CHAPTER 1. ALGEBRAIC METHODOLOGY

During all this work Φ is a unital commutative ring of scalars with $\frac{1}{2}$ ϵ Φ.

Previously, in the introduction, we have established the topics that are covered in this work. In this section we go a step further, laying the foundations of this thesis and establishing its main concepts. Firstly, we will define the basics fundamentals related to associative algebras and superalgebras. Next, we will review some relevant concepts and results to better understand the structure underlying such algebras and superalgebras such as, for example, the notions of primeness and semiprimeness. Afterwards, we will introduce the extended centroid and how it behaves in prime or semiprime associative algebras and superalgebras with involution and superinvolution. Finally, we will recall basic notions on nonassociative algebras and superalgebras, in particular, about Lie and Jordan superalgebras.

In Chapter 2, we will study ad-nilpotent elements belonging to semiprime associative algebras R over Φ with or without involution. In particular, the extended centroid will be a crucial tool (e.g., it allows us to define what is a pure ad-nilpotent element). In Chapter 3 we will work on the super setting, i.e, on ad-nilpotent elements in prime associative superalgebras R over Φ with or without superinvolution. Finally, throughout Chapter 5 we will deal with Lie superalgebras and Jordan superstructures.

1.1. Basic notions on associative algebras and superalgebras

1.1.1. Let R be an algebra over Φ. We say that R is a superalgebra if R is ℤ₂-graded, i.e., R = R₀ ⊕ R₁ such that R_i · R_j ⊆ R_i+j with i, j ϵ ℤ₂. R_o it is the even part and it is a subalgebra of R and R₁ is the odd part and it is a bimodule over R₀. Any element of R₀ ∪ R₁ is called a homogeneous element and we define the parity of a homogeneous element as |a| = 0 if a ϵ R₀ and |a| = 1 if a ϵ R₁.

Let f : R → R′ be a linear map where R and R′ are both superalgebras. We say that f is homogeneous of degree γ ϵ ℤ₂ if f (R_i) ⊂ R′_i+γ. In addition, we say that f is a superalgebra homomorphism if it is an algebra homomorphism and it is homogeneous of degree 0, i.e., f (R_i) ⊂ R′_i.

1.1.2. Let X = {x₁, x₂, …} be a countable infinite set of variables and let Φ 〈X〉 be the free unital associative algebra generated by X over Φ. If I is the two-sided ideal of Φ (X) generated by the set of elements {x_ix_j + x_jx_i | i, j ≥ 1}, we set G := Φ 〈X/I. We call G the (infinite dimensional) Grassmann algebra. We denote by ξ_i := x_i + I. With this notation G has the following presentation:

G = 〈1, ξ₁, ξ₂, … | ξ_iξ_j + ξ_jξ_i = 0, for all i, j ≥ 1〉.

Notice that ${\xi }_i^2=0$ since $\frac{1}{2}$ ϵ Φ. The set $B=\{1,{\xi }_{i_1},\dots {\xi }_{i_k}\mid 1<i_1<\dots <i_k,\text{for all}k\in \mathrm{\mathbb{N}}\}$ is a basis of G over Φ. In addition, G is a ℤ₂-graded module over Φ :

$$G_0:=⟨1,{\xi }_{i_1}\cdots {\xi }_{i_{2k}}\mid 1\le i_1<\dots <i_{2k},k\ge 1⟩,$$

$$G_1:=⟨{\xi }_{i_1}\cdots {\xi }_{i_{2k+1}}\mid 1\le i_1<\dots <i_{2k+1},k\ge 0⟩.$$

Thus, G is an associative superalgebra. Moreover, if R = R₀ ⊕ R₁ is a superalgebra over Φ, we can define the Grassmann envelope of R, G(R), as the even part of the tensor product G ⊗ R, i.e., G(R) = (G ⊗ R)₀ = G₀ ⊗ R₀ + G₁ ⊗ R₁. Notice that G(R) is an algebra.

The Grassmann envelope allows us to define varieties of superalgebras. Let R = R₀ + R₁ be a superalgebra. We say that R belongs to a certain variety of superalgebras (Lie, Jordan, associative,…) if G(R) belongs to the same variety of algebras.

Notice that if R = R₀ ⊕ R₁ is a superalgebra (i.e., ℤ₂-graded) such that it is associative as algebra then it is easy to check that G(R) is associative as well. Hence R is an associative superalgebra if and only if R is an associative ℤ₂-graded algebra. But in general a Lie or Jordan superalgebra is not a Lie or Jordan ℤ₂-graded algebra.

1.1.3. Let R = R₀ ⊕ R₁ be an associative superalgebra over Φ. In these conditions the map σ : R → R defined by σ(x₀ + x₁) = x₀ − x₁, for every x₀ ϵ R₀, x₁ ϵ R₁, is an algebra automorphism with σ² = id. Conversely, given an associative algebra R, every algebra automorphism σ : R → R with σ² = id defines a ℤ₂-graduation on R given by R₀ = {a ϵ R | σ(a) = a} and R₁ = {a ϵ R | σ(a) = −a}. Therefore, a ℤ₂-graduation on R is equivalent to an algebra automorphism σ with σ² = id.

Notice that a Φ-module S of R is graded if and only if σ(S) ⊂ S.

1.1.4. Let R be an associative algebra or superalgebra. We say that * is an involution if it is a linear map * : R → R such that, for every a, b ϵ R, (a*)* = a and (ab)* = b*a*, and we say that * is a superinvolution if it is a homogeneous, 0-degree, linear map such that for every homogeneous a, b ϵ R, (a*)* = a and (ab)* = (−1)^|a||b|b*a*. We denote the symmetric and skew-symmetric sets with respect an involution or a superinvolution * as H := Sym(R, *) = {a ϵ R | a* = a} and K := Skew(R, *) = {a ϵ R | a* = −a} respectively.

1.1.5. An associative algebra R is semiprime (resp. *-semiprime) if for every nonzero ideal (resp. *-ideal) I of $R,I^2:=\{\sum _i{x}_i{y}_i\mid x_i,y_i\in I\}\ne 0$, and it is prime (resp. *-prime) if $IJ:=\{\sum _i{x}_i{y}_i\mid x_i\in I,y_i\in J\}\ne 0$ for every pair of nonzero ideals (resp. *-ideals) I, J of R.

We recall that a *-ideal is an ideal I such that I* ⊂ I.

It is easy to prove that R is semiprime if and only if is *-semiprime: It is clear that if R is semiprime then is *-semiprime. Conversely, let R be a *-semiprime algebra and let I be an ideal of R such that I² = 0. Notice that I ∩ I* is a *-ideal whose square is zero. Then I ∩ I* = 0, hence II* = I*I = 0. Thus (I + I*)² = 0, and since I + I* is a *-ideal, we have that I = 0. Therefore R is semiprime. However, an algebra can be *-prime but not prime: Let S be a prime associative algebra over Φ and let us consider R = S × S with involution (a, b)* = (b, a). Then R is a *-prime algebra but it is not prime. It is interesting to remark that the symmetric elements are of the form (a, a) and the skew-symmetric are of the form (a, −a).

We can prove that an associative algebra R is prime if and only if $aRb\ne 0$ for arbitrary nonzero elements a, b ϵ R, and it is semiprime if and only if it is nondegenerate, i.e., $aRb\ne 0$ for every nonzero element a ϵ R (see [53, §10]).

We are going to study these concepts in super setting. Let R = R₀ ⊕ R₁ be an associative superalgebra and let σ be the automorphism associated to the ℤ₂-graduation. We say that an ideal I is graded if I = I₀ ⊕ I₁ where I₀ = I ∩ R₀ and I₁ = I ∩ R₁ or, as we remarked in 1.1.3, if σ(I) ⊂ I.

An associative superalgebra R is semiprime if for every nonzero graded ideal I of $R,I^2\ne 0$. And it is prime (as a superalgebra) if it does not have nonzero orthogonal graded ideals.

Notice that a *-ideal or a graded ideal satisfies I* ⊂ I or σ(I) ⊂ I, respectively. Then, arguing as before, the concepts of semiprime associative superalgebra and semiprime associative algebra coincide. An associative superalgebra can be prime but not prime as an algebra: for instance, let S be a prime associative algebra over Φ. Then R = S × S with R₀ = {(a, a) | a ϵ S} and R₁ = {(a, −a) | a ϵ S} is a prime associative superalgebra, which is not prime as an algebra (see [25]). We can say more: If R is prime as a superalgebra but not as an algebra we can consider a nonzero ideal P of R with P ∩ σ(P) = 0. Then P ⊕ σ(P) is an essential graded ideal of R, where (P ⊕ σ(P))₀ = {x + σ(x) | x ϵ P} ≅ P as an algebra and (P ⊕ σ(P))₁ = {x − σ(x) | x ϵ P}. Hence

P ⊕ σ(P) ◃_ess R ↪ R/P ⊕ R/σ(P).

Primeness in associative superalgebras can be also characterized by elements: for any two elements a, b of a prime associative superalgebra R where a and b are homogeneous, the condition aRb = 0 implies that either a or b is zero (see [25, pag. 693]). As we said before, semiprime associative superalgebras are semiprime as algebras hence the property aRa ≠ 0 for every nonzero homogeneous element a ϵ R holds in semiprime superalgebras.

Moreover, when dealing with superalgebras we can always consider the algebra R₀. In the next two lemmas F. Montaner states what happens on the even part when the whole superalgebra is semiprime or prime:

Lemma 1.1.6. [60, Lemma 1.2] If R = R₀ ⊕ R₁ is a semiprime associative superalgebra, then R and R₀ are semiprime algebras.

Lemma 1.1.7. [60, Lemma 1.3] If R = R₀ ⊕ R₁ is a prime associative superalgebra, then either R or R₀ are prime as algebras.

1.1.8. An ideal I of an associative algebra R (resp., an associative algebra with involution *) is prime (resp., *-prime) if R/I is a prime (resp. *-prime) associative algebra. If R is a semiprime associative algebra then there exists a family of prime ideals {I_α}_αϵ∆ such that ∩_αϵ∆ I_α = {0} and therefore R can be seen as a subdirect product of prime associative algebras (see [53, §12]). Similarly, if R is a semiprime associative algebra with involution * there exists a family of *-prime ideals {I_α}_αϵ∆ such that ∩α_ϵ∆ I_α = {0} and therefore R can be seen as a subdirect product of *-prime associative algebras. This is also true for superalgebras.

Moreover, if R is semiprime and free of n-torsion then the intersection of all prime ideals I_α such that R/I_α is free of n-torsion is zero (notice that the intersection of all prime ideals I_α such that R/I_α has n-torsion contains the essential ideal nR) and therefore R is a subdirect product of prime associative algebras, all of them free of n-torsion.

1.2. The extended centroid of associative algebras and superalgebras

1.2.1. Given an ideal I of R, we can define the ideal Ann_R(I) := {z ϵ R | zI = Iz = 0}, which is called the annihilator of I in R. Moreover, when R is semiprime, Ann_R(I) = {z ϵ R | zIz = 0}. An ideal I of R is essential (for every nonzero ideal J of R, I ∩ J ≠ 0) if and only if Ann_R(I) = 0 (see [23, Proposition 1.6(1)]).

1.2.2. Given an associative algebra R, we define a permissible map of R as a pair (I, f) where I is an essential ideal of R and f : I → R is a homomorphism of right R-modules. For permissible maps (I, f) and (J, g) of R, define a relation ≡ by (I, f) ≡ (J, g) if there exists an essential ideal K of R, contained in I ∩ J, such that f(x) = g(x) for all x ϵ K. It is easy to see that this is an equivalence relation. If R is a semiprime associative algebra then $Q_m^r(R)$ has an associative algebra structure coming from the addition of homomorphisms and from the composition of restrictions of homomorphisms, see [7, Chapter 2]:

$$•[I,f]+[J,g]:=[I\cap J,f+g],$$

$$•[I,f]·[J,g]:=\left[(I\cap J{)}^2,f\circ g\right]\text{. }$$

The quotient set $Q_m^r\left(R\right)$ with the operations defined above is called the Martindale algebra of quotients of R. Note that if R is a semiprime associative algebra then the map $f:R\to Q_m^r\left(R\right)$ defined by f(r) := [R, _λr], where _λr : R → R is defined by _λr(x) := rx, is a monomorphism of associative algebras, i.e., R can be considered as a subalgebra of its right Martindale algebra of quotients. The right Martindale algebra of quotients of R satisfies that for all $q\in Q_m^r\left(R\right)$ there exists an essential ideal I of R such that qI ⊆ R. This facts allow us to prove that every subalgebra S of $Q_m^r\left(R\right)$ which contains R is semiprime. Otherwise, if I is a nonzero nilpotent ideal of S and pick 0 ≠ q ϵ I. There exists an essential ideal J of R such that qJ ⊆ R, i.e., qJ = qJ ∩ I ⊆ R ∩ I is a nonzero nilpotent ideal of R which is a contradiction with the semiprimeness of R.

The symmetric Martindale algebra of quotients of R is defined as

$Q_m^s\left(R\right):=\{q\in Q_m^r(R)\mid \mathrm{\exists }$ an essential ideal I of R such that ql + Iq ⊂ R}

(if R has an involution one can replace the filter of essential ideals by the filter of essential *-ideals in the definition of the symmetric Martindale algebra of quotients, see [3, p. 858–859]). If R is semiprime then $Q_m^s\left(R\right)$, which is a subalgebra of $Q_m^r\left(R\right)$ containing R, is also a semiprime algebra.

When R has an involution *, this involution can be extended to $Q_m^s\left(R\right)$ as follows: let us consider q ϵ q ϵ $Q_m^s\left(R\right)$ and I an essential *-ideal such that qI + Iq ⊆ R. We define f : I → R by the rule f(x) = (x*q)*. We set q* := [I, f] and note that q*x* = (xq)* for all x ϵ I (see [7, 2.5.4]).

The extended centroid C(R) of a semiprime algebra R is defined as the center of $Q_m^s\left(R\right)$. The extended centroid of a prime algebra is a field (see [7, p. 70]), the set of symmetric elements of the extended centroid of a *-prime algebra is again a field (see [3, Theorem 4(a)]), and the extended centroid of a semiprime algebra is a commutative and unital von Neumann regular algebra (see [7, Theorem 2.3.9(iii)]). In particular, if R is semiprime, C(R) is a semiprime algebra without nilpotent elements.

The central closure of R, denoted by R̂, is defined as the unital subalgebra of $Q_m^s\left(R\right)$ generated by R and C(R), i.e., R̂ := C(R)R + C(R), and can be seen as a C(R)-algebra. Therefore we can consider R contained in R̂. Moreover, since R̂ contains R and it is contained in $Q_m^s\left(R\right)$, if R is semiprime then R̂ is semiprime. The algebra R̂ is centrally closed, i.e., it coincides with its central closure. In particular its center equals its extended centroid, Z(R̂) = C(R̂).

1.2.3. The notion of extended centroid for semiprime associative superalgebras was studied by M. Fošner, see [25]. Let R be a semiprime associative superalgebra. Since R is semiprime as algebra we can consider the symmetric Martindale algebra of quotients $Q_m^s\left(R\right)$. Let σ : R → R be the automorphism associated to the ℤ₂-grading of R (σ² = id). This automorphism, by [7, Proposition 2.5.3], can be extended to $Q_m^s\left(R\right)$ and we denote this extension by $\hat{\sigma }$. Therefore $Q_m^s\left(R\right)$ is an associative superalgebra such that $R_i\subset {\left(Q_m^s\left(R\right)\right)}_i$ with i = 0, 1. Moreover, if R is endowed with a superinvolution *, this can be also extended to $Q_m^s\left(R\right)$ as follows: let us consider $q\in Q_m^s(R{)}_i$, with i = 0, 1, and I an essential graded *-ideal such that qI + Iq ⊆ R. We define f : I → R by f(x) = (−1)^|q||x|(x*q)*. We set q* := [I, f] and note that q*x* = (−1)^|x||q|(xq)* for all x ϵ I homogeneous. Indeed, * is a superinvolution on $Q_m^s\left(R\right)$: Let us consider $q_i\in {\left(Q_m^s\left(R\right)\right)}_i$ and $q_j\in {\left(Q_m^s\left(R\right)\right)}_j$ with i, j = 0, 1. Choose an essential graded *-ideal J of R such that Jq_i, q_iJ, Jq_j, q_jJ, Jq_iq_j, q_iq_jJ are all contained in R and let I = J². Then Iq_i, q_iI, Iq_j, Iq_i ⊆ J. For every homogeneous x ϵ I we have

$$\begin{array}{rl}{\left(q_i{q}_j\right)}^{*}x& =(-1{)}^{|x|(i+j)}{\left(x^{*}{q}_i{q}_j\right)}^{*}=(-1{)}^{|x|(|+i)+(|x|+i)j}{q}_j^{*}{\left(x^{*}{q}_i\right)}^{*}\\ & =(-1{)}^{|x|(i+j)+(|x|+i)j+|x|}{q}_j^{*}{q}_i^{*}x=(-1{)}^j{q}_j^{*}{q}_i^{*}x.\end{array}$$

Hence (q_iq_j)* = (−1)^ijq*_jq*_i for all $q_i\in {\left(Q_m^s\left(R\right)\right)}_i$ and $q_j\in {\left(Q_m^s\left(R\right)\right)}_j$ with i, j = 0, 1.

On the other hand, since R is semiprime as an algebra, we can consider the extended centroid C(R) of R, which it is also ℤ₂-graded because C(R) = Z(Q^r_m(R)). Let R̂ = C(R)R + C(R) be the central closure of R. We will say that R is centrally closed if R = R̂.

1.2.4. Let R be a prime associative superalgebra such that R is not prime as an algebra. Let σ denote the automorphism associated to the ℤ₂-grading of R and consider a nonzero ideal P of R with P ∩ σ(P) = 0. Then P ⊕ σ(P) is a graded essential ideal of R, where (P ⊕ σ(P))₀ = {x + σ(x) | x ϵ P} ≌ P as an algebra and (P ⊕ σ(P))₁ = {x − σ(x) | x ϵ P}. Since P ⊕ σ(P) is essential in R,

$$C\left(R\right)\cong C(P\oplus \sigma (P\left)\right)=C\left(P\right)\oplus \hat{\sigma }\left(C\right(P\left)\right),$$

where the isomorphism is given by the restriction of permissible maps (for any λ = [I, f] ϵ C(R) we define $\hat{\lambda }=\left[(I\cap (P\oplus \sigma \left(P\right)){)}^2\right.$, g] where g : (I∩(P⊕σ(P))² → P⊕σ(P) is the restriction of f to the essential ideal (I ∩ (P ⊕ σ(P)))² of P ⊕ σ(P)). Notice that the ℤ₂-grading of $C\left(P\right)\oplus \hat{\sigma }\left(C\right(P\left)\right)$ comes from the ℤ₂-grading of $P\oplus \sigma \left(P\right) \left(C\right(P)\oplus \hat{\sigma }(C\left(P\right)){)}_0=\{\lambda +\hat{\sigma }\left(\lambda \right)\mid \lambda \in C\left(P\right)\}$ and $\left(C\right(P)\oplus \hat{\sigma }(C\left(P\right)){)}_1=\{\lambda -\hat{\sigma }\left(\lambda \right)\mid \lambda \in C\left(P\right)\}$. In particular,

$$C(R{)}_0\cong \{\lambda +\hat{\sigma }\left(\lambda \right)\mid \lambda \in C\left(P\right)\}\cong C(P).$$

On the other hand, by Lemma 1.1.7, R₀ is prime as an algebra, and therefore its nonzero ideals are essential. By restricting permissible maps from R₀ to (P ⊕ σ(P))₀ we get $C\left(R_0\right)\cong C\left((P\oplus \sigma (P){)}_0\right)\cong C\left(P\right).$

We have obtained that C(R)₀ ≌ C(R₀).

Lemma 1.2.5. Let R = R₀ ⊕ R₁ be a prime associative superalgebra, and let a ϵ R₀. If there exists λ ϵ C(R) such that a − λ is nilpotent of index n and R has no n-torsion then λ ϵ C(R)₀.

Proof. Let us consider a ϵ R₀ and suppose that there exists λ = λ₀ + λ₁ ϵ C(R) such that a − λ is nilpotent of index n. If λ₁ ≠ 0, it is invertible by Lemma 1.2.6 and there exists μ1 ϵ C(R)₁ such that λ₁μ₁ = 1. From the nilpotency of a − λ₀ − λ₁ we get that μ₁a − λ₀ − 1 is again nilpotent of index n, i.e., the element b = μ₁a − μ₁λ₀ ϵ R₁ satisfies a polynomial of the form p(X) = (X − 1)ⁿ ϵ C(R)₀[X]. Since C(R)₀ is a field, p(X) ϵ C(R)₀[X] is the minimal polynomial of b over C(R)₀. In particular

$$b^n-\left(\genfrac{}{}{0ex}{}{n}{1}\right)b^{n-1}+\left(\genfrac{}{}{0ex}{}{n}{2}\right)b^{n-2}+\cdots =0$$

and by homogeneity

$$\left(\genfrac{}{}{0ex}{}{n}{1}\right)b^{n-1}+\left(\genfrac{}{}{0ex}{}{n}{3}\right)b^{n-3}+\cdots =0$$

i.e., b satisfies the polynomial $q\left(X\right)=\sum _{i=1}^{\left[\frac{n+1}{2}\right]}\left(\genfrac{}{}{0ex}{}{n}{2i-1}\right)X^{n-2i+1}$. But n − 1 = degq(X) < deg p(X) = n, a contradiction with the minimality of p(X). Therefore λ₁ = 0 and λ ϵ C(R)₀.

Lemma 1.2.6. [25, Lemma 3.1] Let R be a semiprime associative superalgebra. Then the following assertions are equivalent:

(i) R is a prime superalgebra.

(ii) all nonzero homogenous elements on C(R) are invertible.

(iii) C(R)₀ is a field.

1.3. Basic notions on Lie and Jordan algebras and superalgebras

1.3.1. We will work with Lie algebras and superalgebras arising from associative algebras and superalgebras. A Lie algebra L over a ring of scalarsΦ is a Φ-module with a bilinear product [ , ] satisfying, for every x, y, z ϵ L, the anticommutativity property and the Jacobi identity:

(i) [x, y] = −[y, x],

(ii) [x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0 (Jacobi identity).

Let L = L₀ + L₁ be a superalgebra over Φ with bilinear product denoted by [ , ]_s. By using the Grassmann envelope, L is a Lie superalgebra if G(L) is a Lie algebra. Let us suppose that L = L₀ + L₁ is a Lie superalgebra, i.e., G(L) = L₀ ⊗ G₀ + L₁ ⊗ G₁ is a Lie algebra. We can deepen into which identities L satisfies: let us pick x ⊗ ξ_i, y ⊗ ξ_j ϵ (L₀ ⊗ G₀) ∪ (L₁ ⊗ G₁), then

$$[x,y{]}_s\otimes {\xi }_i{\xi }_j=\left[x\otimes {\xi }_i,y\otimes {\xi }_j\right]=-\left[y\otimes {\xi }_j,x\otimes {\xi }_i\right]=-[y,x{]}_s\otimes {\xi }_j{\xi }_i=-(-1{)}^{\left|x\right|\left|y\right|}[y,x{]}_s\otimes {\xi }_i{\xi }_j$$

so we can assure, by linearity, that [x, y]_s = − (−1)^|x||y| [y, x]_s for every x, y ϵ L₀ U L₁. Notice that the factor (−1) in the identity naturally arises from the property ξ_iξ_j + ξ_jξ_i = 0, i.e, ξ_iξ_j = − ξ_jξ_j of the generators of the Grassman algebra. Therefore, the identities (i) and (ii) can be translated to super setting as follows: Let L be a ℤ₂- graded module over Φ with a bilinear product [ , ]_s such that for every homogeneous x, y, z ϵ L:

(i) [x, y]_s = − (−1)^|x||y|[y, x]_s (super-anticommutativity),

(ii) [x, [y, z]_s]_s + (−1)^|x|(|y|+^|z|) [z, [x, y]_s]_s + (−1)^|z|(|x+|y|) [y, [z, x]_s]_s = 0 (Jacobi superidentity).

Conversely, a superalgebra is a Lie superalgebra if both identities above are satisfied (see [?, Section 1]).

Recall that the adjoint map determined by any a ϵ L (resp. any homogeneous a ϵ L) is ad_a(x) := [a, x] (resp. ad_a(x) := [a, x]_s in super setting) for every x ϵ L. We say that an element a ϵ L is ad-nilpotent of index n ≥ 1 if ${ad}_a^{n}L=0$ L = 0 and ${ad}_a^{n–1}L\ne 0$. We say that an element a in L is a Jordan element if ${ad}_a^{3}L=0$ (see [23, Chapter 4]). Since in superalgebras we will always consider homogeneous elements, we will define Jordan element in superalgebras for even elements as an even element which is ad-nilpotent of index less or equal to 3 of the whole Lie superalgebra. For odd elements we will work with ad-nilpotency of index less or equal to 4.

Typical examples of Lie algebras and superalgebras come from the associative setting: if R is an associative algebra (resp. superalgebra) over a ring of scalars Φ, then R with product, called bracket, [x, y] := xy − yx for every x, y ϵ R (resp. [x, y]_s = xy − (−1)^|x||y|yx, called super-bracket, for every homogeneous x, y ϵ R) is a Lie algebra (resp. a Lie superalgebra) denoted by R⁻. When dealing with R⁻ as a superalgebra, if a ϵ R₀ then ad_a behaves as the usual adjoint map in the non-super setting; when a ϵ R₁, ${ad}_a^2={ad}_{a^2}.$

We will deal with Jordan algebras and superalgebras in Chapter 5. A linear Jordan algebra J over a ring of scalars Φ, with $\frac{1}{2}\in \Phi$, is a Φ-module with a bilinear product • satisfying, for every x, y ϵ J, the commutativity property and Jordan identity:

(i) x • y = y • x,

(ii) ((x • x) • y) • x = (x • x) • (y • x) (Jordan identity).

We already know that a superalgebra is a Jordan superalgebra if its Grassmann envelope is a Jordan algebra. But to translate the Jordan identity to super setting first we need to linearize it because the generatos in the Grassman algebra satisfy ${\xi }_i^2=0$. We can prove that a ℤ₂-graded module J over Φ with a bilinear product •_s is a Jordan superalgebra if it satisfies

(i) x •_s y = (−1)^|x||y|y •_s x (super-commutativity),

(ii) $\begin{array}{rl}& \left(x{\bullet }_{s}y\right){\bullet }_s\left(z{\bullet }_{s}t\right)+(-1{)}^{\left|y\right|\left|z\right|}\left(x{\bullet }_{s}z\right){\bullet }_s\left(y{\bullet }_{s}t\right)+(-1{)}^{\left|y\right|\left|t\right|+\left|z\right|\left|t\right|}\left(x{\bullet }_{s}t\right){\bullet }_s\left(y{\bullet }_{s}z\right)=\\ & =\left(\left(x{\bullet }_{s}y\right){\bullet }_{s}z\right){\bullet }_{s}t+(-1{)}^{\left|y\right|\left|z\right|+\left|y\right|\left|t\right|+\left|z\right|\left|t\right|}\left(\left(x{\bullet }_{s}t\right){\bullet }_{s}z\right){\bullet }_{s}y+(-1{)}^{\left|x\right|\left|y\right|+\left|x\right|\left|z\right|+\left|x\right|\left|t\right|+\left|z\right|\left|t\right|}((y{\bullet }_s \text{t)}{\bullet }_{s}z){\bullet }_{s}x\end{array}$ (Jordan super-identity)

for every homogeneous x, y, z, t ϵ J. As above, if R is an associative algebra (resp. superalgebra) over a ring of scalars Φ, then R with product, called bullet, x • y = xy + yx for every x, y ϵ R (resp. x •_s y = xy + (−1)^|x||y|yx, called super-bullet, for every homogeneous x, y ϵ R) is a Jordan algebra (resp. Jordan superalgebra) denoted by R⁺.

The algebras R⁻ and R⁺ are well-known and it was I.N. Herstein the first one to study the relations between R and both of them in the non-super case (see for example [43]). Moreover, K is a Lie subalgebra (resp. subsuperalgebra) of R⁻ and H is a Jordan subalgebra (resp. subsuperalgebra) of R⁺. We refer the reader to [25], [35], [36], [37], [52], [60] and [62] for further information on associative superalgebras and on the Herstein theory on superalgebras. Although we have denoted super bracket as [ , ]_s, in Chapter 3, in order to simplify the notation, we will denote it as [ , ] (we will just work with the super bracket and there will not be any confusion).

1.3.2. If R is a centrally closed *-prime algebra and Skew(C(R), *) ≠ 0 then for any 0 ≠ λ ϵ Skew(C(R), *) we have R = H + K = λ²H + K ⊆ λK + K ⊆ R because 0 ≠ λ² is invertible, so R = λK + K for every 0 ≠ λ ϵ Skew(C(R), *). This occurs in particular when R is *-prime but not prime, because in this situation there exists a nonzero ideal I of R such that I ∩ I * = 0, and so we can define a nonzero skew element λ : I ⊕ I* → R in C (R) given by λ(x + y) := x − y.

If R is a centrally closed semiprime ring then R⁻ is a Lie algebra over the ring of scalars C(R); if in addition R has an involution *, then K is a Lie algebra over H(C(R), *).

Lemma 1.3.3. ([13, Lemma 2.11]) Let (R, *) be a semiprime associative algebra with involution and let a ϵ R. If there exist λ ϵ C(R) such that a − λ is nilpotent then λ is the unique element of C(R) such that a − λ is nilpotent. Moreover, if a ϵ K then λ ϵ Skew(C(R), *).

Proof. If a−λ and a−μ are nilpotent elements of the central closure R̂ of R, a−λ−(a− μ) = μ − λ is a nilpotent element in the semiprime commutative ring C(R). Therefore λ = μ. Now, if a ϵ K and a − λ is nilpotent then (a − λ)* = − (a + λ*) is nilpotent and therefore a + λ* is nilpotent, which implies that λ = − λ* ϵ Skew(C(R), *).

We will need also this result in superalgebras. With the same argument as in the above lemma we have:

Lemma 1.3.4. Let R = R₀ ⊕ R₁ be a semiprime associative superalgebra with superinvolution *, and let a ϵ R₀ ∪ R₁. If there exists λ ϵ C(R) such that a − λ is nilpotent then λ is the unique element of C(R) such that a − λ is nilpotent. Moreover, if a ϵ K then λ ϵ Skew(C(R), *).

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