Loewy filtration and quantum de Rham cohomology over quantum divided power algebra

Abstract

Abstract. The paper explores the indecomposable submodule structures of quantum divided power algebra Aq(n) defined in [22] and its truncated objects Aq(n, m). An "intertwinedly-lifting" method is established to prove the indecomposability of a module when its socle is non-simple. The Loewy filtrations are described for all homogeneous subspaces A (s)q (n, m), the Loewy layers and dimensions are determined. The rigidity of these indecomposable modules is proved. An interesting combinatorial identity is derived from our realization model for a class of indecomposable uq(sln)-modules. Meanwhile, the quantum Grassmann algebra Ωq(n) over Aq(n) is constructed, together with the quantum de Rham complex (Ωq(n), d • ) via defining the appropriate q-differentials, and its subcomplex (Ωq(n, m), d • ). For the latter, the corresponding quantum de Rham cohomology modules are decomposed into the direct sum of some sign-trivial uq(sln)-modules.

Realization of U_q(sp_2n) within the Differential Algebra on Quantum Symplectic Space

“…We use induction on j to prove (4.7). For j = 2, this is obvious by Proposition 4.2 (3). Now suppose that (4.7) holds for some j with 2 < j < n. Then Proposition 4.2(2) and induction yield…”

“…Especially, this discussion of q-derivatives resulted in the definition of the quantum universal enveloping algebras of abelian Lie algebras for the first time, and even the new Hopf algebra structure so-called the n-rank Taft algebra (see [7,11]) in root of unity case. Based on the realization in [6], Gu and Hu [3] gave further explicit results of the module structures on the quantum Grassmann algebra defined over the quantum divided power algebra, the quantum de Rham complexes and their cohomological modules, as well as the descriptions of the Loewy filtrations of a class of interesting indecomposable modules for Lusztig's small quantum group u q (sl n ).…”

Quantum (dual) Grassmann superalgebra as $\mathcal U_q(\mathfrak{gl}(m|n))$-module algebra and beyond

“…(see [6]) when char(q) = 0 and dim k V = n, while dim k A (see Corollary 2.6 [7]) when char(q) = ℓ > 0. Then the results follow from the proof of Theorem 33.…”

“…1010) (his motivation mainly from [33] & [34]), but not yet sufficient. It should be noticed that the design of our QDO in subsections 2.6, 3.4 (also see [13] & [37]) leads to our quantum differential (form) d satisfying the twisted Leibniz rule (see [7], pp. 5), which is different from both [33] and [34].…”

“…We do it in two steps. Firstly, we lift vector x (α) ⊗ x µ to x (ν ) ⊗x µ , which is due to the special case appeared in Claim (B) of the proof of Proposition 3.5 in [7] for α ∈ A q (m, 1) (t ′ ) with t ′ = t−s = | α | = | ν | and u q (sl(m)) when m = 1 (the restricted case). Secondly, we lift vector x (ν ) ⊗ x µ to highest weight vector x (t) ⊗ 1.…”