate symmetric bilinear form $\mathcal{B},$ where $\cdot$ and $\circ$ are the products defined on $\mathcal{A}$ and $\mathcal{A}^{\ast},$ respectively, and $ α$ and $α^{\ast}$ stand for the corresponding algebra homomorphisms. Such a double construction, also called Hom-Frobenius algebra, is interpreted in terms of an infinitesimal Hom-bialgebra. The same procedure is applied to character…
We show that given a Frobenius algebra there is a unique notion of its second quantization, which is the sum over all symmetric group quotients of n--th tensor powers, where the quotients are given by symmetric group twisted Frobenius algebras. To this end, we consider the setting of Frobenius algebras given by functors from geometric categories whose objects are endowed with geometric group actions and prove structural results, which in turn yield a constructive realization in the case of n--th tensor powers and the natural permutation action. We also show that naturally graded symmetric grou…
Frobenius algebras associated with the $α$-induction for equivariantly braided tensor categories
本研究在G-辫子多张量范畴中给出G-等变α-归纳的范畴定义，在中性double范畴中构造G-等变α-归纳Frobenius代数，还定义G-等变全中心并证明其与该代数一致，推广了已有相关定理与构造。
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