CHAPTER 4. EXAMPLES OF AD-NILPOTENT ELEMENTS

In this chapter we are going to construct examples of all types of ad-nilpotent elements appearing in Theorems 2.3.6 and 2.2.4 (non-super setting), and all types of ad-nilpotent homogeneous elements appearing in Theorem 3.1.2, and in Theorems 3.2.4 and 3.2.5. The examples of even ad-nilpotent elements of R and of K are based on the examples of ad-nilpotent elements in the non-super setting, see [11]; here we have rewritten those examples to have one example for both non-super and super setting.

4.0.1. Let Φ be a ring of scalars and let r, s be natural numbers. Following the notation of [46], the matrix algebra ℳ_r+s (Φ) with

$\begin{aligned} M (r ∣ s)_{0} := \{[\begin{array}{ll} A & 0 \\ 0 & D \end{array}] : A \in M_{r} (Φ), D \in M_{s} (Φ)\} and \\ M (r ∣ s)_{1} := \{[\begin{array}{ll} 0 & B \\ C & 0 \end{array}] : B \in M_{r, s} (Φ), C \in M_{s, r} (Φ)\} \end{aligned}$

becomes an ℤ₂-graded associative algebra. It will be denoted ℳ(r|s) = ℳ(r|s)₀ + ℳ(r|s)₁. We will use the notation ℳ(r) = ℳ(r|r).

4.0.2. Let r and s be two natural numbers with odd r > 1 and even s, let 𝔽 be a field with involution (a second-order automorphism) denoted by ᾱ for any α ϵ 𝔽, and let R be the superalgebra ℳ(r|s) over 𝔽. Let {e_i,j} denote the matrix units, and define

$H = \sum_{i = 1}^{r} (- 1)^{i} e_{i, r + 1 - i} \in M_{r} (F) (notice H = H^{t} = H^{- 1})$

$J = \sum_{i = 1}^{s} (- 1)^{i} e_{i, s + 1 - i} \in M_{s} (F) (notice J^{t} = - J = J^{- 1}) .$

The map * : R → R given by

${[\begin{array}{ll} A B \\ C D \end{array}]}^{*} = {[\begin{array}{ll} H 0 \\ 0 J \end{array}]}^{- 1} \bar{[\begin{array}{cc} A & - B \\ C & D \end{array}]} [\begin{array}{ll} H 0 \\ 0 J \end{array}]$

defines a superinvolution in R. In particular

$e_{i, j}^{*} = (- 1)^{j - i} e_{r - j + 1, r - i + 1} for every i, j \in {1, \dots, r},$

$e_{r + i, r + j}^{*} = (- 1)^{j - i} e_{r + s - j + 1, r + s - i + 1} for every i, j \in {1, \dots, s} and$

$e_{i, r + j}^{*} = (- 1)^{i - j + 1} e_{r + s + 1 - j, r + 1 - i} for every i \in {1, \dots, r} and j \in {1, \dots, s}$

Notice the superinvolution restricted to R₀ is an involution ⋆ such that K₀ = Skew(R, *)₀ = Skew(R₀,⋆).

The associative superalgebra R is a simple superalgebra with superinvolution, and its extended centroid C(R), which coincides with Z(R), is isomorphic to 𝔽. Moreover, the restriction of the superinvolution * to Z(R) is isomorphic to the involution – of 𝔽.

4.1. Examples in the non-super setting and of even ad-nilpotent elements of R⁻ and of K.

Let k be an even number( k ≥ 2), let r = 3k + 3 and s = 2k, and let us consider the associative superalgebra R = ℳ(r|s) over 𝔽 with the superinvolution defined in 4.0.2. Let us denote by K the skew-symmetric elements of R with respect to *. Consider the following nilpotent matrices:

$T := \sum_{i = k + 2}^{2 k + 1} e_{i, i + 1} \in R_{0}$ (nilpotent of index k + 1)

$S := \sum_{i = 1}^{k - 1} (e_{i, i + 1} + e_{r - i, r - i + 1}) \in R_{0}$ (nilpotent of index k)

$U := \sum_{i = 1}^{k - 1} e_{r + i, r + i + 1} + \sum_{i = k + 1}^{2 k - 1} e_{r + i, r + i + 1} \in R_{0}$ (nilpotent of index k)

By Proposition 3.1.1(1), T is ad-nilpotent of R and of R₀ of index 2k + 1, and S and U are ad-nilpotent elements of R and of R₀ of index 2k − 1.

Notice that T* = −T, S* = −S and U* = −U so T,S,U ϵ K₀. Let us calculate their indexes of ad-nilpotence in K:

(a) If Skew(𝔽, –) ≠ 0, by Proposition 3.2.3 the index of ad-nilpotence of T in K coincides with its index of ad-nilpotence in R, i.e., 2k + 1.

(b) If Skew(𝔽, –) = 0, for any B = Σ_{i j} λ_i,je_i,j ϵ K we have that λ_2k+2,k+2 = 0 and λ_2k+1,k+2 = λ_2k+2,k+3, so

$\begin{array}{l} {ad}_{T}^{2 k - 1} (B) = (\binom{2 k - 1}{k}) (T^{k - 1} B T^{k} - T^{k} B T^{k - 1}) = \\ = (\binom{2 k - 1}{k}) ((e_{k + 2, 2 k + 1} + e_{k + 3, 2 k + 2}) B (e_{k + 2, 2 k + 2})) - \\ - (\binom{2 k - 1}{k}) (e_{k + 2, 2 k + 2}) B (e_{k + 2, 2 k + 1} + e_{k + 3, 2 k + 2})) = \\ = (\binom{2 k - 1}{k}) (λ_{2 k + 1, k + 2} e_{k + 2, 2 k + 2} + λ_{2 k + 2, k + 2} e_{k + 3, 2 k + 2}) - \\ - (\binom{2 k - 1}{k}) (λ_{2 k + 2, k + 2} e_{k + 2, 2 k + 1} + λ_{2 k + 2, k + 3} e_{k + 2, 2 k + 2}) = 0 \end{array}$

Furthermore,

${ad}_{T}^{2 k - 2} (e_{2 k + 1, k + 2} - e_{2 k + 1, k + 2}^{*}) = {ad}_{T}^{2 k - 2} (e_{2 k + 1, k + 2} + e_{2 k + 2, k + 3}) \neq 0$

Thus T is ad-nilpotent of K of index 2k − 1.

(c) S is ad-nilpotent of K of index 2k − 1: by its ad-nilpotence in R, we have ${ad}_{S}^{2 k - 1} (K) = 0$ . Moreover, 0 ≠ C = e_k,1 − e_*_k,1 = e_k,1 + e_r,r−k+1 ϵ K and

$\begin{matrix} {ad}_{S}^{2 k - 2} (C) = - (\binom{2 k - 2}{k - 1}) S^{k - 1} (e_{k, 1} + e_{r, r - k + 1}) S^{k - 1} = \\ = - (\binom{2 k - 2}{k - 1}) (e_{1, k} + e_{r - k + 1, r}) (e_{k, 1} + e_{r, r - k + 1}) (e_{1, k} + e_{r - k + 1, r}) = \\ = - (\binom{2 k - 2}{k - 1}) (e_{1, k} + e_{r - k + 1, r}) \neq 0, \end{matrix}$

so S is also ad-nilpotent of K of index 2k − 1.

(d) U is ad-nilpotent of K of index 2k − 1: by its ad-nilpotence in R, we have ${ad}_{U}^{2 k - 1} (K) = 0$ . Moreover, $0 ∊ C = e_{r + k, r + 1} - e_{r + k, r + 1}^{*} = e_{r + k, r + 1} + e_{r + 2 k, r + k + 1}$ ∊ K and

$\begin{matrix} {ad}_{U}^{2 k - 2} (C) = {ad}_{U}^{2 k - 2} (e_{r + k, r + 1} + e_{r + 2 k, r + k + 1}) = \\ = - (\binom{2 k - 2}{k - 1}) U^{k - 1} (e_{r + k, r + 1} + e_{r + 2 k, r + k + 1}) U^{k - 1} = \\ = - (\binom{2 k - 2}{k - 1}) (e_{r + 1, r + k} + e_{r + k + 1, r + 2 k}) \neq 0 \end{matrix}$

Let us use these matrices T, S and U to get examples of any of models of ad- nilpotent elements in Theorems 2.3.6 and 2.2.4 from non-super setting and of even ad-nilpotent elements in Theorems 3.1.2 and 3.2.4. Here is important to point out in Theorems 3.1.2 and 3.2.4 we gave the index of ad-nilpotency of R₀ and K₀ as well, therefore if an even element is ad-nilpotent of R or K it will be always ad-nilpotent of the same index of R₀ and K₀ but in the case n ≡₄ 0 and ad-nilpotent of K then could be of index n − 1 of K₀. Thus, we will give examples of even homogeneous elements ad-nilpotent of R and K and will give examples for non-super setting ad-nilpotent of R₀ and K₀.

(i). Suppose Skew(𝔽, −) ≠ 0. For any λ ϵ Skew(𝔽, −), the element T + λid is ad-nilpotent of R of index 2k + 1, and by Proposition 3.2.3 its index in K is again n = 2k + 1. This is an example that fits case (2) of Theorem 3.2.4 and of Theorem 2.3.6 (a skew element a in K₀ with nilpotent (a − λ) of index k + 1 such that a is ad-nilpotent of index n ≡₄ 1 in K, K₀ and the same index in R). It also provides an example of case (1) in Theorem 3.1.2 and of Theorem 2.2.4.

(ii). Suppose Skew(𝔽, −) ≠ 0. For any λ ϵ Skew(𝔽, −), S + λid is an ad-nilpotent element of R and of K of index n = 2k − 1. This is an example that fits case (1) of Theorem 3.1.2 and case (4.2) of Theorem 3.2.4 and of Theorem 2.3.6 (a skew element in K₀, which is ad-nilpotent of index n ≡₄ 3 in K₀ and in K, and ad-nilpotent of the same index in R and R₀).

(iii). Suppose Skew(𝔽, −) = 0. T is an element of K₀ which is ad-nilpotent of K of index n = 2k − 1. This is an example that fits case (4.1) of Theorem 3.2.4 (an element in K₀ which is ad-nilpotent of index n ≡₄ 3 in K and in K₀, and ad-nilpotent of index n + 2 in R and R₀).

(iv). Suppose Skew(𝔽, −) = 0. The matrix A = T + S, which is an orthogonal sum of T and S, is nilpotent of index t + 1 and ad-nilpotent of R and of R₀ of index 2k + 1. Let us see that it is ad-nilpotent of K of index 2k: from the indexes of nilpotence of T and S, their indexes of ad-nilpotence in K and the fact that TS = 0 = ST we get that ${ad}_{A}^{2 k} (K) = 0$ . Moreover, $C = e_{k, k + 2} - e_{k, k + 2}^{*} = e_{k, k + 2} + e_{2 k + 2, 2 k + 4} ∊ K$ and one can check that ${ad}_{A}^{2 k} (K) = 0$ . Moreover, $C = e_{k, k + 2} - e_{k, k + 2}^{*} = e_{k, k + 2} - e_{2 k + 2, 2 k + 4} \in K$ and one can check that ${ad}_{A}^{2 k - 1} (C) = - (\binom{2 k - 1}{k}) (e_{1, 2 k + 2} + e_{k + 2, 3 k + 3}) \neq 0$ . This is an example that fits case (1) of Theorem 3.2.4 (a skew element in K₀ which is ad-nilpotent of index n ≡₄ 0 in K₀ and in K, and ad-nilpotent of index n + 1 in R and R₀).

(v). Suppose Skew(𝔽, −) = 0. Let us consider A = T + U, which is an orthogonal sum of T and U. The nilpotence of T + U implies that the index of ad-nilpotence of A in R (and in R₀) is 2k + 1 (by Proposition 3.1.1(1)). Since both T and U are ad-nilpotent elements of K₀ of indexes 2k − 1, A is ad-nilpotent of K₀ of index 2k − 1. Nevertheless, its index of ad-nilpotence in K is higher: for any B = Σλ_i,j e_i,j ϵ K we have that

$\begin{aligned} {ad}_{A}^{2 k} (B) = (\binom{2 k}{k}) A^{k} B A^{k} = (\binom{2 k}{k}) e_{k + 2, 2 k + 2} B e_{k + 2, 2 k + 2} = \\ = (\binom{2 k}{k}) λ_{2 k + 2, k + 2} e_{k + 2, 2 k + 2} = 0 \end{aligned}$

because λ_2k+2,k+2 = 0. Moreover, if we consider the element C = e_2k+2,r+1 −

$e_{2 k + 2, r + 1}^{*} = e_{2 k + 2, r + 1} - e_{r + s, k + 2} \in K$ one can check that

$\begin{matrix} {ad}_{A}^{2 k - 1} (C) = (\binom{2 k - 1}{k}) (A^{k - 1} C A^{k} - A^{k} C A^{k - 1}) = \\ = - (\binom{2 k - 1}{k}) (e_{r + k + 1, 2 k + 2} + e_{k + 2, r + k}) \neq 0 \end{matrix}$

because

$A^{k - 1} = T^{k - 1} + U^{k - 1} = e_{k + 2, 2 k + 1} + e_{k + 3, 2 k + 2} + e_{r + 1, r + k} + e_{r + k + 1, r + s}$

This means that the index of ad-nilpotence of A in K is n = 2k. This gives an example of an element in the conditions of Theorem 3.2.4 (1) and a case, again, (4.1) of Theorem 2.3.6 (a skew element in K₀, which ad-nilpotent of K of index n ≡₄ 0, ad-nilpotent of K₀ of index n − 1, and ad-nilpotent of R index n + 1).

4.2. Examples of odd ad-nilpotent elements of R⁻ and of K.

Let 𝔽 be a field with identity involution, let r > 1 be an odd number, let s = r − 1, and consider the superalgebra R = ℳ(r|s) with the superinvolution given in 4.0.2. Again, let us denote by K the skew-symmetric elements of R with respect to *.

Let us consider $T := \sum_{i = 1}^{r - 1} e_{i, r + i} \in R_{1}$ . Then

$A = T - T^{*} = \sum_{i = 1}^{r - 1} e_{i, r + i} + \sum_{i = 2}^{r} e_{r + i - 1, i} \in K_{1}$ (nilpotent of index 2r − 1).

We have that

$\begin{array}{l} A^{2} = \sum_{i = 1}^{r - 1} e_{i, i + 1} + \sum_{i = 2}^{r - 1} e_{r + i - 1, r + i} \\ A^{2 r - 7} = e_{1, 2 r - 3} + e_{2, 2 r - 2} + e_{3, 2 r - 1} + e_{r + 1, r - 2} + e_{r + 2, r - 1} + e_{r + 3, r}, \\ A^{2 r - 6} = e_{1, r - 2} + e_{2, r - 1} + e_{3, r} + e_{r + 1, 2 r - 2} + e_{r + 2, 2 r - 1} \end{array}$

$\begin{array}{l} A^{2 r - 3} = e_{1, 2 r - 1} + e_{r + 1, r}, \\ A^{2 r - 2} = e_{1, r} and \\ A^{2 r - 1} = 0 \end{array}$

By Proposition 3.1.1(2b) A is ad-nilpotent in R of index m = 4r − 3. For every $B = \sum_{i, j} λ_{i, j} e_{i, j} \in K_{0} \cup K_{1},$

$\begin{matrix} {ad}_{A}^{4 r - 5} (B) = {ad}_{A^{2}}^{2 r - 3} {ad}_{A} (B) = \\ = (\binom{2 r - 3}{r - 1}) ({(A^{2})}^{r - 2} {ad}_{A} (B) {(A^{2})}^{r - 1} - {(A^{2})}^{r - 1} {ad}_{A} (B) {(A^{2})}^{r - 2}) = \\ = (\binom{2 r - 3}{r - 1}) (A^{2 r - 3} B A^{2 r - 2} + (- 1)^{| B |} A^{2 r - 2} B A^{2 r - 3}) = \\ = (\binom{2 r - 3}{r - 1}) ((e_{1, 2 r - 1} + e_{r + 1, r}) B e_{1, r} + (- 1)^{| B |} e_{1, r} B (e_{1, 2 r - 1} + e_{r + 1, r})) = \\ = (\binom{2 r - 3}{r - 1}) (λ_{2 r - 1, 1} e_{1, r} + λ_{r, 1} e_{r + 1, r} + (- 1)^{| B |} λ_{r, 1} e_{1, 2 r - 1} + (- 1)^{| B |} λ_{r, r + 1} e_{1, r}) = 0 \end{matrix}$

because when B ϵ K₀ we always have that λ_2r−1,1 = λ_r,r+1 = 0 (by grading) and λ_r,1 = 0, and when B ϵ K₁, λ_r,1 = 0 (by grading) and λ_2r−1,1 = λ_r,r+1. Moreover, by Theorem 3.2.5, the index of ad-nilpotence of A in K can be m, m − 1 or m − 2, so it is m − 2 = 4r − 5.

(i). The element A ϵ K₁ is an example of an element in the conditions of Theorem 3.2.5(6) (a nilpotent element of index 2r − 1, which is ad-nilpotent of index n = 4r − 5 ≡₈ 7 in K and ad-nilpotent of index n + 2 in R, and such that A^2r−3BA^2r−2 + (−1)^|B|A^2r−²BA^2r−3 = 0 for every B ϵ K₀ U K₁).

To produce examples for the rest of the cases of Theorem 3.2.5, let us consider A⁵ ϵ K₁ for some particular cases of odd r > 1.

(ii). Fix r = 10t + 1 for some t ϵ ℕ. Then

$\begin{array}{l} \begin{aligned} {(A^{5})}^{4 t + 1} = A^{2 r + 3} = 0, \\ {(A^{5})}^{4 t} = A^{2 r - 2} \in H_{0}, \end{aligned} \\ {(A^{5})}^{4 t - 1} = A^{2 r - 7} . \end{array}$

In particular, A⁵ is nilpotent of index 4t + 1 and ad-nilpotent of R of index 8t + 1. Notice that for every B = Σ_i,j λ_i,je_i,j ϵ K

${(A^{5})}^{4 t} B {(A^{5})}^{4 t} = e_{1, r} B e_{1, r} = λ_{r, 1} e_{1, r} = 0$

because every B ϵ K has λ_r,1 = 0. Therefore, for every B ϵ K we have

${ad}_{A^{5}}^{8 t} (B) = {ad}_{A^{10}}^{4 t} (B) = (\binom{4 t}{2 t}) {(A^{10})}^{2 t} B {(A^{10})}^{2 t} = 0$

Furthermore, considering C = e_r,r+1 − e*_r,r+1 = e_r,r+1 + e_2r−1,1 ϵ K₁

$\begin{array}{l} {ad}_{A^{5}}^{8 t - 1} (C) = {ad}_{A^{5}}^{8 t - 2} ({ad}_{A^{5}} (e_{r, r + 1} + e_{2 r - 1, 1})) = \\ = {ad}_{A^{10}}^{4 t - 1} ({ad}_{A^{5}} (e_{r, r + 1} + e_{2 r - 1, 1})) = \\ = (\binom{4 t - 1}{2 t}) {(A^{10})}^{2 t - 1} ({ad}_{A^{5}} (e_{r, r + 1} + e_{2 r - 1, 1})) {(A^{10})}^{2 t} - \\ - (\binom{4 t - 1}{2 t}) {(A^{10})}^{2 t} ({ad}_{A^{5}} (e_{r, r + 1} + e_{2 r - 1, 1})) {(A^{10})}^{2 t - 1} = \\ = (\binom{4 t - 1}{2 t}) (A^{20 t - 5} (e_{r, r + 1} + e_{2 r - 1, 1}) A^{20 t}) - (A^{20 t} (e_{r, r + 1} + e_{2 r - 1, 1}) A^{20 t - 5}) = \\ = (\binom{4 t - 1}{2 t}) (e_{3, r} - e_{1, r - 2}) \neq 0 \end{array}$

The element A⁵ gives an example of an element in the conditions of Theorem 3.2.5(1) (a nilpotent element of index 4t + 1, ad-nilpotent element in K₁ of index n = 8t ≡₈ 0, ad-nilpotent in R of index n + 1 = 8t +1 and such that (A⁵)^4tK(A⁵)^4t = 0).

(iii). Fix r = 10t + 3 for some t ϵ ℕ. Then

$\begin{array}{l} {(A^{5})}^{4 t + 1} = A^{2 r - 1} = 0 \\ {(A^{5})}^{4 t} = A^{2 r - 6} . \end{array}$

In particular, A⁵ is nilpotent of index 4t + 1 and ad-nilpotent of R of index 8t + 1 (see Proposition 3.1.1(2b)). In this case the index of ad-nilpotence of A⁵ in K is the same as in R because for $C = e_{r, r + 1} - e_{r, r + 1}^{*} = e_{r, r + 1} + e_{2 r - 1, 1} ∊ K_{1}$ we have

$\begin{array}{l} {ad}_{A^{5}}^{8 t} (C) = {ad}_{A^{10}}^{4 t} (e_{r, r + 1} + e_{2 r - 1, 1}) = \\ = (\binom{4 t}{2 t}) {(A^{10})}^{2 t} (e_{r, r + 1} + e_{2 r - 1, 1}) {(A^{10})}^{2 t} = \\ = (\binom{4 t}{2 t}) (e_{3, 2 r - 2} + e_{r + 2, r - 2}) \neq 0 \end{array}$

The element A⁵ gives an example of an element in the conditions of Theorem 3.2.5(2) (a nilpotent element in K₁ of index 4t +1, ad-nilpotent of K and of R of the same index n = 8t + 1 ≡₈ 1).

(iv). Fix r = 10t + 5 for some t ϵ ℕ. Then A⁵ is nilpotent of index 4t + 2. Since the index of nilpotence of A⁵ is even, we know by Proposition 3.1.1(2a) that A⁵ is ad-nilpotent of R of index 2(4t + 2) − 2 = 8t + 2. Moreover, from the fact that A⁵ is ad-nilpotent of R of index 8t + 2 ≡₈ 2 we get from Theorem 3.2.5 that its index of ad-nilpotence in K is the same as in R. The element A⁵ gives an example of an element in the conditions of Theorem 3.2.5(3) with λ = 0 (a nilpotent element of K₁ of index 4t + 2 which is ad-nilpotent of K and of R of the same index n = 8t + 2 ≡₈ 2.)

(v). Fix r = 10t + 7 for some t ϵ ℕ. Then A⁵ is nilpotent of index 4t + 3. Since the index of nilpotence of A⁵ is odd, we know by Proposition 3.1.1(2a) that A⁵ is ad-nilpotent of R of index 2(4t + 3) − 1 = 8t + 5. Moreover, from the fact that A⁵ is ad-nilpotent of R of index 8t + 5 ≡₈ 5 we get from Theorem 3.2.5 that its index of ad-nilpotence in K is the same as in R. The element A⁵ gives an example of an element in the conditions of Theorem 3.2.5(4) (a nilpotent element of K₁ of index 4t + 3 which is ad-nilpotent of K and of R of the same index n = 8t + 5 ≡₈ 5).

(vi). Fix r = 10t + 9 for some t ϵ ℕ. Then A⁵ is nilpotent of 4t + 4. Since the index of nilpotence of A⁵ is even, we know by Proposition 3.1.1(2a) that A⁵ is ad-nilpotent of R of index 2(4t + 4) − 2 = 8t + 6. Moreover, from the fact that A⁵ is ad-nilpotent of R of index 8t + 6 ≡₈ 6 we get from Theorem 3.2.5 that its index of ad-nilpotence in K is the same as in R. The element A⁵ gives an example of an element in the conditions of Theorem 3.2.5(5) with λ = 0 (a nilpotent element of K₁ of index 4t + 4 which is ad-nilpotent of K and of R of the same index n = 8t + 6 ≡₈ 6).

The matrices given in (i), (ii), (iii) and (v) provide examples of (2.a) in Theorem 3.1.2. Moreover, the matrices of (iv) and (vi) fit in case (2.b) of Theorem 3.1.2 with λ = 0.

4.2.1. Some other examples of odd ad-nilpotent elements of K and of R.

The examples (iv) and (vi) in the previous section are ad-nilpotent elements of K of indexes n ≡₈ 2 and n ≡₈ 6, and fit in Theorem 3.2.5(3) and (5) with λ = 0. To get examples of such types of elements with nonzero λ’s, we will work with matrices over a field with nontrivial involution.

Let r be a natural number, let ℂ be the field of complex numbers with involution given by conjugation, and let us consider the simple superalgebra R = Ϻ(r) over ℂ. The map trp given by

${[\begin{array}{ll} A B \\ C D \end{array}]}^{trp} = [\begin{array}{cc} D^{t} & - B^{t} \\ C^{t} & A^{t} \end{array}]$

where A,B,C,D ϵ Ϻ_r(ℂ) and ( )^t denotes the usual matrix transposition, defines a superinvolution in R known as the transpose superperinvolution (see [36, Example 2.2]).

Let us denote by K the set of skew-symmetric elements of Ϻ(r) with respect trp. Note that any element of K₁ has the form $[\begin{array}{ll} 0 B \\ C 0 \end{array}]$ where B is a symmetric matrix and C is a skew-symmetric matrix in Ϻ_r (ℂ) with respect to the usual transposition.

Let us consider a symmetric matrix B ϵ Ϻ_r (ℂ) with B^r = 0 and B^r−¹ ≠ 0 (it is shown in [51, Corollary 5] that for every r there exist symmetric nilpotent matrices in Ϻ_r(ℂ) of rank r − 1). Let 0 ≠ A E ℝ and let i denote the square root of −1. Then

$a = [\begin{array}{cc} 0 & B + id \\ (λ i) id & 0 \end{array}] \in K_{1} and a^{2} = [\begin{array}{cc} (λ i) B + (λ i) id & 0 \\ 0 & (λ i) B + (λ i) id \end{array}]$

i.e., (a² − λi) is nilpotent of index r.

When r is odd a is an example for Theorem 3.2.5 (3), and when r is even a is an example for Theorem 3.2.5 (5). Both cases are examples of elements of the form (2.b) of Theorem 3.1.2.

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CHAPTER 4. EXAMPLES OF AD-NILPOTENT ELEMENTS

4.1. Examples in the non-super setting and of even ad-nilpotent elements of R− and of K.

4.2. Examples of odd ad-nilpotent elements of R− and of K.

BIBLIOGRAPHY

4.1. Examples in the non-super setting and of even ad-nilpotent elements of R⁻ and of K.

4.2. Examples of odd ad-nilpotent elements of R⁻ and of K.