Adv Syst Sci Appl 2025; 1:108–119 Published online at https://ijassa.ipu.ru. Grading Structure for Derivations of Group Algebras Andronick Arutyunov*1, Igor Zhiltsov 1V.A. Trapeznikov Institute of Control Sciences Russian Academy of Sciences, Moscow, Russia Abstract: In this paper we give a way of equipping the derivation algebra of a group algebra with the structure of a graded algebra. The derived group is used as the grading group. For the proof, the identification of the derivation with the characters of the adjoint action groupoid is used. These results also allow us to obtain the analogous structure of a graded algebra for outer derivations. A non-trivial graduation is obtained for all groups that are not perfect. Keywords: derivations, bimodule, outer derivations, graded algebras Calculation of derivations in a group algebra is a well-known problem. Present work elaborates results from articles [1–3] focused on studying derivations in terms of characters of adjoint action gruppoid. An important result of this research are handy formulas for quick calculation of derivations. These articles explore derivations’ link to combinatorial properties of the group. Among applications note use in coding theory (see [4, 5]), Novikov algebras (see recent work [6]) and more general constructions, like (σ, τ)−derivations (see [7]). 1. INTRODUTION Aim of the present work is grading the derivation algebra by identifying derivations and characters on a certain groupoid (all the necessary definitions are given in Section 2). The main result of the paper follows. Let N be a fixed normal subgroup in G such that G/N is abelian. Theorem 1.1. If |G/N | > 1, Der is graded with G/N , that is Der = ⊕ k∈G/N Derk, ∀k, l ∈ G/N : [Derk,Derl] ⊂ Derkl. Here Derk is a subalgebra of derivation whose characters’ support is localised entirely in one coset aN = k. The structure of the work follows. Section 2 provides main definitions and propositions. Section 3 describes the construction of grading and contains the main result and its proof. Section 4 provides an example of grading for G equal to discrete Heisenberg group along with an example of localising central derivations for such groups G that G is not a stem group. Fix an infinite finitely-generated group G for the rest of the text. ∗Corresponding author: andronick.arutyunov@gmail.com GRADING STRUCTURE FOR DERIVATIONS OF GROUP ALGEBRAS 109 2. PRELIMINARIES Recall that Group algebra C[G] is an algebra of formal finite sums of type (a1, . . . , an ∈ C, g1, . . . , gn ∈ G) a1g1 + · · ·+ angn We define derivation as a linear operator d that satisfies the Leibniz rule (for all a, b ∈ C[G]) d(ab) = d(a) · b+ a · d(b) Derivations over a group algebra form a Lie algebra with respect to commutator. We will denote this algebra as Der or Der(C[G]). 2.1. Characters We use the technique of characters following [1–3]. Definition 2.1. For a given group G consider a small groupoid Γ: 1. objects (Obj) — elements of G, 2. arrows (Hom) — pairs of elements of G. For an arrow (u, v) its source S(u, v) is given by v−1u, and its target T(u, v) — by uv−1 (Hom(a, b) denotes a set of all arrows for which the source is a and target is b), 3. Consider two arrowsφ = (u2, v2) ∈ Hom(b, c), ψ = (u1, v1) ∈ Hom(a, b) (we will call a(n ordered) pair of arrows φ, ψ such that S(φ) = T(ψ) composable). The composition for these two arrows is given by: (u2, v2) ◦ (u1, v1) := (v2u1, v2v1) The formula for composition does not comprise u2 since if we have a pair of composable arrows (u2, v2), (u1, v1), u2 can be expressed in terms of u1, v1, v2. The reader may consider this as an exercise. Γ is the groupoid of group’s inner action on itself. Fix an element a of G. Define following symbols: Definition 2.2. • [a] = {xax−1 : x ∈ G} is a’s conjugacy class in G, • GG := {[g] : g ∈ G} • Γ[a] is Γ’s subgroupoid, informally, a connected component in Γ, given by: Obj(Γ[a]) := [a] = {x ∈ Obj : x ∈ [a]} Hom(Γ[a]) := {(u, v) ∈ Hom : u, v ∈ [a]} Definition 2.3. A character on Γ is a function χ : Hom→ C, such that: • (Composition) for each pair of composable arrows φ, ψ: χ(φ ◦ ψ) = χ(φ) + χ(ψ) • (Locally finite) ∀y ∈ G there is a finite set of x ∈ G, such that χ(x, y) ̸= 0. Holds the following decomposition: Lemma 2.1. (Decomposition) Γ = ⊔ [a]∈GG Γ[a] Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) 110 A. ARUTYUNOV, I. ZHILTSOV Remark. Although characters being locally finite may seem as an alien and a bit too technical detail, it is deliberately placed in the definition to stress that we will not consider ”non-locally finite characters”. The reasons will become clear, among the rest, in Theorem 2.1. We will need the following statement: Statement 2.1. Let (u, v) = φ ∈ Hom, a ∈ G. Then the following statements are equivalent φ ∈ Hom(Γ[a]), S(φ) ∈ [a], T(φ) ∈ [a]. It is proved by direct calculation. 2.2. Connection between Characters and Derivations The following theorem motivates to consider (locally finite) characters when studying derivations. Informally, Theorem 2.1 shows that characters may be seen as a generalization of linear operator’s matrix. Theorem 2.1 (Derivation formula and derivation character, [1, section 2]). For each derivation d there exists a unique character χ such that for each x ∈ G holds d(x) = ∑ k∈G χ(k, x)k (2.1) Consider d, χ from Theorem 2.1. We will say that character χ gives derivation d (derivation d is given by character χ; we will omit words ”derivation” and ”character”). For a derivation d let χd be a character such that χd gives d. Theorem 2.1 implies: Corollary 2.1. Let d, ∂ be derivations given by characters χd, χ∂ correspondingly, Then d+ ∂ be given by χd + χ∂ . Definition 2.4. Let d be given by α, ∂ be given by β. Then {α, β} is the character that gives [d, ∂]. Statement 2.2 (”Matrix” product, [2, Proposition 2.4]). Let α, β be characters. Then {α, β} satisfies (a, b ∈ G) {α, β}(a, b) = ∑ k∈G α(a, k)β(k, b)− β(a, k)α(k, b) Two examples of derivations follow. Example 2.1 will be needed to prove Theorem 1.1. Let a ∈ G. Recall that derivation da is called inner if for any x ∈ C[G] da(x) = [x, a] = xa− ax Example 2.1 (Character of inner derivation [3, Proposition 3]). Let a ∈ G. Then character χa given by formula χa(φ) =  1, a = S(φ), −1, a = T(φ), 0, otherwise. (2.2) gives da(x) = [x, a]. Example 2.2. Another possible example of derivations are central derivations. We will call derivation d central if there exists such central element z ∈ Z(G) and homomorphism τ : G→ (C,+) such that for all basis elements g ∈ G: d(g) = τ(g)gz Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) GRADING STRUCTURE FOR DERIVATIONS OF GROUP ALGEBRAS 111 Such an operator is indeed a derivation, see [3, Proposition 4]. [3, Proposition 5] shows that non-trivial central derivations are not inner. Moreover, [3, Proposition 6] shows that central derivations form a Lie subalgebra in Der(G). Definition 2.5. For a given character χ we define support of χ as following supp χ = {φ ∈ Hom : χ(φ) ̸= 0} For the given subset M ⊂ G denote by DerM the set of derivations d such that for character χ that gives d: supp χ ⊂M . Example 2.3. Recall character χa from Example 2.1 (where a ∈ G.) Its support is easily calculated supp χa = {φ : S(φ) = a} ∪ {ψ : T(ψ) = a} By Statement 2.1, we can localize supp χa in a single conjugacy class a (we will need such a localisation later in Theorem 1.1) supp χa ⊂ Γ[a] 2.3. Applying Decomposition Lemma 2.1 establishes decomposition of groupoid Γ. The current section presents decompositions for (locally finite) characters and derivations. The following two lemmas are equivalent. We prove the first one. Lemma 2.2. Let χ be a character. Then there exists finitely many a1, . . . , aN ∈ G such that supp χ ≤ N⋃ k=1 Γ[ak] Lemma 2.3 (Derivation decomposition). Then for each u ∈ G such that χu is a character, holds decomposition d = ∑ [u]∈GG du, and the set {[u] ∈ GG : ∃x ∈ G : du(x) ̸= 0} is finite. Proof for Lemma 2.2 Let G = ⟨X | R⟩, where X =: {x1, . . . , xk} is finite (G is assumed to be finitely-generated throughout the text). Consider (u, v) ∈ Γ. Let n = n(v) be minimal nonnegative integer such that ∃y0, . . . , yn ∈ X ∪X−1 : v = y0y1 . . . yn 1. Show that ∃z0, . . . , zn ∈ G : (z0, y0) ◦ · · · ◦ (zn, yn) = (u, v) (2.3) Subproof Induction by n = n(v). Base: n = 0 — z0 = u. Step: Consider z0 = uv−1y0. Then (z, y0), (y −1 0 u, y−1 0 v) are composable since S(z0, y0) = y−1 0 z0 = y−1 0 uv−1y0 = = y−1 0 u(y−1 0 v)−1 = T(y−1 0 u, y−1 0 v) Moreover, (z0, y0) ◦ (y−1 0 u, y−1 0 v) = (u, v) Notice that y−1 0 v = y1 . . . yn, thus n(y−1 0 v) < n = n(v). Therefore, applying the step of induction, get eq. (2.3). □ Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) 112 A. ARUTYUNOV, I. ZHILTSOV 2. Since χ is locally finite, the set B = (G× (X ∪X−1)) ∩ supp χ is finite. By ?? for each arrow φ there exists a unique element a ∈ G such that φ ∈ Hom(Γ[a]); thus, there exists a finite setA = {a1, . . . , aN} such that for any a /∈ A:B ∩Hom(Γ[a]) = Ø. Thus, by item 1 for any a /∈ A: supp χ ∩Hom(Γ[a]) = Ø. Therefore: supp χ ≤ N⋃ k=1 Γ[ak] ■ A very nice alternative proof for Lemma 2.2 was submitted in an anonymous review. Alternative proof for Lemma 2.2 Let d be the derivation given by character χ. Let P be a union of conjugacy classes such that supp χ ⊂ ⋃ g∈P Γ[g] =: U The following statements are equivalent: • (x, y) ∈ Hom(U), • y−1x = S(x, y) ∈ P Consider an element y ∈ G such that y−1d(y) ∈ ⟨P ⟩ Here ⟨X⟩ forX ⊂ G denotes a set of all finite sums a1x1 + · · ·+ anxn such that a1, . . . , an ∈ C and x1, . . . , xn ∈ X . Let’s calculate y−1d(y) by Statement 2.2. y−1d(y) = y−1 ∑ x∈G χ(x, y)x = ∑ x∈G χ(x, y)y−1x = ∑ x∈G χ(x, y)S(x, y) ∈ ⟨P ⟩ Therefore, for each x such that χ(x, y) ̸= 0 : y−1x ∈ P . Consider a set H = { y : y−1d(y) ∈ ⟨P ⟩ } As a simple exercise, check H’s being a subgroup in G. To summarize, if y is in subgroup H ≤ G (that is y−1d(y) ∈ ⟨P ⟩) then for each x such that χ(x, y) ̸= 0 : y−1x ∈ P . To finish the proof let’s choose such P that P is a union of a finite number of conjugacy classes and H = G. To achieve this, consider a finite generating set S for G and a finite subset M ⊂ G such that for any s ∈ S there exist complex numbers am,m ∈M such that s−1d(s) = ∑ m∈M amm Informally, calculate all s−1d(s), s ∈ S, which are finite sums, and store all elements in G present in at least one of finite sums. Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) GRADING STRUCTURE FOR DERIVATIONS OF GROUP ALGEBRAS 113 Since M is finite, P = ⋃ m∈M [m] is a union of finite number of conjugacy classes. Moreover, for such P S ⊂ H Thus, G = H . All in all, there exists a union P of a finite number of conjugacy classes such that supp χ ⊂ ⋃ g∈P Γ[g] =: U ■ Let d be the derivation given by character χ, and χu(φ) := { χ(φ), φ ∈ Hom(Γ[u]), 0, otherwise . We will denote the derivation given by χu as du. 3. CONSTRUCTING GRADED ALGEBRA Grading with Abelian Quotients Definition 3.1. Let A be an abelian group, e be a neutral element in A, A be a Lie algebra, A can be expressed as a direct sum A = ⊕ n∈A An, such that ∀n, l ∈ A : [An,Al] ⊂ Anl such that Ae ̸= A. Then A is called graded (with A). The direct sum Def 3.1 is called A’s grading with A. Notice that trivial gradings are excluded. Definition 3.2. A commutator subgroup of group G is G′ := {xyx−1y−1 : x, y ∈ G} Recall a few well-known definitions and statements that will be required below. Statement 3.1. For any group G a subgroup G′ is normal, G′ ◁ G Let N be a fixed normal subgroup in G such that G/N is abelian. Conceptually, the latter condition derives from the following Def 3.1, however it is also needed for more technical, yet crucial details, like Lemma 3.1. Statement 3.2. G′ ⊂ N . Lemma 3.1. Let a ∈ N . Then [a] ⊂ aN . Proof A calculation for any a, t ∈ G proves lemma: tat−1N = tat−1a−1Na = Na = aN The first and the last equivalences hold since N is normal; the second equivalence holds by Statement 3.2 since G/N is abelian (tat−1a−1 ∈ G′ ⊂ N .) ■ Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) 114 A. ARUTYUNOV, I. ZHILTSOV Note that Lemma 3.1 would have been false if we did not requireG/N to be abelian. Consider G = S4, N = V4, a = (12) for counterexample. Lemma 3.1 motivates the following symbols: ΓaN = ⋃ k∈aN Γ[k], DeraN = { d ∈ Der : supp χd ⊂ ΓaN } . Lemma 3.2. Let a, b ∈ G, d is given by character α : supp α ⊂ ΓaN , ∂ is given by character β : supp β ⊂ ΓbN . Then supp {α, β} ⊂ ΓabN . Proof Statement 2.2 implies: {α, β}(h, g) = ∑ k∈G α(h, k)β(k, g)− β(h, k)α(k, g) Consider an arrow (h, g) : {α, β}(h, g) ̸= 0. There exists k ∈ G:[ α(h, k)β(k, g) ̸= 0 β(h, k)α(k, g) ̸= 0 { α(h, k) ̸= 0 β(k, g) ̸= 0{ β(h, k) ̸= 0 α(k, g) ̸= 0 (3.4) Expressing eq. (3.4) in terms of supp : { (h, k) ∈ supp α ⊂ Hom(ΓaN) (k, g) ∈ supp β ⊂ Hom(ΓbN){ (k, g) ∈ supp α ⊂ Hom(ΓaN) (h, k) ∈ supp β ⊂ Hom(ΓbN) (3.5) By Statement 2.1, an arrow φ belongs to Γ[x] ⊂ ΓxN iff its target T(φ) belongs to [x] ⊂ xN . Thus, eq. (3.5) implies:  { hk−1 = u ∈ aN kg−1 = v ∈ bN{ kg−1 = u ∈ aN hk−1 = v ∈ bN (3.6) Multiplying the equations in eq. (3.6):[ hg−1 = uv hg−1 = vu , for u ∈ aN, v ∈ bN Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) GRADING STRUCTURE FOR DERIVATIONS OF GROUP ALGEBRAS 115 By definition, T(h, g) = hg−1. By Statement 2.1:[ (h, g) ∈ Hom(Γ[uv]) (h, g) ∈ Hom(Γ[vu]) Since [uv] = [u · vu · u−1] = [vu] and uv ∈ abN , then by Lemma 3.1: [uv] ⊂ uvN = abN Thus: (h, g) ∈ Γ[uv] ⊂ ΓabN , for u ∈ aN, v ∈ bN All in all, {α, β}(h, g) ̸= 0 ⇒ (h, g) ∈ ΓabN , therefore supp {α, β} ⊂ ΓabN . ■ Theorem 1.1. If |G/N | > 1, Der is graded with G/N , that is Der = ⊕ k∈G/N Derk, ∀k, l ∈ G/N : [Derk,Derl] ⊂ Derkl. Proof Consider the sum ∑ k∈G/N Derk. 1. First, show that ∑ k∈G/N Derk is equal to Der. • ∑ k∈G/N Derk ⊂ Der — since Der is closed under finite sums. • Now we show an opposite inclusion. Consider an arbitrary d ∈ Der. Lemmas 2.3 and 3.1 imply (see Lemma 2.3 for definition of du) d = ∑ [u]⊂G du = ∑ k∈G/N ( ∑ [u]⊂k du ) Consider for given k ∈ G/N sk := ∑ [u]⊂k du ∈ Derk By Lemma 2.3, there is only a finite number of such [u] ⊂ G that du is not constant 0. Thus, there exists an integer N and k1, . . . , kN ∈ G/N such that for any k ∈ G/N, k ̸= k1, . . . , kN : sk is constant 0. Thus, d = N∑ i=1 ski Thus, Der ⊂ ∑ k∈G/N Derk Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) 116 A. ARUTYUNOV, I. ZHILTSOV • All in all, Der = ∑ k∈G/N Derk. 2. ∑ k∈G/N Derk is direct. Subproof Let dk ∈ Derk be given by χk. Suppose that∑ k∈G/N dk ≡ 0 Since constant 0 is a derivation given by constant character (equal to 0), by Corollary 2.1 for any φ ∈ Hom ∑ k∈G/N χk(φ) = 0 Since supp χk, supp χl are disjoint (for k ̸= l), for each φ ∈ Hom exists no more than one k ∈ G/N such that χk(φ) ̸= 0. Thus, for each φ ∈ Hom and for each k ∈ G/N : χk(φ) = 0, and for each k ∈ G/N : dk ≡ 0. Therefore, ∑ k∈G/N Derk = ⊕ k∈G/N Derk is direct by definition of direct sum. □ 3. We established that Der = ⊕ k∈G/N Derk Since there exists such m ∈ G that mN ̸= N . Thus, by Examples 2.1 and 2.3 the inner derivation [x,m] is given by a character χm such that supp χm ⊂ [m] ⊂ mN by Lemma 3.1. Thus, Der ̸= DerN (i.e. the grading is not trivial.) Finally, check that (∀k, l ∈ G/N ) ∀k, l ∈ G/N : [Derk,Derl] ⊂ Derkl Proof for item 3 Let dk ∈ Derk, dl ∈ Derl. Let character χk give dk, character χl give dl. Thus, by definition of Derk,Derl supp χk ≤ Γk, supp χl ≤ Γl Therefore, by Lemma 3.2 supp {χk, χl} ≤ Γkl Finally, [dk, dl] ∈ Derkl □ ■ Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) GRADING STRUCTURE FOR DERIVATIONS OF GROUP ALGEBRAS 117 Example 3.1. Let G be a perfect group (|G/G′| = 1), that is G′ = G. Theorem 1.1 yields a trivial grading(which we do not regard as a grading in this text) for Der(C[G]) since |G/G′| = 1. And vice versa: Corollary 3.1. IfG is NOT a perfect group (G ̸= G′) then Der admits a (non-trivial) grading with G/G′. Example 3.2. Let G be a knot group (i.e. let there exist some knot K such that G is the knot group ofK). It is well-known that in this caseG/G′ = Z, thereforeDer(G) admits a grading with Z. 4. EXAMPLES Discrete Heisenberg Group Consider discrete Heisenberg group (a group of 3× 3 upper unitriangular matrices with integer entries). Following [3], we use this group as a handy example since it admits easy calculations. Definition 4.1. Consider a group of integer unitriangular matrices with respect to matrix multiplication: H = {( 1 a c 0 1 b 0 0 1 ) : a, b, c ∈ Z } ( 1 a c 0 1 b 0 0 1 )( 1 x z 0 1 y 0 0 1 ) := ( 1 a+ x c+ z + ay 0 1 b+ y 0 0 1 ) Since all the matrices in H have determinant 1, the inverse is well-defined and given by( 1 a c 0 1 b 0 0 1 )−1 = ( 1 −a ab− c 0 1 −b 0 0 1 ) Our goal is to grade Der(H). Definition 4.2. The centre of G is Z(G) = {z ∈ G : ∀g ∈ G : gz = zg} The follwing statements are well-known and trivial. Statement 4.1. H ′ = Z(H) = {( 1 0 a 0 1 0 0 0 1 ) : a ∈ Z } Statement 4.2. H/H ′ ≃ Z⊕ Z Let ψ : Z⊕ Z → H/H ′ be an isomorphism. Recall the symbols: ΓaN = ⋃ k∈aN Γ[k], DeraN = { d ∈ Der : supp χd ⊂ ΓaN } . Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) 118 A. ARUTYUNOV, I. ZHILTSOV Define: Der(i,j) := Derψ(i,j) Corollary 4.1 (From Statement 4.2, Theorem 1.1). Der(H) is graded with Z⊕ Z, that is Der(H) = ⊕ (i,j)∈Z⊕Z Der(i,j) ∀(i, j), (k, l) ∈ Z⊕ Z : [Der(i,j),Der(k,l)] ⊂ Der(i+k,j+l) Example 4.1. Statement 4.1 implies: if d is given by such χ that supp χ ≤ Γ[z] for z ∈ Z(H), then d ∈ Der(0,0). Definition 4.3. G is a stem group if Z(G) ≤ G′ Example 4.1 can be generalised under the assumption that G is a stem group. Note that H is a stem group since H ′ = Z(H) by Statement 4.1. See [8] for more details on (finite, which is not our case) stem groups. Proposition 4.1. Let G be a stem group. If d is given by such χ that supp χ ≤ Γ[z] for z ∈ Z(G), then d ∈ DerG′ . Central derivations were introduced in [3] as operators given on a group algebra generators g ∈ G with the homomorphism τ : G→ (C,+) and the central element z ∈ Z(G) by formula dτ,z : g 7→ τ(g)gz. Recall that [3, Proposition 6] shows that central derivations form a Lie subalgebra in Der(G) (denote as ZDer(G)). Proposition 4.2. Let G be NOT a stem group. Then there is an ”induced” nontrivial grading of ZDer(G) with G′. Note that if G is a stem group, the grading is trivial, i.e. not a grading at all in our terms. Another example of an induced grading follows. Let a ∈ G. Recall that derivation da is called inner if for any x ∈ C[G] da(x) = [x, a] = xa− ax Let InnerDer be the set of all derivations of the form (a1, . . . , an ∈ C; y1, . . . , yn ∈ G) G ∋ x 7→ a1[x, y1] + · · ·+ an[x, yn] = [x, a1y1 + · · ·+ anyn] A direct calculations shows that InnerDer is an ideal in Der (i.e. for any d ∈ InnerDer and for any ∂ ∈ Der : [d, ∂], [∂, d] ∈ InnerDer; recall that [d, ∂](x) = d(∂(x))− ∂(d(x)).) Therefore, there exists a factor-algebra OuterDer := Der/InnerDer. Corollary 4.2. If G is NOT a stem group, there is an induced (non-trivial) grading of OuterDer with G′ of the form OuterDer = ⊕ k∈G/G′ Derk/InnerDerk, where InnerDerk := Derk ∩ InnerDer Copyright © 2025 ASSA. 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