Monoidal Hom–Hopf Algebras

Abstract

Abstract. Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in literature recently. We study Homstructures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal category such that (certain classes of) Hom-algebras coincide with algebras in this monoidal category. Similar properties for Hom-coalgebras, Hopf and Lie algebras hold.

Braided Mixed Datums and Their Applications on Hom-Quantum Groups

“…Finally, by Corollary 3.11, [19], (3) is equivalent to (4). Similarly, by Corollary 5.11, [19], (5) is equivalent to (6). …”

“…Furthermore, if B M is a braided category with the braiding t, then there is an R-matrix R = t B,B (1 B ⊗ 1 B ) on B such that (B, R) is quasitriangular. But in the Hom case, recall from Remark 2.7 [6], if (H, α) is a Hom-bialgebra (the monoidal Hom-bialgebra case can be discussed in the same way), H is not a generator in its representation category. That means, if H M is the category of left H-Hom-modules, and if we define f m : H → M by f m (h) = h · m for any M ∈ H M, m ∈ M, then f m is not H-linear.…”

Milnor–Moore categories and monadic decomposition

“…Lemma 9.13). As particular cases of this situation we prove that the category H (M) of [CG,Proposition 1.1] is an MM-category, see Remark 9.10. Note that an object in Lie M , for M = H (M), is nothing but a Hom-Lie algebra.…”

“…which is the Hom-version of the universal enveloping algebra, see [CG,Section 8]. Note that, as a by-product, we have that…”

“…Let C be an ordinary category. Following [CG,Section 1], we associate to C a new category H (C) as follows. Objects are pairs (M, f M ) with M ∈ C and…”