Symmetric algebras, skew category algebras and inverse semigroups

Constantin-Cosmin Todea

Babeş-Bolyai University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu, 1, 400084 Cluj-Napoca, Romania

Abstract

We use results from the theory of inverse semigroups to identify and define a class of symmetric algebras, which we call inverse-symmetric algebras. For a finite inverse category C we define the skew category algebra associated to a C -algebra, which satisfies some additional properties, and we prove that remains inverse-symmetric. The classical examples of symmetric algebras: matrix algebra, Iwahori-Hecke algebra and the (twisted) group algebra are inverse-symmetric.

Keywords: inverse semigroup, symmetric algebra, category algebra, inverse category 2000 MSC:16S99, 20M18, 18B99

1. Introduction

There are different examples of symmetric algebras: group algebra, Iwahori-Hecke algebra, matrix algebra. More recently in [6], M. Linckelmann proved that the category algebra associated to a finite inverse category is a symmetric algebra. Inverse categories are defined in a similar way to inverse semigroups and this suggest that inverse semigroups could have applications for symmetric algebras. In Section 2 we recall some basic results for inverse semigroups from [5] and we analyze some well known examples of symmetric algebras, which are also of finite rank and G-algebras, where G is a finite group. These examples and their basis leads to the definition of inverse-symmetric algebras. In the main result of this section we prove that an inverse-symmetric algebra is principally symmetric. We also notice a different proof for [6, Lemma 4.4] using the general setting of inversesymmetric algebras. That is, we prove that the category algebra of a finite inverse category is inverse-symmetric. In Section 3 we extract the concept of C -algebras (C is a finite

January 11, 2014

Email address: Constantin.Todea@math.utcluj.ro (Constantin-Cosmin Todea)

category) from a definition given in [7, Section 1]. By gluing the concept of an inversesymmetric algebra with that of C -algebra we define in this case a ”skew category algebra” and show that it remains inverse-symmetric. In particular we obtain conditions for a Galgebra which is symmetric to induce a symmetric skew group algebra.

We consider k to be a commutative ring, A an associative algebra with identity element, which usually is free of finite rank as k-module, unless otherwise is specified. If S is a finite set and (b_{s})_{s∈S} is a subset of A indexed by S, which is a k-basis for A we denote it by B_{S}, that is B_{S} = {b_{s} | s ∈ S}. When T is a subset of S we have B_{T} = {b_{t} | t ∈ T } ⊆ B_{S}. In this case we associate the map α_{T} : T × T × T → k , where α_{T}(s,t,u) is the coefficient of b_{u} in the unique decomposition of b_{s}· b_{t} in the k-basis B_{S} if b_{s}· b_{t} ≠ 0, or α_{T}(s,t,u) = 0 if b_{s}· b_{t} = 0, for any s,t,u ∈ T. We call α_{T} the coefficient map associated to T. We denote by k^{×} the group of invertible elements in the ring k.

We denote by C a finite category, that is a category with the set of morphisms HomC a finite set. In this case the set of objects ObC is also finite. We denote by: EndC, E(C) the set of all endomorphisms of C, respectively the set of all idempotent endomorphisms of C.

2. Inverse-symmetric algebras

Following [5] we recall what is an inverse semigroup and we use basic results without comments. A semigroup (S,·) is inverse if for any s ∈ S there is a unique ŝ(named inverse) such that s · ŝ· s = s and ŝ· s · ŝ= ŝ. By [5, 1.1, Theorem 3], if (S,·) is inverse then all idempotents of S commutes and we have ˆ ŝ= s and ŝ · t= t̂· ŝ, for any s ∈ S.

In the following sections we work with inverse semigroups which can have a zero (that is an element denoted 0 ∈ S, such that s · 0 = 0 · s = 0 for any s ∈ S), and in this case we denote by S^{∗} the set S \ {0}, or we work with examples of inverse semigroups which don’t have a zero, and in this case we make the convention that S^{∗} = S. We denote by Idemp(T) the set of all idempotents of T, where T ⊆ S. The following definition is suggested by the ideas from [6] and by methods used to prove that the (twisted) group algebras and the Iwahori-Hecke algebras are symmetric. The multiplication in a semigroup and in an algebra is denoted with the same symbol ”·”, but from the context the reader should understand the difference.

Definition 2.1. A k-algebra A is inverse-symmetric if there is a triplet (S,B_{S^{∗}},α_{S^{∗}}) where S is a finite inverse semigroup, B_{S^{∗}} is a k-basis of A and α_{S^{∗}} is the coefficient map associated to S^{∗} such that:

(1) For s,t ∈ S^{∗},e ∈ Idemp(S^{∗}) we have that α_{S^{∗}}(s,t,e) ≠ 0 if and only if t = ŝ and e = s · ŝ; (2) For any s ∈ S^{∗} we have α_{S^{∗}}(s,ŝ,s · ŝ) = α_{S^{∗}}(ŝ,s,ŝ· s) ∈ k^{×}.

Example 2.2. a) If A = kG is the group algebra over a finite group G then the finite set B_{G} = G is a k-basis and S = S^{∗} = G may be considered as an inverse semigroup, with ĝ = g^{−1} for g ∈ G. Since Idemp(G) = {1_{G}} and 1_{G} can be in a decomposition of g · h in the basis G if and only if h = g^{−1} , then condition (1) from Definition 2.1 is satisfied. Moreover for any g ∈ G we have

α_{G}(g,g^{−1} ,1_{G}) = α_{G}(g^{−1} ,g,1_{G}) = 1_{k}.

b) If A = End_{k}(M), where M is a kG-lattice (that is a finitely generated, free k-module with a G-stable finite basis X), then B = {b_{x,y}| x,y ∈ X} is a k-basis of A where

b_{x,y}: M → M, b_{x,y}(z) = \{ \begin{matrix}x, z = y \\ 0, z ≠ y \end{matrix} .

Let S = X × X ∪ {0} and the composition law defined on S for (x,y),(x_{1},y_{1}) ∈ S by

(x,y) · (x_{1},y_{1}) = \{ \begin{matrix}(x,y_{1}), y = x_{1} \\ 0, y ≠ x_{1} \end{matrix} .

Then (S,·) is an inverse finite semigroup and { \atop (x,y)}= (y,x). Now Idemp(S^{∗}) = {(x,x) | x ∈ X} and b_{x,y}∘ b_{x_{1},y_{1}} is the zero map if y ≠ x_{1}, or it is b_{x,y_{1}} if y = x_{1}. The conditions from Definition 2.1 are now satisfied.

Moreover the above two examples are also G-algebras with G-stable basis. This suggest that we can define a class of symmetric G-algebras and to analyze the skew group algebra in this case. This will be done in Section 3 in a more general situation.

Remark 2.3. Let A be an inverse-symmetric k-algebra with (S,B_{S^{∗}},α_{S^{∗}}) its corresponding triplet and let s,t ∈ S^{∗},e ∈ Idemp(S^{∗}) with t ≠ ŝ. Then α_{S^{∗}}(s,t,e) = 0 and α_{S^{∗}}(t,s,e) = 0.

Following [1] we recall the definition of a symmetric algebra. A k-algebra A is called symmetric if it is finitely generated and projective as k-module and there is τ : A → k a central form (that is k-linear map with τ(a · a′) = τ(a′· a) for all a,a′∈ A), which induces an isomorphism of A − A-bimodules

τ̂: A → A^{∗}, τ̂(a)(b) = τ(a · b),

for a,b ∈ A, where A^{∗} is the k-dual. τ is called symmetric form of A and A is principally symmetric if τ is onto.

Theorem 2.1. If A is an inverse-symmetric k-algebra then A is principally symmetric. In particular, it is symmetric.

Proof. By Definition 2.1 we have that A is a finitely generated k-module and free, thus projective. There is a triplet (S,B_{S^{∗}},α_{S^{∗}}) which satisfy Definition 2.1. We define the following k-linear form on the basis B_{S^{∗}}

τ_{B_{S^{∗}}}: A → k, τ_{B_{S^{∗}}}(b_{s}) = \{ \begin{matrix}1_{k}, s ∈ Idemp(S^{∗}) \\ 0, s ∉ Idemp(S^{∗}) \end{matrix}

We prove that it is a central form, that is τ_{B_{S^{∗}}}(b_{t}· b_{s}) = τ_{B_{S^{∗}}}(b_{s}· b_{t}) where b_{t},b_{s} ∈ B_{S^{∗}}, by considering the cases:

-If t ≠ ŝ, by Remark 2.3 it follows that

τ_{B_{S^{∗}}}(b_{t}· b_{s}) = 0 = τ_{B_{S^{∗}}}(b_{s}· b_{t}).

-If t = ŝlet bŝ· b_{s} = ∑_{u∈S^{∗}} α_{S^{∗}}(ŝ,s,u)b_{u}. Then

τ_{B_{S^{∗}}}(bŝ· b_{s}) = ∑_{e∈Idemp(S^{∗})}α_{S^{∗}}(ŝ,s,e) = α_{S^{∗}}(ŝ,s,ŝ· s),

where the last equality follows from Definition 2.1, (1). Similarly we have τ_{B_{S^{∗}}}(b_{s}· bŝ) = α_{S^{∗}}(s,ŝ,s · ŝ), which gives us the first part of the proof.

τ_{B_{S^{∗}}} induces the following A − A-bimodule homomorphism τ̂_{B_{S^{∗}}}: A → A^{∗} defined by

τ̂_{B_{S^{∗}}}(b_{t})(b_{s}) = τ_{B_{S^{∗}}}(b_{t}· b_{s})

for any b_{t},b_{s} ∈ B_{S^{∗}}.

Now let λ ∈ A^{∗} and define a = ∑_{t∈S^{∗}}[α_{S^{∗}}(t,t̂,t ·t̂)]^{−1} λ(b_{t})bt̂ ∈ A. Then for b_{s} ∈ B_{S^{∗}}

τ̂_{B_{S^{∗}}}(a)(b_{s}) = ∑_{t∈S^{∗}}[α_{S^{∗}}(t,t̂,t ·t̂)]^{−1} λ(b_{t})τ_{B_{S^{∗}}}(bt̂· b_{s}).

Since τ_{B_{S^{∗}}}(bt̂· b_{s}) = α_{S^{∗}}(s,ŝ,s · ŝ) = α_{S^{∗}}(ŝ,s,ŝ· s) if t = s, and it is zero otherwise, we obtain that

τ̂_{B_{S^{∗}}}(a)(b_{s}) = λ(b_{s})[α_{S^{∗}}(s,ŝ,s · ŝ)]^{−1} τ_{B_{S^{∗}}}(bŝ· b_{s}) =
λ(b_{s})[α_{S^{∗}}(s,ŝ,s · ŝ)]^{−1} [α_{S^{∗}}(s,ŝ,s · ŝ)] = λ(b_{s}),

thus τ̂_{B_{S^{∗}}} is surjective. We conclude that it is a bimodule isomorphism since A is of finite rank as k-module.

We end this section with some examples of inverse-symmetric algebras. In [6] M. Linckelmann proves that the category algebra over a finite inverse category (see [4]) is a symmetric algebra. Definition 2.1 and Theorem 2.1 allow us to give an alternative proof for this. In [6, Section 3] results from [8] are transposed into the language of inverse categories. If C is a finite inverse category then HomC becomes a finite partial order set with the partial order defined by s ≤ t if there is e ∈ E(C) such that s = t ∘ e. We can associate the Möbius function μ : HomC ×HomC → k, defined inductively by μ(s,s) = 1 if s ∈ HomC and if s < t by

μ(s,t) = − ∑_{s<u≤t} μ(u,s).

The Möbius Inversion theorem due to G. Rota states that if we consider the abelian group (kC ,+) and the maps

g: HomC → kC , g(u) = u,
f : HomC → kC , f(s) = \underline{s},

where u,s ∈ HomC , \underline{s} = ∑_{u≤s} μ(u,s)u, then g(s) = s = ∑_{u≤s} \underline{u}. It is now easy to check the following Remark which is a restatement of existing results in [6] and [8], so we don’t give the proof.

Remark 2.4. Let C be a finite inverse category. Then the set \{\underline{s}\}_{s∈HomC} is a finite k-basis for kC. If we consider \underline{t},\underline{s} ∈ kC then the product in this basis is \underline{t}· \underline{s} = \underline{t ∘ s}if s ∘ ŝ= t̂∘ t, and \underline{t}· \underline{s} = 0 otherwise. If t,s ∈ HomC such that s ∘ ŝ= t̂∘ t and t ∘ s ∈ E(C) then t = ŝ. Moreover, if t,s ∈ HomC we have \underline{t}= \underline{s} if and only if t = s.

Example 2.5. In [6, Theorem 4.1] M. Linckelmann proves that kC is isomorphic to a category algebra over a groupoid associated to the inverse category, that is kG(C). Now it is clear that clear that kG(C) is a trivially inverse-symmetric k-algebra. An alternatively, more directly proof is to consider as k-basis of kC the set from the Remark 2.4. By taking S to be Hom(C) ∪ {0} we see that kC is an inverse-symmetric algebra.

Remark 2.6. Let G be a finite group and λ : G × G → k^{×} a normalized 2-cocycle. Then the twisted group algebra k_{λ}G is an inverse-symmetric algebra. We take S = S^{∗} to be the group G, so Idemp(G) = \{1_{G}\}. We have the k-basis B_{G} = \{\overline{s} | s ∈ G\} with the product given by \overline{s}· \overline{t}= λ(s,t)\overline{s · t}, for any \overline{s},\overline{t}∈ B_{G}.

Remark 2.7. For Iwahori-Hecke algebras we follow the definitions from [3]. Let (W,X) be a finite Coxeter system and H = H_{k}(W,X,{a_{x},b_{x}| x ∈ X}) be the associated Iwahori-Hecke algebra. If all a_{x}(x ∈ X) are invertible in k then H is inverse-symmetric. The Coxeter group (W,·) is considered as the finite inverse semigroup S = S^{∗}. Then B_{W}= {T_{s} | s ∈ W } is a finite k-basis of H (see [3, Theorem 4.4.6]) and Idemp(W) = {1}, since W is a group. Using [3, Definition 4.4.1] and the symmetric form, which is defined in [3, Proposition 8.1.1], it is easy to prove that the Definition 2.1 is satisfied.

3. Skew category algebras

In this section if f ∈ HomC with f : x → y, where x,y ∈ ObC, we denote by d(f) the domain x of f and by c(f) the codomain y. Let k − Alg be the category of k-algebras and k − alg (k − mod) be the category of finitely generated k-algebras (k-modules). Moreover C^{[2]}represents the category of functors from the category [2] (with 3 objects and only one morphism between any two objects in [2]) to C. We are mainly interested in its objects, that is chains of two composable morphisms. In general we have an object (f,g) ∈ C^{[2]}if and only if d(f) = c(g). We denote by 1_{x} the identity morphism of x ∈ ObC and by id_{A} the identity morphism of an algebra A. In this section if C is an inverse category we keep the notation f̂for the inverse of f ∈ HomC. So we will have the same notation for inverse semigroups and inverse categories.

In [7] J. Oïner and P. Lundström defined the crossed product associated to a crossed system, see [7, §1]. From this we extract only the part that it is useful to define a concept of ”skew category algebra” associated to an inverse category.

Definition 3.1. Let C be a finite category. A C -algebra is a pair (A,F), where A = (A_{x})_{x∈ObC} is a family of k-algebras and F = (F_{f})_{f∈HomC} is a family of homomorphisms of k-algebras such that:

(1) F_{f} : A_{d(f)} → A_{c(f)}, for any f ∈ HomC);

(2) F_{1_{x}} = id_{A_{x}}, for any x ∈ ObC;

(3) F_{f∘g} = F_{f} ∘ F_{g}, for any (f,g) ∈ C^{[2]}.

Remark 3.2. Equivalently , (A,F) is a C -algebra if we view F as a covariant functor F : C → k − Alg such that F (x) = A_{x} for any x ∈ ObC, and F (f) = F_{f} for any f ∈ HomC.

Example 3.3. Let C = Ĝbe a group G viewed as a category with one object (the identity 1_{G}) and HomĜ= G. Then (A,F) is a Ĝ-algebra if and only if A = A_{1_{G}} is a family with one k-algebra and F_{g} : A → A are homomorphism of k-algebras which satisfy Definition 3.1, for any g ∈ HomĜ. We obtain that F induces a group homomorphism F : G → Aut_{k}(A), thus A is actually a G-algebra.

We construct in the next example a C -algebra for C a groupoid, which can have more than one object. We use without comments explicit descriptions and results regarding category algebras and modules over category algebras from [9].

Example 3.4. Let C be a groupoid and M be a kC -module obtained from a covariant functor M : C → k − mod. That is M = ⊕_{x∈ObC}M(x) as kC -module. The C -algebra ((End_{k}(M(x))_{x∈ObC},M) is defined by the functor M : C → k − Alg, where M (x) = End_{k}(M(x)). For f ∈ Hom_{C}(x,y) there is a well defined homomorphism of k-algebras

M_{f} : End_{k}(M(x)) → End_{k}(M(y)), M_{f} (α) = M(f) ∘ α ∘ M(f^{−1} ),

where α ∈ End_{k}(M(x)).

Definition 3.5. Let (A,F) be a C -algebra. We say that (A,F) is C -inverse-symmetric if C is a finite inverse category and we have:

(1) A_{x} is inverse-symmetric with its corresponding triplet (S_{x},S_{x}^{∗},α_{S_{x}^{∗}}) for any x ∈ ObC;

(2) F_{f} (S_{d(f)}^{∗}) = S_{c(f)}^{∗} and F_{f} restricted to S_{d(f)}^{∗} is a bijective map with its inverse Ff̂ for any f ∈ HomC.

Remark 3.6. In particular, if (A,F) is a C -inverse-symmetric algebra then F_{f} is an isomorphism of k-algebras with its inverse Ff̂. Intuitively Definition 3.5 says that a C - inverse-symmetric algebra is a C -algebra that is obtained from particular inverse-symmetric algebras. These algebras have their basis (with the multiplication in the algebra) identified with the sets obtained from the indexing semigroups by removing the zero (if it is the case). Moreover in this case we have Im(α_{S_{x}^{∗}}) ⊆ {0,1_{k}} for any x ∈ ObC.

Example 3.7. Let C be a finite groupoid. We consider the regular kC -module M = kC. As a functor

kC : C → k − mod, kC (x) = kHom_{C}(−,x),

for any x ∈ ObC. By Example 3.4, A = (End_{k}(M(x)_{x∈ObC},M) is a C -algebra, with the defining functor M. For x ∈ ObC we have that M (x) = End_{k}(kHom_{C}(−,x)) is a kalgebra which is inverse-symmetric with S_{x} = {b_{m,n}| (m,n) ∈ T_{x}^{∗}} ∪ {0}, having its maps defined similarly to Example 2.2, where T_{x} = (Hom_{C}(−,x) × Hom_{C}(−,x)) ∪ {0}. In Example 2.2 we defined T_{x} as an inverse semigroup and we can prove similarly that S_{x} is an inverse semigroup for any x ∈ ObC. Using the explicit definition of an element b_{m,n}∈ S_{x}^{∗} (see Example 2.2), if f ∈ Hom_{C}(x,y) then M_{f} (b_{m,n}) = b_{f∘m,f∘n}. In this case since C is a groupoid we have that

Mf̂(M_{f} (b_{m,n}) = b_{f̂∘f∘m,f̂∘f∘n} = b_{m,n}.

It follows now that the conditions from Definition 3.5 are satisfied.

The following definition is similar to the definition from [7, §1], using Definition 3.1 but restricted only for C -inverse-symmetric algebras. It can be given more generally for any C -algebra but we prefer to use the basis from Remark 2.4 since it will help us to prove the main result of this section (Theorem 3.1).

Definition 3.8. Let (A,F) be an C -inverse-symmetric algebra. The skew category algebra is the algebra denoted A ⋊^{F} C given by the formal finite sums ∑_{f∈HomC} a_{f} ⋆ \underline{f} with a_{f} ∈ A_{c(f)} and the product

(a_{f} ⋆ \underline{f} ) · (b_{g}⋆ \underline{g}) = \{ \begin{matrix}a_{f} · F_{f} (b_{g}) ⋆ \underline{f ∘ g}, g ∘ ĝ = f̂∘ f \\ 0, g ∘ ĝ ≠ f̂∘ f \end{matrix} ,

where a_{f} ∈ A_{c(f)},b_{g} ∈ A_{c(g)}.

Then A ⋊^{F} C is a k-algebra which has the identity element ∑_{x∈ObC} ∑_{u_{x}≤1_{x}} 1_{A_{x}}⋆ \underline{u_{x}}, since ObC is a finite set.

Theorem 3.1. Let (A,F) be a C -inverse-symmetric algebra. Then the skew category algebra A ⋊^{F} C is an inverse-symmetric algebra.

Proof. Form the Definitions 3.5, 3.8 and Remark 2.4 we can verify that the set

B = \{s ⋆ \underline{f} | f ∈ HomC ,s ∈ S_{c(f)}^{∗}\}

is a k-basis for the k-algebra A ⋊^{F} C.

We denote by S the set B ∪ {0} and then it is easy to check that S is a finite semigroup, with the operation induced from the multiplication in the skew category algebra. That is, for s ⋆ \underline{f} ,t ⋆ \underline{g} ∈ S^{∗} we have

(s ⋆ \underline{f} ) · (t ⋆ \underline{g}) = \{ \begin{matrix}s · F_{f} (t) ⋆ \underline{f ∘ g}, g ∘ ĝ = f̂∘ f \\ 0, g ∘ ĝ ≠ f̂∘ f \end{matrix} ,

where f,g ∈ HomC and s ∈ S_{c(f)}^{∗},t ∈ S_{c(g)}^{∗}. Next we prove that (S,·) is an inverse semigroup. Let s ⋆ \underline{f} ∈ S^{∗} and suppose that there is t ⋆ \underline{g} ∈ S^{∗} such that

(s ⋆ \underline{f} ) · (t ⋆ \underline{g}) · (s ⋆ \underline{f} ) = s ⋆ \underline{f} ,
(t ⋆ \underline{g}) · (s ⋆ \underline{f} ) · (t ⋆ \underline{g}) = t ⋆ \underline{g}.

From the above equalities we obtain that \underline{f ∘ g ∘ f} = \underline{f} , s · F_{f} (t · F_{g}(s)) = s, and respectively \underline{g ∘ f ∘ g} = \underline{f} , t · F_{g}(s) · F_{g∘f} (t) = t. Since C is an inverse category it follows that g = f̂ (see also Remark 2.4). We have that

s = s · F_{f} (t) · F_{f∘f̂}(s),
t · Ff̂(s) · F_{f̂∘f} (t) = t.

Applying F_{f} on the second equality, by Definition 3.5, (2), it follows that

s = s · F_{f} (t) · s,
F_{f} (t) · s · F_{f} (t) = t.

Since S_{c(f)} is an inverse semigroup it follows that ŝ= F_{f} (t), hence t = Ff̂(ŝ). We con-

clude that s ⋆ \underline{f} has a unique inverse Ff̂(ŝ) ⋆ \underline{f̂}.

Next we compute the idempotents of S^{∗}. If (s ⋆ \underline{f} ) · (s ⋆ \underline{f} ) = s ⋆ \underline{f} then we obtain

f ∘ f = f, f ∘ f̂= f̂∘ f,
s · F_{f} (s) = s.

It follows that f = f̂. Moreover the second equality gives us F_{f} (s) · s = F_{f} (s), by applying F_{f} = Ff̂. Then s = F_{f} (s), and moreover

s · s = s · F_{f} (s) · s = s · F_{f} (s) = s.

It follows that

Idemp(S^{∗}) = \{s ⋆ \underline{f} | f ∈ E(C),s = F_{f} (s),s ∈ Idemp(S_{c(f)}^{∗})\}  (1)

Let s⋆\underline{f} ,t ⋆\underline{g}∈ S^{∗} and e⋆\underline{h}∈ Idemp(S^{∗}). From (1) we have that h ∈ E(C),e ∈ Idemp(S_{c(h)}^{∗}) with F_{h}(e) = e. We consider the cases:

I. α_{S^{∗}}(s ⋆ \underline{f} ,t ⋆ \underline{g},e ⋆ \underline{h}) ≠ 0 if and only if the following equivalences of systems of conditions are satisfied

\{ \begin{matrix}g ∘ ĝ = f̂∘ f \\ e ⋆ \underline{h}= s · F_{f} (t) ⋆ \underline{f ∘ g} \end{matrix} ⇔ \{ \begin{matrix}g ∘ ĝ = f̂∘ f \\ e = s · F_{f} (t) \\ h = f ∘ g \end{matrix} ⇔ \{ \begin{matrix}g = f̂ \\ e = s · F_{f} (t) \\ h = f ∘ f̂ \end{matrix} ,

where the last system of conditions follows from Remark 2.4. Now since e = s·F_{f} (t) is an idempotent in S_{c(f)}^{∗} and A_{c(f)} is an inverse-symmetric algebra, by Definition 2.1, condition

(1), the last system of conditions is true if and only if \{ \begin{matrix}g = f̂ \\ F_{f} (t) = ŝ \\ e = s · ŝ \\ h = f ∘ f̂ \end{matrix} .

II. The right part of the condition (1) from Definition 2.1 for (S,S^{∗},α_{S^{∗}}) is equivalent to

\{ \begin{matrix}t ⋆ \underline{g} = ŝ ⋆ \underline{g} = Ff̂(ŝ) ⋆ \underline{f̂} \\ e ⋆ \underline{h}= (s ⋆ \underline{f} ) · (Ff̂(ŝ) ⋆ \underline{f̂}) \end{matrix} ⇔ \{ \begin{matrix}t = Ff̂(ŝ) \\ g = f̂ \\ e = s · F_{f} (Ff̂(ŝ)) \\ h = f ∘ f̂ \end{matrix} ⇔ \{ \begin{matrix}F_{f} (t) = ŝ \\ g = f̂ \\ e = s · ŝ \\ h = f ∘ f̂ \end{matrix} .

Condition (2) from Definition 3.5 is easy to check.

Remark 3.9. Let A be a G-algebra which has a G-stable k-basis S^{∗}, where S is a finite inverse subsemigroup of (A,·). By Example 3.3, if S^{∗} satisfies the conditions from Definition 3.5 we have that (A,F) is Ĝ-inverse-symmetric. In this case A ⋊^{F} Ĝ is the skew group algebra (since \underline{g}= g for any g ∈ G). So, by Theorem 3.1 and Theorem 2.1,we obtain that the skew group algebra is a symmetric algebra .

Acknowledgements. This work was possible with the financial support of the Sectoral Operational Program for Human Resources Development 2007-2013, co-financed by the European Social Fund, within the project POSDRU 89/1.5/S/60189 with the title ”Postdoctoral Programs for Sustainable Development in a Knowledge Based Society”.

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