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Register a new userHomology theory valued in the category of bicommutative Hopf algebras
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Authors
Minkyu Kim
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The codomain category of a generalized homology theory is the category of modules over a ring. For an abelian category A, an A-valued (generalized) homology theory is defined by formally replacing the category of modules with the category A. It is known that the category of bicommutative (i.e. commutative and cocommutative) Hopf algebras over a field k is an abelian category. Denote the category by H. In this paper, we give some ways to construct H-valued homology theories. As a main result, we give H-valued homology theories whose coefficients are neither group Hopf algebras nor function Hopf algebras. The examples contain not only ordinary homology theories but also extraordinary ones.
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In this paper, we give a generalization of Kitaevs stabilizer code based on chain complex theory of bicommutative Hopf algebras. Due to the bicommutativity, the Kitaevs stabilizer code extends to a broader class of spaces, e.g. finite CW-complexes ; more generally short abstract complex over a commutative unital ring R which is introduced in this paper. Given a finite-dimensional bisemisimple bicommutative Hopf algebra with an R-action, we introduce some analogues of A-stabilizers, B-stabilizers and the local Hamiltonian, which we call by the (+)-stabilizers, the (-)-stabilizers and the elementary operator respectively. We prove that the eigenspaces of the elementary operator give an orthogonal decomposition and the ground-state space is isomorphic to the homology Hopf algebra. In application to topology, we propose a formulation of topological local stabilizer models in a functorial way. It is known that the ground-state spaces of Kitaevs stabilizer code extends to Turaev-Viro TQFT. We prove that the 0-eigenspaces of a topological local stabilizer model extends to a projective TQFT which is improved to a TQFT in typical examples. Furthermore, we give a generalization of the duality in the literature based on the Poincare-Lefschetz duality of R-oriented manifolds.
This paper investigates the homology of the Brauer algebras, interpreted as appropriate Tor-groups, and shows that it is closely related to the homology of the symmetric group. Our main results show that when the defining parameter of the Brauer algebra is invertible, then the homology of the Brauer algebra is isomorphic to the homology of the symmetric group, and that when the parameter is not invertible, this isomorphism still holds in a range of degrees that increases with n.
The rank $n$ symplectic oscillator Lie algebra $mathfrak{g}_n$ is the semidirect product of the symplectic Lie algebra $mathfrak{sp}_{2n}$ and the Heisenberg Lie algebra $H_n$. In this paper, we study weight modules with finite dimensional weight spaces over $mathfrak{g}_n$. When $dot z eq 0$, it is shown that there is an equivalence between the full subcategory $mathcal{O}_{mathfrak{g}_n}[dot z]$ of the BGG category $mathcal{O}_{mathfrak{g}_n}$ for $mathfrak{g}_n$ and the BGG category $mathcal{O}_{mathfrak{sp}_{2n}}$ for $mathfrak{sp}_{2n}$. Then using the technique of localization and the structure of generalized highest weight modules, we also give the classification of simple weight modules over $mathfrak{g}_n$ with finite-dimensional weight spaces.
This paper studies the homology and cohomology of the Temperley-Lieb algebra TL_n(a), interpreted as appropriate Tor and Ext groups. Our main result applies under the common assumption that a=v+v^{-1} for some unit v in the ground ring, and states that the homology and cohomology vanish up to and including degree (n-2). To achieve this we simultaneously prove homological stability and compute the stable homology. We show that our vanishing range is sharp when n is even. Our methods are inspired by the tools and techniques of homological stability for families of groups. We construct and exploit a chain complex of planar injective words that is analogous to the complex of injective words used to prove stability for the symmetric groups. However, in this algebraic setting we encounter a novel difficulty: TL_n(a) is not flat over TL_m(a) for m<n, so that Shapiros lemma is unavailable. We resolve this difficulty by constructing what we call inductive resolutions of the relevant modules. Vanishing results for the homology and cohomology of Temperley-Lieb algebras can also be obtained from existence of the Jones-Wenzl projector. Our own vanishing results are in general far stronger than these, but in a restricted case we are able to obtain additional vanishing results via existence of the Jones-Wenzl projector. We believe that these results, together with the second authors work on Iwahori-Hecke algebras, are the first time the techniques of homological stability have been applied to algebras that are not group algebras.
Kitaevs quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. Originally it is based on finite groups, and is later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized Kitaev quantum double model. These ribbon operators are important tools to understand quasi-particle excitations. It turns out that there are some subtleties in defining the operators in contrast to what one would naively think. In particular, one has to distinguish two classes of ribbons which we call locally clockwise and locally counterclockwise ribbons. Moreover, this issue already exists in the original model based on finite non-Abelian groups. We show how certain properties would fail even in the original model if we do not distinguish these two classes of ribbons. Perhaps not surprisingly, under the new definitions ribbon operators satisfy all properties that are expected. For instance, they create quasi-particle excitations only at the end of the ribbon, and the types of the quasi-particles correspond to irreducible representations of the Drinfeld double of the input Hopf algebra. However, the proofs of these properties are much more complicated than those in the case of finite groups. This is partly due to the complications in dealing with general Hopf algebras rather than just group algebras.
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