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Representations of

at even roots of unity

### Abstract

We construct all projective modules of the restricted quantum group at an even, 2p th, root of unity. This 64p^{4}-dimensional Hopf algebra is a common double bosonization of two rank-2 Nichols algebras 𝔅(X) with fermionic generator(s). We show that the category of -modules is equivalent to the category of Yetter–Drinfeld 𝔅(X)-modules in for H = ℤ2p ⊗ ℤ2p, where the coaction is defined by a universal R-matrix ρ ∈ H ⊗ H. As an application of the projective module construction, we study the basic algebra of and find the associative algebra structure and the dimension, 5p^{2} − p + 4, of its center.

© 2016 AIP Publishing LLC

Received 21 July 2015
Accepted 05 January 2016
Published online 24 February 2016

Acknowledgments:
We thank B. Feigin, A. Gainutdinov, and I. Heckenberger for the useful discussions and suggestions. The results obtained here and further prospects were also discussed with I. Angiono, A. Kiselev, S. Lentner, I. Runkel, H.-J. Schneider, and C. Schweigert. This paper was supported in part by the RFBR under Grant No. 13-01-00386.

Article outline:

I. INTRODUCTION
II. THE HOPF ALGEBRA **U**(*X*)
A. Generators and relations
1. The Hopf algebra
2. The second Hopf algebra structure
B. PBW basis in **U**(*X*)
C. Casimir elements
D. Quasitriangular and related structures
1. *R*-matrix relations
2. The *R*-matrix *ρ*
3. Drinfeld map
E. Coincidence of the two Drinfeld maps
III. EQUIVALENCE BETWEEN **U**(*X*) MODULES AND YETTER–DRINFELD 𝔅(*X*) MODULES
A. **U**(*X*) as a double bosonization
1. *U*_{<} is an algebra in *C*_{ρ}
2. *U*_{<} is a Hopf algebra in *C*_{ρ}
3. *U*_{<} = 𝔅(*X*)
4. **U**(*X*) modules are objects of *C*_{ρ}
5. **U**(*X*) modules are objects of
6. The functor is braided monoidal
B. The functor
1. Reminder: Radford’s biproduct
2. Radford formula for comodules
C. The functor
D. Composing the functors
IV. SIMPLE MODULES OF **U**(*X*)
A. Constructing simple **U**(*X*)-modules
1. decompositions and bases of simple modules
B. Casimir elements from simple modules
C. Ext^{1} spaces for simple **U**(*X*)-modules
1. Ext^{1} spaces for typical simple modules
2. Ext^{1} spaces for atypical simple **U**(*X*)-modules
3. Linkage classes
V. PROJECTIVE **U**(*X*) MODULES
A. The **U**(*X*) action and Loewy diagrams
B. Constructing the projective modules
1. Convention
2. Simple projective modules , 1 ⩽ *r* ⩽ *p* − 1
3. Projective covers of typical simple modules with 1 ⩽ *s* ⩽ *p* − 1, *r* ≠ 0, *s*
4. : Projective cover of
5. : Projective cover of
6. : Projective cover of for 2 ⩽ *s* ⩽ *p* − 1
7. : Projective cover of for 1 ⩽ *s* ⩽ *p* − 2
8. : Projective cover of
C. Completeness
D. Applications
VI. THE **U**(*X*) CENTER
A. The basic algebra of **U**(*X*)
B. The dimension of the center
1. Central idempotents and nilpotents
2. Proving Theorem 6.2: The strategy
3. Dimension of the center
VII. CONCLUSIONS AND AN OUTLOOK

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