project: pn-ii-id-pce-2011-3-0635, contract nr. 253/5.10...
TRANSCRIPT
Project: PN-II-ID-PCE-2011-3-0635, contract nr. 253/5.10.2011
”Hopf algebras in category theory, representation theory and combinatorics”
Director: S. Dascalescu
SCIENTIFIC REPORT, NOVEMBER 2016
The research activity on the subjects of the project led to the following 36 papers, of which 31
have been published or are accepted for publication in ISI journals, one is accepted for publica-
tion in a BDI journal, 4 are submitted for publication to ISI journals or are almost finalized. We
want to emphasize that all 3 junior researchers of the team have been involved in the research
subjects of the project, and they are authors or coauthors for the papers [2], [5], [11], [17],
[23], [25], [32] and [35]. Also, as a result of their participation to this project, all the 3 junior
researchers finalized and defended the PhD thesis: M. Barascu in November 2013, A. Manea in
September 2016, and L. Nastasescu in November 2016.
• [1] S. Dascalescu, M. Iovanov si C. Nastasescu, Quiver Algebras, Path Coalgebras and
co-reflexivity, Pacific Journal of Mathematics 262 (2013), 49-79.
• [2] M. Barascu si S. Dascalescu, Good gradings on upper block triangular matrix algebras,
Communications in Algebra 41 (2013), 4290-4298.
• [3] S. Dascalescu, S. Predut si L. Van Wyk, Jordan isomorphisms of generalized structural
matrix rings, Linear and Multilinear Algebra 61 (2013), 369-376.
• [4] F. Panaite, Equivalent crossed products and cross product bialgebras, Communications
in Algebra 42 (5) (2014), 1937-1952.
• [5] D. Joita, C. Nastasescu si L. Nastasescu, Recollement of Grothendieck categories. Ap-
plications to schemes, Bull. Math. Soc. Sci. Math. Roumanie 56 (104), 2013, 109-116.
• [6] A. Petrescu Nita si D. M. Staic, Symmetry group of two special types of carbon nanotori,
Acta Crystallographica Section A, Vol. 69, Part 4, 2013, 435-439.
• [7] M. Iovanov, Complete path algebras and rational modules, Bull. Math. Soc. Sci. Math.
Roumanie 56 (2013), 349-364.
• [8] S. Dascalescu, M. Iovanov, S. Predut, Frobenius structural matrix rings, Linear Alg.
Appl. 439 (2013), 3166-3172.
• [9] S. Dascalescu, M. Iovanov, Semiperfect and coreflexive coalgebras, Forum Mathematicum
27 (2015), 2587-2607.
• [10] M. Iovanov, Triangular matrix coalgebras and applications, Linear and Multilinear
Algebra 63 (2015), 46-67.
• [11] M. Barascu, Good Zp2 ×Zp×Zp-gradings on matrix algebras, Annals of the University
of Bucharest (Mathematical Series) 4 (2013), 425-431.
• [12] F. Panaite, Iterated crossed products, Journal of Algebra and its Applications 13 (7)
(2014), 14 pagini.1
2
• [13] D. Bulacu si B. Torrecillas, On Frobenius and separable algebra extensions in monoidal
categories. Applications to wreaths, Journal of Noncommutative Geometry 9 (2015), 707-774.
• [14] A. Makhlouf, F. Panaite, Yetter-Drinfeld modules for Hom-bialgebras, J. Math. Phys.
55, 013501 (2014) (17 pages).
• [15] A. Makhlouf, F. Panaite, Twisting operators, twisted tensor products and smash prod-
ucts for Hom-associative algebras, Glasgow Math. J 58(3), 513–538 (2016).
• [16] M. Hughes, D. M. Staic, Xie Xiangdong, Classification of a class of nonrigid Carnot
groups, Journal of Lie Theory 25 (2015), 717-732.
• [17] S. Dascalescu, C. Nastasescu, L. Nastasescu, Frobenius algebras of corepresentations
and group-graded vector spaces, J. Algebra 406 (2014), 226-250.
• [18] D. M. Staic, A. Stancu, Operations on the Secondary Hochschild Cohomology, Homol-
ogy, Homotopy and Applications 17 (2015), 129-146.
• [19] D. Bulacu, S. Caenepeel si B. Torrecillas, Frobenius and separable functors for the
category of generalized entwined modules. Applications, submitted to Trans. Amer. Math.
Soc.
• [20] Pascual Jara, Javier Lopez Pena, Dragos Stefan, Koszul pairs. Applications, accepted
by Journal of Noncommutative Geometry.
• [21] D. M. Staic, Secondary Hochschild homology, Algebra and Representation Theory 19
(2016), no. 1, 47-56.
• [22] D. Stefan, C. Vay, The cohomology ring of the 12-dimensional Fomin-Kirillov algebra,
Advances in Mathematics 291 (2016), 584-620.
• [23] A. Manea, D. Stefan, Further Properties of Koszul Pairs and Applications, SYMMETRY
INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS 12 (2016), Article
Number:092.
• [24] A. Makhlouf, F. Panaite, Hom-L-R-smash products, Hom-diagonal crossed products
and the Drinfeld double of a Hom-Hopf algebra, J. Algebra 441 (2015), 314–343.
• [25] M. Barascu, Counting good gradings on upper block triangular matrix algebras, in
preparation.
• [26] J. Dello, F. Panaite, F. Van Oystaeyen, Y. Zhang, Structure theorems for bicomod-
ule algebras over quasi-Hopf algebras, weak Hopf algebras and braided Hopf algebras, Comm.
Algebra 44 (2016), 4609–4636.
• [27] M. C. Iovanov, Z. Mesyan, M. L. Reyes, Infinite-dimensional diagonalization and
semisimplicity, accepted by Israel J. Math.
• [28] G. Graziani, A. Makhlouf, C. Menini, F. Panaite, BiHom-associative algebras, BiHom-
Lie algebras and BiHom-bialgebras, Symmetry, Integrability Geom. Methods Appl. (SIGMA)
11 (2015), 086, 34 pages.
3
• [29] L. Daus, F. Panaite, A new way to iterate Brzezinski crossed products, Colloq. Math.
142(1), 51–60 (2016).
• [30] D. Bulacu, B. Torrecillas, Galois and cleft monoidal cowreaths. Applications, submitted
to Proc. London Math. Soc.
• [31] C. Boboc, S. Dascalescu, L. van Wyk, Jordan isomorphisms of 2-torsionfree triangular
rings, Linear Multilinear Algebra 64(2016), 290-296.
• [32] A. Manea, D. Stefan, On Koszulity of finite graded posets, accepted by Journal of
Algebra and Its Applications.
• [33] Bruce R. Corrigan-Salter, Mihai D. Staic, Higher-order and secondary Hochschild co-
homology. C. R. Math. Acad. Sci. Paris 354 (2016), no. 11, 1049-1054.
• [34] F. Panaite, F. Van Oystaeyen, Twisted algebras and Rota-Baxter type operators,
accepted by J. Algebra Appl.
• [35] S. Dascalescu, C. Nastasescu, L. Nastasescu, Symmetric algebras in categories of corep-
resentations and smash products, J. Algebra 465 (2016), 62-80.
• [36] Jacob Laubacher, Mihai D. Staic, Alin Stancu, Bar Simplicial Modules and Secondary
Cyclic (Co)homology, in preparation.
In the paper [1] we investigate the connections between two combinatorial objects associ-
ated to an oriented graph: the quiver algebra and the path coalgebra. For this purpose,
we first extend the construction of the dual coalgebra of an algebra to algebras with enough
idempotents, not necessarily unital. For such an algebra A we consider the set A0 = {f ∈A∗| Ker(f) contains an ideal of A of finite codimension} and we construct a counital coalgebra
structure on A0, associated to the algebra structure of A. We give characterizations of A0 ex-
tending the ones in the unital case. We show that MA0, the category of right A0-comodules, is
isomorphic to the category LocF inAM of locally finite left A-modules.
Let now Γ be a quiver, K[Γ] the associated quiver algebra, which is an algebra with enough
idempotents, and KΓ the path coalgebra associated to Γ. It is easy to see that there exists
an embedding θ : KΓ → K[Γ]0. Let F θ : KΓM → K[Γ]0M the associated scalar corestriction
functor. The next result shows that in certain situations it is possible to reconstruct the path
coalgebra as the finite dual of the quiver algebra.
Theorem.Let Γ be a quiver. The following are equivalent:
(i) Γ does not have oriented cycles and between any two vertices there exists a finite number of
arrows.
(ii) θ(KΓ) = K[Γ]0.
(iii) Any cofinite ideal of K[Γ] contains a monomial cofinite ideal.
(iv) The functor F θ : KΓM → K[Γ]0M is an equivalence.
(v) Any locally finite representation of the quiver Γ is locally nilpotent.
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Conversely, one can ask the question of reconstructing the quiver algebra from the path
coalgebra. For this, let the linear map ψ : K[Γ] → (KΓ)∗, ψ(p)(q) = δp,q for any two paths p
and q. It is clear that ψ is an algebra homomorphism. The next result shows that the quiver
algebra may be reconstructed as a rational part of (KΓ)∗ in certain circumstances.
Theorem. The following are equivalent.
(i) Im(ψ) = (KΓ)∗ratl .
(ii) Im(ψ) = (KΓ)∗ratr .
(iii) For any vertex v of Γ there exists only a finite number of paths beginning in v and only a
finite number of paths ending in v.
(iv) KΓ is a left and right semiperfect coalgebra.
The question of reconstruction may be asked similarly for the incidence coalgebra and the
incidence algebra associated to a locally finite poset. It is well known that the incidence algebra
is isomorphic to the dual of the incidence coalgebra. One can show that always the incidence
coalgebra is isomorphic to the finite dual of a certain subalgebra of the incidence algebra (con-
sisting of all functions of finite support). Also, this subalgebra can be reconstructed from the
incidence coalgebra as a rational part, in certain circumstances.
Next, we treat the problem of the coreflexivity of path or incidence coalgebras and of their
subcoalgebras. We prove first the following general result.
Theorem. Let C be a coalgebra with the property that for any finite dimensional subcoalgebra
V there exists a finite dimensional subcoalgebra W such that V ⊆W and W⊥W⊥ =W⊥. Then
C is coreflexive if and only if its coradical C0 is a coreflexive coalgebra.
As a consequence we prove that if C is a subcoalgebra of a path coalgebra KΓ, such that
there exists only a finite number of paths between any two vertices of Γ, then C is coreflexive
if and only if C0 is coreflexive. This can be applied in particular to subcoalgebras of incidence
coalgebras. We prove also that a tensor product of two coreflexive coalgebras, that embed in
path coalgebras with the above mentioned finiteness property for paths, is coreflexive; this gives
a partial answer to an old open problem in coalgebra theory.
In the paper [2] we consider the k-algebra
A =
Mm1(k) Mm1,m2(k) . . . Mm1,mr(k)
0 Mm2(k) . . . Mm2,mr(k)
. . . . . . . . . . . .
0 0 . . . Mmr(k)
where k is a field and m1, . . . ,mr are nonzero natural numbers. The aim of the paper is to
classify the gradings of this algebra having the property that the matrix units eij that belong
to A are homogeneous elements. We prove that any such grading on A (called a good grading)
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is isomorphic to End(F) for a graded flag F : V1 ⊆ V2 ⊆ . . . ⊆ Vr, where V1, V2, . . . , Vr have
dimensions m1,m1 +m2, . . . ,m1 +m2 + . . .+mr. We prove the following:
Theorem. Let F and F ′ be two G-graded flags of the type (m1, . . . ,mr). Then End(F) ≃End(F ′) as G-graded algebras if and only if there exists σ ∈ G such that F ′ ≃ F(σ) as graded
flags.
The classification of good gradings on A is given by:
Theorem. There exists a bijection between the types of isomorphisms of good G-gradings on
A and the orbits of the biaction of the group Sm1 × . . .× Smr to the left (by permutations) and
of the group G to the right (by translations) on the set Gn.
In the paper [3] we study the Jordan isomorphisms between a ring A of block upper triangular
matrices (as in the paper [2]), where k is an arbitrary ring, and another ring. We prove that
if k is an indecomposable ring then any such Jordan isomorphism is a ring isomorphism or
anti-isomorphism. In fact, we prove the following much more general result:
Theorem. Let M be an indecomposable R,S-bimodule which is faithful as left R-module and
right S-module. We assume that R, S and M are without 2-torsion. Let A and B structural
matrix rings over the rings R and S and let M(P,M) be a structural bimodule of matrices hav-
ing no zero lines or columns and such that the set P that defines the bimodule is indecomposable
in a certain sense. Then any Jordan isomorphism between the ring T =
(A M(P,M)
0 B
)and
another ring is a ring isomorphism or anti-isomorphism.
In the paper [4] we study certain properties of crossed products. In 1997, Brzezinski intro-
duced a very general construction, called crossed product, containing as particular cases several
important constructions introduced before. Given an algebra A, a vector space V endowed with
a distinguished element 1V and two linear maps σ : V ⊗V → A⊗V and R : V ⊗A→ A⊗V sat-
isfying certain conditions, the Brzezinski crossed product is a certain algebra structure on A⊗V ,
denoted in what follows by A⊗R,σ V . In a previous paper we proved a result of the type invari-
ance under twisting for crossed products: if A⊗R,σ V is a crossed product and θ, γ : V → A⊗Vare linear maps, then we can define certain maps σ′ : V ⊗V → A⊗V and R′ : V ⊗A→ A⊗V and
if certain conditions are satisfied then A⊗R′,σ′ V is a crossed product, isomorphic to A⊗R,σ V .
Our first aim was to prove a converse of this fact. We introduce the following concept.
Let (A,µ, 1A) be an algebra, V a vector space endowed with a distinguished element 1V and
A ⊗R,σ V , A ⊗R′,σ′ V two crossed products. We will say that A ⊗R,σ V and A ⊗R′,σ′ V are
equivalent if there exists a linear isomorphism φ : A ⊗R′,σ′ V ≃ A ⊗R,σ V that is a morphism
of algebras and a morphism of left A-modules. With this terminology, the main result of the
paper is the following. Let A ⊗R,σ V and A ⊗R′,σ′ V be two crossed products. Then A ⊗R,σ V
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and A⊗R′,σ′ V are equivalent if and only if there exist linear maps θ, γ : V → A⊗ V , satisfying
certain conditions, such that
R′ = (µ2 ⊗ idV ) ◦ (idA ⊗ idA ⊗ γ) ◦ (idA ⊗R) ◦ (θ ⊗ idA),
σ′ = (µ⊗ idV ) ◦ (idA ⊗ γ) ◦ (µ2 ⊗ idV ) ◦ (idA ⊗ idA ⊗ σ)
◦(idA ⊗R⊗ idV ) ◦ (θ ⊗ θ),
(µ⊗ idV ) ◦ (µ⊗ σ′) ◦ (idA ⊗ γ ⊗ idV ) ◦ (R⊗ idV ) ◦ (idV ⊗ γ)
= (µ⊗ idV ) ◦ (idA ⊗ γ) ◦ σ.
There exists a dual construction to the one of Brzezinski, called crossed coproduct. By using
the two concepts, Bespalov and Drabant introduced the concept of cross product bialgebra,
which is a bialgebra whose algebra structure is a crossed product and whose coalgebra structure
is a crossed coproduct. If A ρW ◃▹ σR C and A ρ′
W ′ ◃▹ σ′R′ C are two cross product bialgebras, we will
call them equivalent if there exists a linear isomorphism φ : A ρ′
W ′ ◃▹ σ′R′ C ≃ A ρ
W ◃▹ σR C which
is a morphism of bialgebras, of left A-modules and of right C-comodules. A natural problem
is to characterize all cross product bialgebras A ρ′
W ′ ◃▹ σ′R′ C that are equivalent to a given cross
product bialgebra A ρW ◃▹ σR C.
In the paper we solve a particular case of this problem. Let (A,µA, 1A,∆A, εA) be a bialgebra
and (C,∆C , εC) a coalgebra. We assume that we have a crossed product A ⊗R,σ C. We define
the maps W0 : A⊗C → C⊗A, W0(a⊗ c) = c⊗a, ρ0 : A⊗C → A⊗A, ρ0(a⊗ c) = a1⊗a2εC(c).It is easy to see that AW0,ρ0 ⊗C is a crossed coproduct, namely it is exactly the tensor product
coalgebra A ⊗ C. We assume that moreover A ρ0W0
◃▹ σR C is a cross product bialgebra. Our
result describes all cross product bialgebras that are equivalent to A ρ0W0
◃▹ σR C. Namely, a cross
product bialgebra A ρ′
W ′ ◃▹ σ′R′ C is equivalent to A ρ0
W0◃▹ σR C if and only if there exist linear maps
θ, γ : C → A ⊗ C, with notation θ(c) = c<−1> ⊗ c<0> and γ(c) = c{−1} ⊗ c{0}, for all c ∈ C,
such that the conditions in the theorem describing equivalent crossed products are fulfilled plus
a number of extra conditions, among which the most important are (for all a ∈ A, c ∈ C):
W ′(a⊗ c) = εA(c<0>1{−1})εC(c<0>2{0}
)c<0>1{0}⊗ ac<−1>c<0>2{−1}
,
ρ′(a⊗ c) = εC(c<0>{0}{0})a1c<−1>1c<0>{−1} ⊗ a2c<−1>2c<0>{0}{−1}
.
As a particular case we obtained that the Drinfeld double of a quasitriangular Hopf algebra is
isomorphic, as a Hopf algebra, to a certain Radford biproduct (this is Majid’s theorem).
In the paper [5] are considered pairs of exact functors F : C → B and G : D → B, whereB, C,D are Grothendieck categories, C/Ker(F ) and D/Ker(G) are equivalent to B, such that
F and G are the canonical functors, and F and G commute with direct sums. We prove that
if F and G commute also with direct products and C, D are locally finite, then C∏BD is locally
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finite. Moreover, if C,D have enough projectives, then C∏BD has the same property. Some
applications to comodule categories are presented.
In the paper [6] we study the symmetry group of the graph associated to a carbon nan-
otorus. More precisely, by using a result of M. Buratti (Cayley, Marty and Schreier Hypergraphs,
1994), we associate to a group G generated by two elements {a, b} of order 3, a hypergraph
Cay3(G, {a, b}). G acts naturally on the hypergraph Cay3(T2, {a, b}). In the case in which
G = Tn =< a, b |a3 = 1, b3 = 1, (ab)3 = 1, (ab2)n = 1 > or G = SL(2, 3) =< a, b |a3 = 1, b3 =
1, aba = bab >, the associated hypergraph can be regarded as sitting on the torus S1 × S1. In
particular, any element of G induces an automorphism of the associated nanotorus. Also, in
the paper we study a subclass of automorphisms of the hypergraph that cannot be extended
to automorphisms of the torus and we present explicit examples of automorphisms having this
property. In the last section of the paper we compute the minimal model of the two types of
nanotori considered in this paper.
In the paper [7] we use quiver algebras to construct large classes of coalgebras C for which the
class of rational modules over the dual algebra C∗ is closed under extensions, or equivalently the
functor Rat from the category of left C∗-modules to that of rational modules is a torsion functor.
Some counterexamples are given and several open problems concerning the closure to extensions
of the class of rational modules are solved. We prove that the properties of coreflexivity, closure
to extensions of finite dimensional C∗-rational modules and of arbitrary modules in C∗M are
Morita invariant and are preserved under taking coalgebras.
In the paper [8] the following problem is considered: when is a structural matrix algebra
Frobenius? The structural matrix algebras are subalgebras of the algebra of n×n matrices over
a base field, consisting of those matrices having 0 in certain prescribed positions. On the other
hand, Frobenius algebras have applications in representation theory, topology, the Yang-Baxter
equation, conformal field theory, Hopf algebras etc. Our method is the following: a structural
matrix algebra is isomorphic to an incidence algebra of a certain finite preordered set, and this is
the dual of the incidence coalgebra corresponding to the same preordered set. So, it is natural to
consider the more general problem: when the incidence coalgebra of a locally finite preordered
set is right co-Frobenius? A complete answer is given by using techniques coming from the
corepresentation theory of (quasi)-co-Frobenius coalgebras.
Theorem. Let C = IC(X) be the incidence coalgebra of a locally finite preordered set X.
The following are equivalent:
(1) C is right co-Frobenius.
(2) C is right quasi-co-Frobenius.
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(3) C is cosemisimple.
(4) For all x, y ∈ X such that x ≤ y, we have also y ≤ x.
(5) IC(X) is right co-Frobenius.
(6) The order relation induced on the poset X is the equality.
As a consequence, a structural matrix algebra is Frobenius if and only if it is isomorphic (up
to a permutation of rows and columns) to an algebra of the type Mn1(k)× . . .×Mnr(k).
In the paper [9] we studied coalgebras without counit and we extended some results for couni-
tal coalgebras to the non-counital case. We proved the fundamental theorem for comodules
(coalgebras), the correspondence between the subcomodules of a C-comodule and the submod-
ules of the associated C∗-module. We constructed a universal counital coalgebra associated to a
non-counital coalgebra. Using these results we constructed the finite dual A0 of a not necessarily
unital algebra A and we gave equivalent characterizations for it. We proved that there exists
an isomorphism of categories between the category of right A0-comodules and the category of
locally finite representations of A. In case H is a bialgebra, H0 is also a bialgebra.
If C is a counital coalgebra, then the rational part RatC∗(C∗) of the left C∗-module C∗ is an
ideal in C∗, so it is a subalgebra (without unit). We can consider the finite dual of this algebra
and the application ϕl : C → (RatC∗(C∗))0 defined by ϕl(c)(c∗) = c∗(c), which is a morphism
of coalgebras. We say that C is left coreflexive if ϕl is bijective. We prove the following results.
Proposition. If C is left coreflexive, then any finite dimensional left RatC∗(C∗)-module M
has the structure of a right C-comodule such that the structure of RatC∗(C∗)-module is obtained
by restricting scalars from the associated structure of left C∗-module.
Theorem. Let C be a coalgebra. Then:
(a) C is right semiperfect if and only if ϕl is injective. In particular, if C is left coreflexive, then
C is right semiperfect.
(b) If C is right semiperfect and the coradical C0 is coreflexive, then C is left coreflexive.
Corollary. Let C be a coalgebra such that the coradical C0 is coreflexive. Then C is left
coreflexive if and only if C is right semiperfect. In this case C is coreflexive.
Theorem. Let C be a coalgebra which is left and right semiperfect. Then C is left and right
coreflexive. Moreover, ϕl is an isomorfism of counital coalgebras.
In this way, we unified some results about path coalgebras associated to some quivers having
certain finiteness properties and results concerning incidence coalgebras of locally finite posets.
The results are of interest for Hopf algebras as well: we prove the following reults.
Proposition. Let G be an algebraic group over C, L its Lie algebra and H the algebra of
functions on G. If L is finite dimensional (in particular, if G is an affine algebraic group), then
the Hopf algebras U(L) and H0 are coreflexive.
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Corollary. Let H be a Hopf algebra with nonzero integrals. Then there exists a coalgebra
isomorphism H ≃ (H∗rat)0.
In the paper [10] it is systematically studied the dual of the concept of generalized matrix
ring. Such rings appear by considering a structure of the type A =
(R M
N S
), where R and
S are rings, M and N are bimodules. This kind of ring structures are in correspondence with
concepts of Morita-type; they are also the source of many examples and counterexamples in
algebra. They also appear in the study of finite dimensinal algebras, of Artin algebras and of
generalized quiver algebras. The dual structure is that of a comatrix coalgebra. We define such
a structure of type n×n and we show how it is related to the Morita-Takeuchi theory. We study
in detail comatrix coalgebras of the type C =
(D M
0 E
), where D,E are coalgebras, and M
is an D − E−bicomodule. By using such coalgebras we completely determine the connections
between the properties of being quasi-finite, strictly quasi-finite, Artinian, co-Noetherian (for a
coalgebra). We also give examples showing the lack of simetry for these concepts.
We also use triangular matrix coalgebras to answer another problem in the theory of coalge-
bras, that of rational splitting for finitely generated modules: the rational part of any finitely
generated left C∗-module splits off if and only if C is a generalized triangular coalgebra C =(D M
0 E
), where D is a serial coalgebra whose Ext-quiver is a disjoint union of cycles, E is a
finite dimensional coalgebra and M is an D − E-bicomodule.
In the paper [11] is used the result obtained in [2], where the isomorphism types of good
gradings by a group G on a block superior triangular algebra are classified as orbits of a cer-
tain biaction. These orbits (and so the isomorphism types) are counted, in the case when
G = Zp2 × Zp × Zp.
In the paper [12] we obtained a result that generalizes in the same time two previously
obtained results for iterated twisted tensor products of algebras respectively for two-sided quasi-
Hopf smash products. The paper begins by presenting a ”mirror version” of the Brzezinski
crossed product, namely, given an algebra B, a vector space W endowed with a distingushed
element 1W and two linear maps ν : W ⊗W → W ⊗ B and P : B ⊗W → W ⊗ B, satisfying
certain conditions, one can define a certain algebra structure (also called Brzezinski crossed
product) on W ⊗B, denoted by W⊗P,νB. The main result is that under certain circumstances
the two versions of Brzezinski crossed product can be iterated. More precisely, if W⊗P,νD and
D ⊗R,σ V are two Brzezinski crossed products, and Q : V ⊗W → W ⊗D ⊗ V is a linear map
satisfying certain conditions, then one can define certain applications σ, R, ν, P such that we
10
have the Brzezinski crossed products (W⊗P,νD)⊗R,σ V and W⊗P ,ν(D⊗R,σ V ) which moreover
coincide as algebras (this algebra will de called an iterated Brzezinski crossed product). We
prove also that this result admits a certain converse.
In the paper [13] we introduced and studied extensions of Frobenius, respectively separable,
algebras in monoidal categories. A meaningful definition for these concepts was obtained by
studying forgetful functors. More precisely, if i : R→ S is a morphism of algebras in a monoidal
category C then this is Frobenius/separable if and only if the forgetful functor F : CS → CR is
Frobenius/separable. But this is possible only if the unit object 1 of the monoidal category has
what we called the property of being ⊗-generator. For many of the known monoidal categories,
this means that the morphisms from 1 to an object X are in a bijective correspondence with the
elements of the set X. Comparing to the version reported before, in order for the paper to be
accepted it had to be completed with a sufficient number of examples of categories having this
property or containing ⊗-generator objects. In this sense, we proved:
(1) In the category of sets and that of topological spaces the unit object is ⊗-generator.
(2) In a category of modules over a commutative ring the unit object is ⊗-generator; in
particular, the category of abelian groups has this property.
(3) In the category of bimodules associated to a k-algebra R the unit object is ⊗-generator
if and only if R is an Azumaya k-algebra; in particular this happens if R is a separable
k-algebra.
(4) In the category of finite dimensional Hilbert spaces and respectively in Zunino’s category
the unit object is ⊗-generator.
(5) In Turaev’s category the unit object is not ⊗-generator, but the object ({0, 1}, {k, k})is.
(6) In the category of k-vector spaces graded by a group G, with monoidal structure given
by a normalized 3-cocycle ω ∈ H3n(G, k
∗), the group algebra k[G] is ⊗-generator.
(7) In the category of representations of a Hopf (quasi-Hopf, respectively weak Hopf) algebra
H a ⊗-generator is exactly H.
The characterization of Frobenius/separable extensions with the help of forgetful functors
allowed to prove that i is Frobenius/separable if and only if S is a Frobenius/separable algebra
in the monoidal category of of R-bimodules, RCR. Consequently, we obtained a set of charac-
terizations for the extension i to be Frobenius/separable, specializing the ones existing in the
literature for a monoidal algebra. We have also proved that these concepts behave well with
respect to Frobenius/separable monoidal functors. Actually we proved that the forgetful functor
U : RCR → C, equipped with a canonical monoidal structure, is a Frobenius/separable monoidal
11
functor if and only if R is a Frobenius/separable algebra in C and moreover in this case the op-
monoidal structure of U is completely determined by a Frobenius structure of R. This allowed
us to prove the following result.
Theorem. If R is a Frobenius separable algebra in a sovereign monoidal category C then an
extension of algebras i : R→ S in C is Frobenius if and only if S is a Frobenius algebra and the
restriction to R of the Nakayama automorphism associated to S coincide with the Nakayama
automorphism associated to R.
The above mentioned results have been applied to extensions of algebras that are wreath
products in a 2-category K. These ones, as we proved, are pairs (A, s) with A monad in K and
s algebra in the Eilenberg-Moore category EM(K)(A). Thus, the canonical monad extension
induced by (A, s) is Frobenius/separable if and only if the pair (A, s) is Frobenius/separable, ifand only if s is a Frobenius/separable algebra in EM(K)(A). All these characterizations have
been specialized for C a monoidal category, regarded as a 2-category with a unique 0-cell. We
have obtained from this point of view characterizations at the level Frobenius/separable for
many of the extensions of algebras produced by various actions and coactions of a Hopf algebra
or generalizations (quasi-Hopf, weak Hopf, bialgebroid, braided, etc.). These results have been
improved later in the paper [19].
In the paper [14] we introduced the concept of Yetter-Drinfeld module over a Hom-bialgebra
and we proved that a (co)module over a (co)quasitriangular Hom-bialgebra becomes a Yetter-
Drinfeld module. If (H,µH ,∆H , αH) is a Hom-bialgebra with bijective αH , we proved that
Yetter-Drinfeld modules over H produce solutions of the Hom-Yang-Baxter equation. If we
denote by HHYD the category of those Yetter-Drinfeld modules (M,αM ) with bijective αM , we
proved that HHYD may be organized, in two different ways, as a quasi-braided pre-monoidal
category.
As is well known, an example of a Brzezinski crossed product is the twisted tensor product
of two associative algebras. In the paper [15] we introduced a similar construction for Hom-
associative algebras. Namely, if (A,µA, αA) and (B,µB, αB) are two Hom-associative algebras
and R : B ⊗ A → A ⊗ B is a linear map satisfying certain conditions, then we can define a
new Hom-associative algebra, denoted by A ⊗R B, called Hom-twisted tensor product. This
construction is introduced as a particular case of a more general construction, namely that of
twisting the multiplication of a Hom-associative algebra via a linear map called ”pseudotwistor”.
We proved also that under certain circumstances Hom-twisted tensor products may be iterated.
If H is a Hom-bialgebra and A is a left H-module algebra, we defined the smash product
A#H, which is a Hom-associative algebra, about which we prove that is a Hom-twisted tensor
product between A and H. We present a concrete example, obtained via a so-called Yau twisting
12
of the quantum group Uq(sl2). We prove that the smash product A#H is a right H-comodule
Hom-algebra, and if A is moreover a left H-comodule Hom-algebra and a left Yetter-Drinfeld
module over H, then the smash product A#H is an H-bicomodule Hom-algebra. We define as
well a two-sided smash product A#H#B, which turns out to be a particular case of an interated
Hom-twisted tensor product.
In the last section of the paper we present a new method to obtain Hom-associative algebras
from associative algebras, generalizing both the Yau twisting and a method we introduced before
to twist the multiplication of an associative algebra to obtain a new associative algebra.
In the paper [16] we classify up to isomorphism a family of nilpotent Lie algebras of dimension
10 over R. More precisely, we consider two copies of the Heisenberg Lie algebra H1C (generated
over C by X, Y and Z with the only nontrivial relation [X,Y ] = Z), and we identify the cen-
tres of the two Lie algebras by using a linear map f . In this way we obtain a Lie algebra Rf
of dimension 10 over R. The isomorphism classes in this family are parametrized by a scalar
α ∈ (0, 1] (the value α = 1 corresponds to the Heisenberg Lie algebra H2C). In the article we
used this characterization in order to classify (up to quasi-isomorphisms) a class of nilpotent Lie
groups.
In the paper [17] we investigated Frobenius algebras in certain monoidal categories. The study
of Frobenius algebras in monoidal categories was initiated by Muger, Street, Fuchs and Stigner,
Yamagami, etc. We consider the monoidal category MH of comodules (i.e. corepresentations)
over a Hopf algebra H. If A is a finite dimensional algebra in this category, i.e. A is a right H-
comodule algebra, then A and A∗ have natural structures of objects in the category MHA of right
Doi-Hopf modules. We say that A is right H-Frobenius if these two objects are isomorphic. On
the other hand, A∗ has a natural structure of left Doi-Hopf module over the right H-comodule
algebra A(S2), which is A as an algebra and has the H-coaction obtained via S2. We say that
A is left H-Frobenius if A∗ and A(S2) are isomorphic as left Doi-Hopf modules. We present
equivalent characterizations for these two Frobenius properties. Actually the property right
H-Frobenius is equivalent to being Frobenius algebra in the category MH . We prove that A is
left H-Frobenius if and only if A(S2) is a Frobenius algebra in the category MH . Also, we prove
that if A is right H-Frobenius then it is also left H-Frobenius, and that if S is injective, the two
properties are equivalent.
For the rest of the paper we specialize the study of Frobenius algebras to the case when
H is the Hopf group algebra kG of a group G. In this situation the corepresentations of H
are the G-graded vector spaces, and algebras in this category are the G-graded algebras. One
of the main results gives the structure of Frobenius algebras in this category. These are the
graded algebras A for which the homogeneous component of degree e (the unit element of G)
13
is a Frobenius algebra in the usual sense and A is e-faithful. Among gr-Frobenius algebras we
emphasize objects with more symmetry: gr-symmetric algebras. We prove that under certain
circumstances a graded field is a gr-symmetric algebra. We discuss the concept of gr-Frobenius
in relation with Frobenius functors. We prove that if the matrix algebra Mn(A) is Frobenius,
then A is also Frobenius. As an application we give a new proof of the fact that if H is a finite
dimensional Hopf algebra acting on an algebra A, then the smash product A#H is Frobenius if
and only if A is Frobenius.
In the paper [18] we continue the study of the secondary cohomology associated to a B-algebra
A. The main result obtained in this paper generalizes the cup product and the bracket from the
Hochschild cohomology H∗(A,A). More precisely, we have:
Theorem. If A is a k-algebra, B is a commutative k-algebra, ε : B → A is a morphism
of k-algebras such that ε(B) ⊂ Z(A), then the complex C∗((A,B, ε), A), defining the secondary
Hochschild cohomology, admits a structure of multiplicative operad.
In particular we have a structure of Gerstenhaber algebra on H∗((A,B, ε), A).
Another result in this paper shows that there exists a bijection between the elements in
H2((A,B, ε);M) and the equivalence classes of extensions of B-algebras of the type 0 → M →X → A → 0 where M2 = 0 and π : X → A is a morphism of B-algebras. This result
is a generalization of a well-known theorem about the second Hochschild cohomology group
H2(A,M).
We also prove in the article the existence of a Hodge-type decomposition for the secondary
Hochschild cohomology and we present a method to compute the group Hn((A,B, ε);A) for
n = 0, 1 si 2.
The paper [19] was initiated in 2013, and continued in 2014 and 2015. It can be regarded as
a sequel to the paper [13], because its aim was to characterize from the point of view Frobe-
nius/separable the forgetful functors from the category of entwined modules CXA to CA, where(A,X) is a co-wreath in a monoidal category C. In a first phase, we proved that EM(C)(A)acts on CA and so a (generalized) entwined module may be introduced as a comodule over the
coalgebra X in EM(C)(A) := T #A , in CA. This categorical point of view allowed us to prove that
the functor F is Frobenius, respectively separable, if and only if X is a co-Frobenius, respec-
tively coseparable, coalgebra, in the monoidal category EM(C)(A). We have to mention that
this result works only if the unit object of the category C is an ⊗-generator.
When X admits a right dual ∗X we proved that ∗X has a canonical structure of algebra in
EM(C)(A) := #AT (C is the reversed monoidal category associated to C) and the category of
representations over the wreath product associated to it is isomorphic to CXA . Moreover, we
obtained the following necessary and sufficient conditions for F to be a Frobenius functor:
14
Theorem. Let C be a monoidal category with co-equalizers such that any object is coflat
and robust to the left. If (A,X) is a co-wreath in C and X admits a right dual ∗X in C then the
following are equivalent:
(i) (X,ψ) is a co-Frobenius coalgebra in T #A ;
(ii) ∗X ⊗A is a Frobenius A-ring in C;(iii) (∗X,ψ) is a Frobenius algebra in #
AT ;
(iv) The extension of algebras A ↪→ ∗X#ψ,ζ,σA is Frobenius;
(v) A ⊗X and ∗X ⊗ A are isomorphic as left A-modules and as right ∗X#ψ,ζ,σA-modules
in C.(vi) A⊗X and ∗X ⊗A are isomorphic as left A-modules and as objects in CXA ;
(vii) There exists a morphism t : 1 → X in T #A (i.e. a co-Frobenius element for the coalgebra
(X,ψ) in T #A ) such that
Φ := (mA(IdA⊗mA)⊗IdX)(IdA⊗A⊗ψ(IdX⊗ev′X⊗IdA))((IdA⊗).t⊗Id∗X⊗A) :∗X⊗A→ A⊗X
is an isomorphism in C;(viii) There exists a morphism B : X ⊗ X → 1 in T #
A that is a Casimir morphism for the
coalgebra (X,ψ) in T #A and such that
Ψ := (Id∗X ⊗mA)(Id∗X⊗A ⊗B)(Id∗X ⊗ ψ ⊗ IdX)(coev′X ⊗ IdA⊗X) : A⊗X → ∗X ⊗A
is an isomorphism in C.(ix) The A-coring A⊗X is co-Frobenius, i.e. A⊗X is a co-Frobenius coalgebra in ACA.
If, moreover, 1 is left ⊗-generator in C then the above statements are also equivalent to
(x) The forgetful functor F : CXA → CA is a Frobenius functor.
(xi) The forgetful functor U : CA⊗X → CA is a Frobenius functor.
Similar characterizations have been obtained as well for the separability of the functor F. For
this, we first generalized a result of Larson to the level of monoidal categories:
Lemma. For a coalgebra C in a monoidal category C the following are equivalent:
(i) C is a coseparable coalgebra in C;(ii) The comultiplication ∆C : C → C ⊗ C co-splits in the category of C-bicomodules, i.e.
there exists a C-bicolinear morphism γ : C ⊗ C → C such that γ∆C = IdC .
(iii) There exists a morphism B : C ⊗ C → 1 in C such that
B∆C = εC and (IdC ⊗B)(∆C ⊗ IdC) = (B ⊗ IdC)(IdC ⊗∆C).
Together with the next result, all these lead to necessary and sufficient conditions for F to be
a separable functor.
Theorem. Let (A,X) be a co-wreath in a monoidal category C for which 1 is a left ⊗-
generator. The following are equivalent:
15
(i) The functor F : CXA → CA is separable;
(ii) (X,ψ) is a coseparable coalgebra in T #A ;
If, moreover, C admits co-equalizers and A,X are left coflat and C := A ⊗ X is the A-coring
associated to (A,X) then (i)-(ii) are also equivalent to
(iii) C is a coseparable A-coring in C, i.e. a coseparable coalgebra in !ACA,
and respectively to
(iv) The forgetful functor U : CC → CA is separable.
But if X admits right dual we can add the following equivalences:
(v) (∗X,ψ) is a separable algebra in #AT ;
(vi) A ↪→ ∗X#ψ,ζ,σA is a separable extension of algebras in C;(vii) ∗X ⊗A is a separable algebra in ACA, i.e. a separable A-ring in C.
The above results have been applied to co-wreaths that are obtained from actions and coac-
tions of quasi-Hopf algebras. They can be also applied to bialgebroids but the lenght of the
paper forced us to postpone this. From the important results obtained in this direction we
mention:
Theorem. Let H be a quasi-Hopf algebra and C a right H-module coalgebra. Then the
forgetful functor F : MCH → MH is
(i) Frobenius if and only if C is a co-Frobenius coalgebra in MH .
(ii) separable if C is a coseparable coalgebra in MH . The converse is not true, as shown by
the connection between the separable structures of F and the ones of the coalgebra C in MH .
Theorem. LetH be a quasi-Hopf algebra and C anH-bimodule coalgebra, that is a coalgebra
in the category of H-bimodules HMH . Then the functor F : HMCH → HMH is
(i) Frobenius if and only if C is a co-Frobenius coalgebra HMH . For C = H this happens if
and only if H is finite dimensional and unimodular.
(ii) separable if C is a coseparable coalgebra in HMH ; the converse is not true. For C = H
the separability of F is strongly connected to the unimodularity and cosemisimplicity of H.
As the referees asked, we had to include in the previous version the separable case for the
quasi-Hopf bimodule categories, as well as the study when the forgetful functor from a category
of Yetter-Drinfeld modules to the category of modules is Frobenius/separable. Thus the content
of the manuscript was improved by adding the
Theorem. Let H be a finite dimensional quasi-Hopf algebra. Then the forgetful functor
F : HMHH → HMH is separable if and only if H is unimodular, if and only if F is a Frobenius
functor,
and of a new subsection entitled “Yetter-Drinfeld modules over quasi-Hopf algebras” who
contains the following central results:
16
• The forgetful functor F : HHYD → HM is Frobenius if and only if H is finite dimensional
and H is a coFrobenius coalgebra within the monoidal category of H-bimodules;
• The forgetful functor F : HHYD → HM is separable if and only if H is a coseparable
coalgebra within the monoidal category of H-bimodules, if and only if H is unimodular and
cosemisimple.
• The algebra extension defined by the embedding of a finite dimensional quasi-Hopf algebra
H into its quantum double D(H) is Frobenius (resp. separable) if and only if H is finite dimen-
sional and unimodular (resp. unimodular and cosemisimple).
The study of Koszul rings is continued in [20] by using Koszul pairs as a main tool. Thus
• New bicomplexes are constructed for the computation of the Hochschild (co)homology
of a Koszul ring.
• In the case of twisted tensor products, the new complexes lead to the construction of
some spectral sequences connecting the Hocschild (co)homology of the product to the
(co)homologies of the factors.
• A method for computing the Hochschild dimension of a Koszul ring is indicated.
• New examples of Koszul rings are constructed and their Hochschild dimension and
Hochschild homology are computed.
• It is given a general criterion for a braided commutative bialgebra to be Koszul. An im-
portant particular case which is investigated is the one of symmetric braided bialgebras,
including quantum affine spaces.
• The method allows us to describe the coring structure of TorA∗ (R,R) for a Koszul R-ring
A.For example, the dual of a trivial extension is the cofree coalgebra cogenerated by the
homogeneous component of degree 1 of the extension.
• We show that the incidence algebra of the power set of a set with d elements is a Koszul
ring of dimension 3d as a vector space and of Hocschild dimension d.
• It is extended a result of Froberg, by showing that a quotient of a path algebra of a
quiver by an ideal generated by paths of length 2 is Koszul. It is indicated a formula for
the Hochschild dimension of the quotient ring and its Koszul dual is characterized.
In [21] we introduce a cohomology theory Hn((A,B, ε);M) associated to a B-algbera A and
an A-bimodule M . This cohomology describes the simultaneous deformation of the product on
A[[t]] and of the B-algebra structure on A[[t]]. The main idea is that a B-algebra structure on
A is determined by a family of multiplications {mαA ⊗ A → A}α∈B that satisfy a generalized
associativity condition mαβ(idA ⊗mα) = mβγ(mα ⊗ idA). We also give examples which show
that in general the natural map Hn((A,B, ε);M) → Hn(A,M) is not injective nor surjective.
17
When B = k we recover the usual Hochschild cohomology Hn(A,M).
In [22] the 12-dimensional Fomin-Kirillov FK3 is defined as the quadratic algebra with gen-
erators a, b and c and relations a2 = b2 = c2 = 0 and ab + bc + ca = 0 = ba + cb + ac. This
algebra is a member of a family {FKn}n∈N of algebras introducd by Fomin and Kirillov in order
to explain in a combinatorial way the fact that in the cohomology ring of a flag manifold, the
product of two Schubert classes is a linear combination with positive coefficients of Schubert
cells.
The cohomology ring E(A) := Ext∗A(K,K) of an algebra A over a field k plays an important
role in the study of the representations of A by using Algebraic Geometry. This method can be
applied for those algebras for which the ring E(A) is commutative (in a graded sense) and finitely
generated. For a Hopf algebra H, the first condition is always satisfied, and it was conjectured
that E(H) is finitely generated if and only if H is finite dimensional. The conjecture was proved
for several classes of Hopf algebras, including finite dimensional pointed Hopf algebras with
abelian coradical.
In this paper we initiate the computation of the cohomology of pointed Hopf algebras with
noncommutative coradical. More precisely, E(H) is computed for H = FK3#kΣ3, the 72-
dimensional Hopf algebra obtained by bosonizing FK3.
Briefly, the method is the following. We first determine E(FK3) by using the fact, proved
by A. Milinski and H.-J. Schneider, that the algebras FKn have a bialgebra structure in the
category of Yetter-Drinfeld modules, and by noting that FK3 is a twisted tensor product of an
algebra A of dimension 6 and R := k[X]/(X2). By using a version of the spectral sequence of
Cartan and Eilenberg, the computation reduces to knowing E(A). This last step is solved by
constructing a projective resolution of k regarded as an A-module. The result is the following.
Theorem. The algebra E(FK3) is isomorphic to the polynomial ring S[X], where S is the
symmetric braided algebra of the Yetter-Drinfeld module associated to the conjugacy class of the
transposition (1, 2) ∈ Σ3 and to the sign representation of Σ3.
One main step in the proof is the uncovering of the Hilbert series of the N-graded ring E(A).
Using the above result and an older result of the first author about the cohomology of a smash
product, we also obtain the following.
Theorem. The cohomology ring E(H) is isomorphic to k[X,U, V ]/(U2V −V U2), where degU =
deg V = 2 and degX = 4.
[23] Starting from the definitions and the characterizations of Koszul rings obtained in [2], one
obtains and studies new structures, namely Koszul R-corings. For these, we obtain 7 equivalent
characterizations. Namely, we have the following.
18
Theorem. Let C be a connected, strongly graded R-coring. The following are equivalent:
(1) The coring C is Koszul.
(2) The pair (E(C), C) is Koszul.
(3) The pair (C !, C) is Koszul.
(4) The canonical morphism E(C) → C ! is an isomorphism.
(5) The R-ring E(C) is strongly graded.
(6) The canonical map QE(C) → E1(C) is an isomorphism.
(7) The relation En,m(C) = 0 holds for all n = m.
In particular, we obtain that if C is a Koszul R-coring, then it is quadratic, i.e. Ext2,mC (R,R) =
0, for all m ≥ 3. This result constitutes the dual for the characterization of quadratic Koszul
R-rings, which ensures the nullity of TorA2,m(R,R).
Also, in the article, we study some concrete cases. First, in the case of graded left (right)
locally finite R-rings, we show that taking the left (right) graded linear dual keeps the Koszulity
property.
Theorem. Let (A,C) be a Koszul pair. If A and C are left locally finite, then (∗-grC, ∗-grA) is
Koszul. Similarly, if A and C are right locally finite, then (C∗-gr, A∗-gr) is Koszul.
In particular, the left (right) linear graded dual for a Koszul R-ring that is left (right) locally
finite is a Koszul Rop-coring. Moreover, we have the following result.
Theorem. If? (A,C) is a Koszul pair, then the ring E(A) = Ext∗A(R,R) is Koszul and:
E(A) ≃ ∗−grT (A) ≃ E(∗−grA) ≃ (∗−grA)!.
Another application concerns the incidence (co)rings of finite graded partially ordered sets.
In this case one proves that the incidence algebra of a finite graded poset is Koszul if and only
if the incidence coring of the same poset is Koszul as well.
In [24] we introduce the Hom-analog of the L-R-smash product. This is used to introduce
the Hom-analog of the diagonal crossed product. If H is a finite dimensional Hom-Hopf algebra
with bijective antipode and bijective structural map, we define the Drinfeld double of H; its
algebra structure is a diagonal Hom-crossed product, it is Hom-Hopf quasitriangular, and the
modules over the Drinfeld double are the same with the left-right Yetter-Drinfeld modules over
H.
The aim of [25] is to count the isomorphism types of good gradings on a complete upper tri-
angular blocked matrix algebra by one of the following groups: a finite cyclic group, Zp2 ×Zp2 ,
Zp × Zp2 , Zp × Zp × . . .× Zp, where p is a prime number. Combinatorial techniques and group
theoretical methods are used. The lattice of subgroups is investigated for certain finite abelian
19
groups.
In [26] we prove that if B is a bicomodule algebra over a quasi-Hopf algebra H such that
there is a morphism of H-bicomodule algebras v : H → B, then there exists an algebra A in the
category of Yetter-Drinfeld modules over H such that B ≃ A#H.
[27] We characterize the diagonalizable subalgebras of End(V ), the full ring of linear opera-
tors on a vector space V over a field, in a manner that directly generalizes the classical theory of
diagonalizable algebras of operators on a finite-dimensional vector space. Our characterizations
are formulated in terms of a natural topology (the finite topology) on End(V ), which reduces
to the discrete topology in the case where V is finite-dimensional. We further investigate when
two subalgebras of operators can and cannot be simultaneously diagonalized, as well as the
closure of the set of diagonalizable operators within End(V ). Motivated by the classical link
between diagonalizability and semisimplicity, we also give an infinite-dimensional generalization
of the Wedderburn-Artin theorem, providing a number of equivalent characterizations of left
pseudocompact, Jacoboson semisimple rings that parallel various characterizations of artinian
semisimple rings. This theorem unifies a number of related results in the literature, including
the structure of linearly compact, Jacobson semisimple rings and of cosemisimple coalgebras
over a field.
[28] In a previous work, G. Graziani extended the construction of the Hom-category H(C)introduced by Caenepeel and Goyvaerts to include the action of a given group G. Namely, given
a monoidal category C, a group G, two elements c, d ∈ Z(G) and ν an automorphism of the unit
object of C, the group Hom-category Hc,d,ν(G, C) has as objects pairs (A, fA), where A is an
object in C and fA : G → AutC(A) is a group homomorphism. The associativity constraint of
Hc,d,ν(G, C) is naturally defined by means of c, d, ν and it is, in general, non trivial. A braided
structure is also defined on Hc,d,ν(G, C), turning it into a braided category which is symmetric
whenever C is. When G = Z, c = d = 1Z and ν = id1 one gets the category H(C), while for
c = 1Z, d = −1Z and ν = id1 one gets the category H(C).We first investigated the case when G = Z× Z, c = (1, 0), d = (0, 1), ν = id1 and C = kM.
If M ∈ kM, a group homomorphism fM : Z× Z → Autk (M) is completely determined by
fM ((1, 0)) = αM and fM ((0, 1)) = β−1M .
Thus, an object inH(Z×Z, kM) identifies with a triple (M,αM , βM ), where αM , βM ∈ Autk (M)
and αM ◦ βM = βM ◦ αM . For (X,αX , βX), (Y, αY , βY ), (Z,αZ , βZ) objects in the category
20
H(1,0),(0,1),1(Z× Z, kM), the associativity constraint in H(1,0),(0,1),1(Z× Z, kM) is given by(ac,d,ν
)(X,αX ,βX),(Y,αY ,βY ),(Z,αZ ,βZ)
= aX,Y,Z ◦[(αX ⊗ Y )⊗ β−1
Z
],
and the braiding is
γc,d,ν
(X,αX ,βX),(Y,αY ,βY ) = τ[(αXβ
−1X
)⊗(α−1Y βY
)],
where τ : X ⊗ Y → Y ⊗ X denotes the usual flip in the category of linear spaces. Being
H(1,0),(0,1),1(Z× Z, kM) an additive braided monoidal category, all the concepts of algebra, Lie
algebra and so on, can be introduced in this case.
By writing down the axioms for an algebra in H(1,0),(0,1),1(Z × Z, kM) and discarding the
invertibility of α and β if not needed, we arrived at the following concept. A BiHom-associative
algebra over k is a linear space A endowed with a multiplication µ : A⊗A→ A, µ(a⊗ b) = ab,
and two commuting multiplicative linear maps α, β : A→ A satisfying what we call the BiHom-
associativity condition:
α(a)(bc) = (ab)β(c), ∀ a, b, c ∈ A.
Thus, a BiHom-associative algebra with bijective structure maps is exactly an algebra in
H(1,0),(0,1),1(Z× Z, kM).
If A is an associative algebra and α, β : A → A are two commuting algebra maps, then A
with the new multiplication defined by a ∗ b = α(a)β(b) is a BiHom-associative algebra, called
the Yau twist of A.
Take now the group G to be arbitrary. It is natural to describe how an algebra in the monoidal
category Hc,d,ν(G, kM) looks like. By writing down the axioms, it turns out that an algebra in
such a category is a BiHom-associative algebra with bijective structure maps having some extra
structure (like an action of the group on the algebra).
We initiated in this paper the study of what we called BiHom-structures. The next structure
we introduced is that of a BiHom-Lie algebra; for this, we used also a categorical approach.
There exists also a Yau twisting in this setting: if (L, [−]) is a Lie algebra over a field k and
α, β : L → L are two commuting multiplicative linear maps and we define the linear map
{−} : L ⊗ L → L, {a, b} = [α (a) , β (b)] , for all a, b ∈ L, then L(α,β) := (L, {−} , α, β) is a
BiHom-Lie algebra.
We defined representations of BiHom-associative algebras and BiHom-Lie algebras and found
some of their basic properties. Then we introduced BiHom-coassociative coalgebras and BiHom-
bialgebras together with some of the usual ingredients (comodules, duality, etc). We defined
smash products, as particular cases of twisted tensor products. We wrote down explicitly such
a smash product, obtained from an action of a Yau twist of the quantum group Uq(sl2) on a
Yau twist of the quantum plane A2|0q .
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[29] The main result of this paper is the following:
Theorem Let A⊗R,σV and A⊗P,νW be two Brzezinski crossed products and Q :W⊗V → V ⊗Wa linear map, with notation Q(w ⊗ v) = vQ ⊗ wQ, for all v ∈ V and w ∈ W . Assume that the
following conditions are satisfied:
(i) Q is unital, i.e. Q(1W ⊗ v) = v ⊗ 1W and Q(w ⊗ 1V ) = 1V ⊗ w, for all v ∈ V, w ∈W ;
(ii) the braid relation for R, P , Q, i.e.
(idA ⊗Q) ◦ (P ⊗ idV ) ◦ (idW ⊗R) = (R⊗ idW ) ◦ (idV ⊗ P ) ◦ (Q⊗ idA);
(iii) we have the following hexagonal relation between σ, P , Q:
(idA ⊗Q) ◦ (P ⊗ idV ) ◦ (idW ⊗ σ) = (σ ⊗ idW ) ◦ (idV ⊗Q) ◦ (Q⊗ idV );
(iv) we have the following hexagonal relation between ν, R, Q:
(idA ⊗Q) ◦ (ν ⊗ idV ) = (R⊗ idW ) ◦ (idV ⊗ ν) ◦ (Q⊗ idW ) ◦ (idW ⊗Q).
Define the linear maps
S : (V ⊗W )⊗A→ A⊗ (V ⊗W ), S := (R⊗ idW ) ◦ (idV ⊗ P ),
θ : (V ⊗W )⊗ (V ⊗W ) → A⊗ (V ⊗W ),
θ := (µA ⊗ idV ⊗ idW ) ◦ (idA ⊗R⊗ idW ) ◦ (σ ⊗ ν) ◦ (idV ⊗Q⊗ idW ),
T :W ⊗ (A⊗ V ) → (A⊗ V )⊗W, T := (idA ⊗Q) ◦ (P ⊗ idV ),
η :W ⊗W → (A⊗ V )⊗W,
η(w ⊗ w′) = (ν1(w,w′)⊗ 1V )⊗ ν2(w,w
′), ∀ w,w′ ∈W.
Then we have a Brzezinski crossed product A⊗S,θ (V ⊗W ) (with respect to 1V⊗W := 1V ⊗ 1W ),
we have a Brzezinski crossed product (A ⊗R,σ V ) ⊗T,η W and we have an algebra isomorphism
A⊗S,θ (V ⊗W ) ≃ (A⊗R,σ V )⊗T,η W given by the trivial identification.
[30] We developed a Hopf-Galois type theory for cowreaths (A,X) in monoidal categories
(regarded as 2-categories with one zero cell) for which A belongs to the associated category of
entwined modulesM(ψ)XA ; we called such a cowreath a pre-Galois cowreath. We have shown that
(A,X) is pre-Galois, provided that the associated A-coring A⊗X admits a grouplike element.
Furthermore, we called (A,X) Galois if it is pre-Galois and a certain canonical morphism is an
isomorphism.
In the case when (A,X) is pre-Galois we can associate to A the subalgebra of coinvariants,
denoted in what follows by B, and so an algebra extension i : B ↪→ A. More generally, to any
object of M(ψ)XA we can associate a right B-submodule, namely the one of the coinvariants.
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Consequently, we have a functor R from M(ψ)XA to MB which is a right adjoint for the extension
of scalars functor L = −⊗B A.
A first important result that we proved is the following (affiness criterion):
Theorem Let (A,X) be a pre-Galois cowreath. Then the following assertions are equivalent:
(1) L is a monoidal equivalence;
(2) R is a monoidal equivalence;
(3) BA is faithfully flat and (A,X) is Galois.
The Galois concept introduced so far was also studied vis-a-vis the so-called cleft extension
notion. More precisely, a pre-Galois cowreath (A,X) was called cleft if there exist morphisms
ϕ, ϕ−1 : X → A satisfying compatibility relations with the comultiplication ofX and the coaction
of X on A. Note that, in general, our definition has nothing to do with the concept of algebra
of convolution, since we cannot always consider this algebra. Nevertheless, with the help of the
concept that we introduced we were able to prove the following
Theorem. A cowreath is cleft if and only if it is Galois and satisfies the normal basis property.
The theoretical part ends with the study of cleft extensions versus wreath algebras. In this
direction, we have characterized cleft extensions as being certain wreath algebras.
The motivation for all this study was based on the fact that at this very moment does not
exist a Hopf-Galois theory for quasi-Hopf algebras. This is why, in the section of applications,
we resumed ourselves in specializing the general results presented above to cowreaths defined by
actions and coactions of a quasi-Hopf algebra only. But it can be also specialized to other many
situations like: bialgebroids, weak Hopf algebras, braided Hopf algebras and so to monoidal
Hom-Hopf algebras, etc.
[31] We construct a class of Jordan isomorphisms from a triangular ring T , and we show that
if T is 2-torsionfree, any Jordan isomorphism from T to another ring is of this form, up to a ring
isomorphism. As an application, we show that for triangular rings in a large class, any Jordan
isomorphism to another ring is a direct sum of a ring isomorphism and a ring anti-isomorphism.
Particular cases are complete upper block triangular matrix rings and indecomposable triangular
rings.
[32] A first goal of the paper is to give some new characterizations of Koszul rings, using as a
main tool the notion of Koszul pairs. More precisely, if R is a semisimple ring, one proves the
following.
Theorem. Let A be a connected strongly graded R-ring. The following are equivalent:
(1) The R-ring A is Koszul;
(2) The pair (A, T (A)) is Koszul;
(3) The pair (A,A!) is Koszul;
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(4) The canonical R-coring morphism ϕA : A! → T (A) is an isomorphism;
(5) The R-coring T (A) is strongly graded;
(6) Any primitive element of T (A) is homogeneous of degree 1;
(7) If n = m, then Tn,m(A) = 0.
The second goal of the paper is to present an algorithm for obtaining posets that have a Koszul
incidence R-ring (which we call Koszul posets). The algorithm is based on the result below, in
which we fix a finite graded poset P together with a maximal element t ∈ P. If Q := P \ {t},one denotes the incidence algebras of P and Q by A and B, respectively.
Theorem. Keeping the notations and the definitions above, the following identity holds true:
TorAn,m(R,R)∼= TorBn,m(S, S)⊕ TorBn−1,m(S,M).
In particular, A is Koszul if and only if B is so and TorBn−1,m(S,M) = 0, for all n = m.
Thus, after the verification of some simple combinatorial conditions, one can apply any of 4 steps
of a constructive algorithm finitely many times such that, starting with a Koszul poset, the final
construction has the same property. In particular, starting from the trivial poset P = {•},Koszul, we can construct a more complicated structure, having the same property. We distin-
guish two types of constructions which we present and study in the article: planar “tilings” and
nested diamonds. We describe for each of these the method of obtaining them, as well as some
general remarks which extend or restrict the applicability of the algorithm we introduced.
[33] In this paper we introduce a generalization of the higher Hochschild cohomology and
study its connection with secondary Hochschild cohomology. More precisely for a simplicial pair
(X,Y ), two commutative k-algebras A and B, a morphism of k-algebras ε : B → A and an A-
bimodule M , we introduce the group Hn(X,Y )((A,B, ε);M). When X = Y , A = B and ε = idA
we recover the higher Hochschild cohomology introduced by Pirashvili. When (X,Y ) = (D2, S1)
(with the usual simplicial structure) we recover the exact definition of the secondary Hochschild
cohomology.
[34] The concept of pseudotwistor (with a particular case called twistor) was introduced
in a previous paper as a general device for twisting (or deforming) the multiplication of an
algebra in a monoidal category, obtaining thus a new algebra structure on the same object.
Namely, if A is an algebra with multiplication µ : A ⊗ A → A in a monoidal category C, apseudotwistor for A is a morphism T : A⊗A→ A⊗A in C, such that there exist two morphisms
T1, T2 : A ⊗ A ⊗ A → A ⊗ A ⊗ A in C, called the companions of T , satisfying some axioms
ensuring that (A,µ ◦ T ) is also an algebra in C. There are many classes of examples of such
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pseudotwistors, arising for instance from twisted tensor product of algebras, braidings, Fedosov
products or bialgebras twisted by convolution invertible 2-cocycles.
However, there exist natural examples of ”twisted algebras” that are not given by pseu-
dotwistors, prominent examples being bialgebras twisted by non-convolution invertible 2-cocycles
or algebras twisted by a Rota-Baxter operator. Motivated by these examples, in this paper we
introduced the following concept. Assume that (A,µ) is an algebra in a monoidal category C,T : A⊗A→ A⊗A and T : A⊗A⊗A→ A⊗A⊗A are morphisms in C such that:
T ◦ (idA ⊗ (µ ◦ T )) = (idA ⊗ µ) ◦ T ,
T ◦ ((µ ◦ T )⊗ idA) = (µ⊗ idA) ◦ T .
Then (A,µ ◦ T ) is also an algebra in C, denoted by AT ; the morphism T is called a weak
pseudotwistor for A and the morphism T is called the weak companion of T . It turns out
that all the above-mentioned examples of deformed associative multiplications are afforded by
weak pseudotwistors, and we provided as well some other examples, coming especially from
Rota-Baxter type operators (Reynolds operators, Leroux’s TD-operators etc). We presented
also some general properties of weak pseudotwistors.
In the last section of the paper we used weak pseudotwistors in order to introduce an equiv-
alence relation for algebras in a monoidal category C: if A and B are two such algebras, we say
that A and B are twist equivalent (and write A ≡t B) if there exists an invertible weak pseu-
dotwistor T for A, with invertible weak companion T , such that AT and B are isomorphic as
algebras. For example, if A⊗R B is a twisted tensor product of algebras with bijective twisting
map R, then A⊗R B ≡t A⊗B.
In the paper [35] we consider the monoidal category MH of right comodules (or corepresen-
tations) over a Hopf algebra H. If A is an algebra in this category, i.e. a right H-comodule
algebra, then A ∈ AMHA , i.e. A is a left (A,H)-Doi-Hopf module and a right (A,H)-Doi-Hopf
module. On the other hand, A∗ is a right (A,H)-Doi-Hopf module, but not necessarily a left
(A,H)-Doi-Hopf module; however A∗ has a natural structure of a left (A(S2),H)-Doi-Hopf mod-
ule, where A(S2) is the algebra A with the coaction shifted by S2, where S is the antipode of H.
If H is cosovereign, i.e. there exists a character u on H such that S2(h) =∑u−1(h1)u(h3)h2
for any h ∈ H, then A ≃ A(S2) as comodule algebras, and this induces a structure of A∗ as an
object in AMHA , where the left A-action is a deformation of the usual one by u. Then it makes
sense to consider when A and A∗ are isomorphic in this category; in this case we say that A is
symmetric in MH with respect to u, or shortly that A is (H,u)-symmetric. We give explicit
characterizations of this property in MH . We show that the definition of symmetry depends
on the character (i.e. on the associated sovereign structure of MH). Also, we use a modified
version of the trivial extension construction to give examples of (H,u)-symmetric algebras of
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corepresentations. In the case where H is involutory, i.e. S2 = Id, H is cosovereign if we take
u = ϵ, the counit of H, and in this case it is clear that an (H, ϵ)-symmetric algebra is also sym-
metric as a k-algebra. However, we show that in general A may be (H,u)-symmetric, without
being symmetric as a k-algebra.
Given a finite dimensional algebra A in the category MH , where H is a finite dimensional
Hopf algebra, one can construct the smash product A#H∗. Smash products are also called
semidirect products, since the group algebra of a semidirect product of groups is just a smash
product. Smash product constructions are of great relevance since they describe the algebra
structure in a process of bosonization, which associates for instance a Hopf algebra to a Hopf
superalgebra. It is known that A is Frobenius if and only if so is A#H∗. On the other hand,
we show in an example that such a good connection does not hold for the symmetric property.
We show that if A is a Frobenius algebra in MH , then A#H∗ is a Frobenius algebra in MH∗,
but the converse does not hold. Also we uncover a good transfer of the symmetry property
between A and A#H∗, more precisely we show that A is (H,α)-symmetric if and only if A#H∗
is (H∗, g)-symmetric, where g and α are the distinguished grouplike (or modular) elements of
H and H∗, provided that H is cosovereign by α, and H∗ is cosovereign by g.
[36] In this paper we introduce a secondary bar complex B(A,B, ε), and show that it is a
simplicial module over a certain simplicial algebra A(A,B, ε). Then we prove that the complex
which gives the secondary Hochschild cohomology can be identified withHomA(A,B,ε)(B(A,B, ε), C(M)),
in particular this complex admits a simplicial structure. Using this secondary bar complex we
introduce a cyclic cohomology associated to a triple (A,B, ε) and show the existence of Connes’s
long exact sequence. We also study similar results for the homology version of this theory.