Mathematics

We introduce a notion of an integral along a bimonoid homomorphism as a simultaneous generalization of the integral and cointegral of bimonoids. The purpose of this paper is to characterize an existence of a specific integral, called a normalized generator integral, along a bimonoid homomorphism in terms of the kernel and cokernel of the homomorphism. We introduce a notion of a volume on an abelian category as a generalization of the dimension of vector spaces and the order of abelian groups. In applications, we show that there exists a nontrivial volume partially defined on a category of bicommutative Hopf monoids. The volume yields a notion of Fredholm homomorphisms between bicommutative Hopf monoids, which gives an analogue of the Fredholm index theory. This paper gives a technical preliminary of our subsequent paper about a construction of TQFTs.

This paper first introduces the notion of a Rota-Baxter operator (of weight 1 ) on a Lie group so that its differentiation gives a Rota-Baxter operator on the corresponding Lie algebra. Direct products of Lie groups, including the decompositions of Iwasawa and Langlands, carry natural Rota-Baxter operators. Formal inverse of the Rota-Baxter operator on a Lie group is precisely the crossed homomorphism on the Lie group, whose tangent map is the differential operator of weight 1 on a Lie algebra. A factorization theorem of Rota-Baxter Lie groups is proved, deriving directly on the Lie group level, the well-known global factorization theorems of Semenov-Tian-Shansky in his study of integrable systems. As geometrization, the notions of Rota-Baxter Lie algebroids and Rota-Baxter Lie groupoids are introduced, with the former a differentiation of the latter. Further, a Rota-Baxter Lie algebroid naturally gives rise to a post-Lie algebroid, generalizing the well-known fact for Rota-Baxter Lie algebras and post-Lie algebras. It is shown that the geometrization of a Rota-Baxter Lie algebra or a Rota-Baxter Lie group can be realized by its action on a manifold. Examples and applications are provided for these new notions.

This paper computes the generic fusion rules of the Grothendieck ring of Rep(PSL(2,q)), q prime-power, by applying the Schur orthogonality relations on the generic character table. It then proves that this family of fusion rings interpolates to all integers q>1, providing (when q is not prime-power) the first example of infinite family of non group-like simple integral fusion rings. Furthermore, they pass all the known criteria of (unitary) categorification. This provides infinitely many serious candidates for solving the famous open problem of whether there exists an integral fusion category which is not weakly group-theoretical. A braiding criterion is finally discussed.

Let U be a compact semisimple Lie group with complexification G and associated Cartan involution Θ . Let ν be an involutive complex Lie group automorphism of G commuting with Θ , and consider the associated semisimple real Lie group G ν ={g∈G∣ν(g)=Θ(g)} . We consider q -deformed analogues of the U -orbits of the quotient space G ν ∖G , and determine for these the associated von Neumann algebra and invariant state.

We use Khovanov-Rozansky gl(N) link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals. The technical heart of this construction is a proof of the sweep-around property, which makes these link homologies well defined in the 3-sphere.

By using the notion of a rigid R-matrix in a monoidal category and the Reshetikhin--Turaev functor on the category of tangles, we review the definition of the associated invariant of long knots. In the framework of the monoidal categories of relations and spans over sets, by introducing racks associated with pointed groups, we illustrate the construction and the importance of consideration of long knots. Else, by using the restricted dual of algebras and Drinfeld's quantum double construction, we show that to any Hopf algebra H with invertible antipode, one can associate a universal long knot invariant Z H (K) taking its values in the convolution algebra ((D(H) ) o ) ∗ of the restricted dual Hopf algebra (D(H) ) o of the quantum double D(H) of H . That extends the known constructions of universal invariants previously considered mostly either in the case of finite dimensional Hopf algebras or by using some topological completions.

The main aim of this paper is to provide set-theoretical solutions of the Yang-Baxter equation that are not necessarily bijective, among these new idempotent ones. In the specific, we draw on both to the classical theory of inverse semigroups and to that of the most recently studied braces, to give a new research perspective to the open problem of finding solutions. Namely, we have recourse to a new structure, the inverse semi-brace, that is a triple (S,+,⋅) with (S,+) a semigroup and (S,⋅) an inverse semigroup satisfying the relation a(b+c)=ab+a( a −1 +c) , for all a,b,c∈S , where a −1 is the inverse of a in (S,⋅) . In particular, we give several constructions of inverse semi-braces which allow for obtaining solutions that are different from those until known.

We prove that a finite braided tensor category A is invertible in the Morita 4-category BrTens of braided tensor categories if, and only if, it is non-degenerate. This includes the case of semisimple modular tensor categories, but also non-semisimple examples such as categories of representations of the small quantum group at good roots of unity. Via the cobordism hypothesis, we obtain new invertible 4-dimensional framed topological field theories, which we regard as a non-semisimple framed version of the Crane-Yetter-Kauffman invariants, after Freed--Teleman and Walker's construction in the semisimple case. More generally, we characterize invertibility for E_1- and E_2-algebras in an arbitrary symmetric monoidal oo-category, and we conjecture a similar characterization of invertible E_n-algebras for any n. Finally, we propose the Picard group of BrTens as a generalization of the Witt group of non-degenerate braided fusion categories, and pose a number of open questions about it.

A Lie conformal algebra is an algebraic structure that encodes the singular part of the operator product expansion of chiral fields in conformal field theory. A Lie pseudoalgebra is a generalization of this structure, for which the algebra of polynomials k[\partial] in the indeterminate is replaced by the universal enveloping algebra U(d) of a finite-dimensional Lie algebra d over the base field k. The finite (i.e., finitely generated over U(d)) simple Lie pseudoalgebras were classified in our 2001 paper [BDK]. The complete list consists of primitive Lie pseudoalgebras of type W, S, H, and K, and of current Lie pseudoalgebras over them or over simple finite-dimensional Lie algebras. The present paper is the third in our series on representation theory of simple Lie pseudoalgebras. In the first paper, we showed that any finite irreducible module over a primitive Lie pseudoalgebra of type W or S is either an irreducible tensor module or the image of the differential in a member of the pseudo de Rham complex. In the second paper, we established a similar result for primitive Lie pseudoalgebras of type K, with the pseudo de Rham complex replaced by a certain reduction, called the contact pseudo de Rham complex. This reduction in the context of contact geometry was discovered by M. Rumin [Rum]. In the present paper, we show that for primitive Lie pseudoalgebras of type H, a similar to type K result holds with the contact pseudo de Rham complex replaced by a suitable complex. However, the type H case in more involved, since the annihilation algebra is not the corresponding Lie-Cartan algebra, as in other cases, but an irreducible central extension. When the action of the center of the annihilation algebra is trivial, this complex is related to work by M. Eastwood [E] on conformally symplectic geometry, and we call it conformally symplectic pseudo de Rham complex.

Let d \subset d' be finite-dimensional Lie algebras, H = U(d), H'=U(d') the corresponding universal enveloping algebras endowed with the cocommutative Hopf algebra structure. We show that if L is a primitive Lie pseudoalgebra over H then all finite irreducible L' = Cur_H^{H'} L-modules are of the form Cur_H^{H'} V, where V is an irreducible L-module, with a single class of exceptions. Indeed, when L = H(d, \chi, \omega), we introduce non current L'-modules that are obtained by modifying the current pseudoaction with an extra term depending on an element t \in \d' \setminus d, which must satisfy some technical conditions. This, along with results from [BDK1, BDK2, BDK3], completes the classification of finite irreducible modules of finite simple Lie pseudoalgebras over the universal enveloping algebra of a finite-dimensional Lie algebra.