Math Problems Vault

Problem 65.6 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 65 "Special classes of algebras"

(a)(b)

Let $A=A_{0} \oplus A_{1}$ be a $2$ -graded associative algebra over a field $K$ , i.e. $A_{i} A_{j} \subset A_{i+j}$ (indices added modulo 2). Define a new operation on $A$ by setting

[x, y]=x y-(-1)^{i j} y x

where $x \in A_{i}, y \in A_{j}$ .

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(a)

Prove that for any homogeneous elements $x \in A_{i}, y \in A_{j}$ , $z \in A$ we have

\begin{gathered} {[x, y]=(-1)^{i j}[y, x]} \\ {[x,[y, z]]+[y,[z, x]]+(-1)^{i j+1}[z,[x, y]]=0} \end{gathered}

An algebra with a $2$ -grading for which the homogeneous elements satisfy these relations is called a Lie superalgebra.

(b)

Let $V$ be an $n$ -dimensional vector space with basis $\left(e_{1}, \ldots , e_{n}\right)$ over a field $K$ of characteristic other than 2, let $\Lambda (V)$ be the exterior algebra on $V$ , let $I$ be the identity operator on $V$ , let $L_{0}=K \cdot I$ , and let $L_{1}$ be the linear span of the operators $\varphi_{i}$ and $\psi_{i}$ , where

\begin{gathered} \varphi _{i}(w)=w \wedge e_{i} \\ \psi _{i}\left(e_{i_{1}} \wedge \ldots \wedge e_{i_{p}}\right)= \begin{cases} (-1)^{p-k} e_{i_{1}} \wedge \ldots \wedge \widehat{e}_{i_{k}} \wedge \ldots \wedge e_{i_{p}}, & \text{ if } i_{k}=i, \\ 0, & \text{ if } i_{k} \neq i \text{ for all } k=1, \ldots , p. \end{cases} \end{gathered}

Prove that $L=L_{0} \oplus L_{1}$ is a Lie superalgebra with respect to the operation introduced in (a).

Problem 19.1.23 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 19.1 "Chapter Problems"

The $\mathfrak {gl}(1|1)$ superalgebra involves two even (denoted by $H$ and $Z$ ) and two odd (denoted by $E$ , $F$ ) generators. The following commutation and anti-commutation relations hold

[Z,E] = [Z,F] = [Z,H] = 0, \qquad [H,E] = E, \qquad [H,F] = -F, \qquad [E,F]_{+} = Z

and $E^{2} = F^{2} = 0$ . Find a $2 \times 2$ matrix representation.

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Problem 3.98 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

Let $\mathbb {Z}_{2} = \{ 0,1\}$ be the finite group of integers under addition modulo 2. The super-Lie algebra $\mathfrak {g} = \mathfrak {g}_{0}+\mathfrak {g}_{1}$ is a $\mathbb {Z}_{2}$ -graded Lie algebra with a super-Lie bracket $[\, , \, ]$ which satisfies

[A,B] = AB - (-1)^{d(A)d(B)}BA

and (super Jacobi identity)

[A,[B,C]] = [[A,B],C] + (-1)^{d(A)d(B)}[B,[A,C]]

where $d(X) = 0$ (1) if $A \in \mathfrak {g}_{0}$ ( $\mathfrak {g}_{1}$ ). $d(A)$ is called the degree of $A$ . We denote by $[\, , \, ]_{-}$ the commutator and by $[\, , \, ]_{+}$ the anticommutator. The super-Lie algebra $\mathfrak {sl}(m/n)$ is defined by

\mathfrak {sl}(m/n) := \left\{ X = \begin{pmatrix} R & S \\ T & U \end{pmatrix} \, : \, \operatorname {Str}(X) := \tr (R) - \tr (U) = 0 \right\}

where $R$ is an $m \times m$ matrix, $S$ a $m \times n$ matrix, $T$ an $n \times m$ matrix and $D$ an $n \times n$ matrix. We assume that $m \neq n$ . Consider the super-Lie algebra $\mathfrak {sl}(2/1)$ . A basis is given by

E_{1} = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad E_{2} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix},

F_{1} = \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad F_{2} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix},

G_{1} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad G_{2} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix},

H_{1} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad H_{2} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

i.e. we have four even elements $E_{1}, F_{1}, H_{1}, H_{2}$ and four odd elements $E_{2}, F_{2}, G_{1}, G_{2}$ . Find the non-vanishing commutators and anticommutators.

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Problem 4.99 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 4 "Applications"

The $\mathfrak {gl}(1|1)$ superalgebra has two even and two odd generators we denote by $h, z$ and $e, f$ , respectively. The following commutation relations hold

[z,e] = [z,f] = [z,h] = 0, \quad [h,e] = e, \quad [h,f] = -f

and

[e,f]_+ = z, \quad e^2 = f^2 = 0

where $[\, ,\, ]_+$ denotes the anticommutator. Find a representation by $2\times 2$ matrices.

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Problem 4.100 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 4 "Applications"

The Lie superalgebra $\mathfrak {osp}(1/2)$ has five generators $K_0, K_+, K_-, F_+, F_-$ . The commutation relations are given by

[K_0,K_\pm ] = \pm K_\pm , \quad [K_+,K_-] = -2K_0 .

[K_0,F_\pm ] = \pm \frac{1}{2}F_\pm , \quad [K_\pm ,F_\pm ] = 0, \quad [K_\pm ,F_\mp ] = \mp F_\pm .

The anticommutation relations are given by

[F_\pm ,F_\pm ]_+ = K_\pm , \quad [F_+,F_-]_+ = K_0 .

Thus the Lie superalgebra contains the sub Lie algebra $\mathfrak {su}(1,1)$ spanned by $K_0, K_+, K_-$ .

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(i)

Find the Casimir operator of the Lie superalgebra $\mathfrak {osp}(1/2)$ .

(ii)

Find a faithful representation with Bose operators $b^\dagger , b$ and the identity operator $I$ .

(iii)

Introduce the number states (Fock states) $\ket {n}$ ( $n=0,1,\ldots ,$ ) and apply the operators from (ii) to the number states.

(iv)

Let $\beta , \gamma \in \mathbb {C}$ . Introduce the operators

D(\beta ) = \exp (\beta F_+-\beta ^*F_-), \quad S(\gamma ) = \exp (\gamma K_+-\gamma ^*K_-) .

Calculate

D^\dagger (\beta ) \begin{pmatrix} F_- \\ F_+ \end{pmatrix} D(\beta ), \quad S^\dagger (\gamma ) \begin{pmatrix} F_- \\ F_+ \end{pmatrix} S(\gamma ),

D^\dagger (\beta ) \begin{pmatrix} K_- \\ K_+ \end{pmatrix} D(\beta ), \quad S^\dagger (\gamma ) \begin{pmatrix} K_- \\ K_+ \end{pmatrix} S(\gamma ) .

(v)

Consider the coherent-squeezed state

\ket {\beta \gamma } := S(\gamma )D(\beta )\ket {0} .

Calculate the state $\ket {\beta \gamma }$ using the number states $\ket {n}$ and the identity

\sum _{n=0}^{\infty } \frac{(t/2)^n}{n!}H_n(x)H_n(y) \equiv \frac{1}{\sqrt{1-t^2}}\exp ((1-t)^{-1}(2xyt-(x^2+y^2)t))

where $H_n$ ( $n=0,1,\ldots$ ) are the Hermite polynomials.

Problem 4.110 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Section 4.1 "Programming Problems"

Consider the fundamental representation of the superalgebra $\mathfrak {su}(2|1)$ . Let $\sigma_1, \sigma_2, \sigma_3$ be the Pauli spin matrices. Let $\oplus$ denote the direct sum. Its generators are given by the $3 \times 3$ matrices

L_1 = \frac{1}{2}\sigma _1\oplus (0), \quad L_2 = \frac{1}{2}\sigma _2\oplus (0), \quad L_3 = \frac{1}{2}\sigma _3\oplus (0), \quad L_4 = \frac{1}{2}\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}

V_1 = \frac{1}{2}\begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}, \quad V_2 = \frac{1}{2}\begin{pmatrix} 0 & 0 & i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix}

W_1 = \frac{1}{2}\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, \quad W_2 = \frac{1}{2}\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix} .

Let

L_\pm = L_1\pm iL_2, \quad V_\pm = V_1\pm iV_2, \quad W_\pm = W_1\pm iW_2 .

Here $L_1, L_2, L_3, L_4$ are the generators forming the Lie subalgebra $\mathfrak {su}(2)\otimes \mathfrak {u}(1)$ of $\mathfrak {su}(2|1)$ and $V_1, V_2, W_1, W_2$ are the supergenerators. Let $c^\dagger , c$ be Fermi creation and annihilation operators. Let $b_1^\dagger , b_2^\dagger$ be Bose creation operators. A Boson-Fermion realization is given by

L_+ = b_1^\dagger b_2 \otimes I_F, \quad L_- = b_2^\dagger b_1 \otimes I_F

L_3 = \frac{1}{2}(b_1^\dagger b_1 \otimes I_F - b_2^\dagger b_2 \otimes I_F), \quad L_4 = \frac{1}{2}(b_1^\dagger b_1 \otimes I_F + b_2^\dagger b_2 \otimes I_F) + I_B\otimes c^\dagger c

V_+ = b_1^\dagger \otimes c, \quad V_- = b_1 \otimes c^\dagger , \quad W_+ = b_2^\dagger \otimes c, \quad W_- = b_2 \otimes c^\dagger .

Show that the representations are isomorphic. Give a SymbolicC++ implementation for the Bose-Fermi realization.

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Lie superalgebras

References