Abstract
A solution of the classical Yang–Baxter equation associated with the queer Lie superalgebra is constructed in terms of Hermite theta functions.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Let \(\mathfrak {g}\) be any finite-dimensional Lie superalgebra over a complex field \({\mathbb {C}}\,\). Let \(r{\hspace{0.5pt}}(u,v)\) be a meromorphic function of two complex variables u and v which takes values in \(\mathfrak {g}{\hspace{0.5pt}}\otimes \mathfrak {g}\,\). The classical Yang–Baxter equation for the function \(r{\hspace{0.5pt}}(u\,,v)\) is
where w is another complex variable and the function of u, v, w at the left-hand side takes values in \(\mathfrak {g}{\hspace{0.5pt}}\otimes \mathfrak {g}{\hspace{0.5pt}}\otimes \mathfrak {g}\,\). Here we use the standard notation, for example \(r_{{\hspace{0.5pt}}23}(v,w)=1\otimes r{\hspace{0.5pt}}(v,w)\,\). For conventions on the tensor products see Sect. 2.
A solution of the above equation is called nondegenerate if not every value of the function \(r{\hspace{0.5pt}}(u,v)\) is degenerate as a quadratic tensor. For simple Lie algebras \(\mathfrak {g}\) the nondegenerate solutions were classified in [4, 5]. In particular, it was shown in [5] that for any nondegenerate solution \(r{\hspace{0.5pt}}(u,v)\) with a simple Lie algebra \(\mathfrak {g}\) one can find a domain \(D\subset {\mathbb {C}}\) and two holomorphic maps
where \(\psi \) is not constant and the function \(({\hspace{0.5pt}}\omega (u)\otimes {\hspace{0.5pt}}\omega (v))\,r{\hspace{0.5pt}}({\hspace{0.5pt}}\psi (u),\psi (v))\) depends only on the difference \(u-v\,\).
This basic result of [5] does not hold for Lie superalgebras. Solutions \(r{\hspace{0.5pt}}(u,v)\) which depend not only on the difference \(u-v\) were constructed in [11]. There \(\mathfrak {g}\) is the general linear Lie superalgebra \(\mathfrak {gl}_{\,n|n}\) with any positive integer \(n\,\). These solutions are antisymmetric, that is
A general phenomenon underlying this construction was independently described in [1]. See [12] for further discussion of this remarkable phenomenon.
The solutions \(r{\hspace{0.5pt}}(u,v)\) constructed in [11] are rational functions of variables u and \(v\,\). Here we construct a solution \(r{\hspace{0.5pt}}(u,v)\) which is expressed in theta functions of \(u-v\) and \(u+v\,\). Our \(\mathfrak {g}\) is the quotient of the special linear Lie superalgebra \(\mathfrak {sl}_{\,n|n}\) by its one-dimensional centre. This quotient is denoted by \(\mathfrak {psl}_{\,n|n}\,\).
Elliptic solutions \(r{\hspace{0.5pt}}(u,v)\) for the Lie superalgebra \(\mathfrak {g}=\mathfrak {psl}_{\,n|n}\) were constructed in [9]. They have the form \(r{\hspace{0.5pt}}(u,v)=s{\hspace{0.5pt}}(u-v)\) where \(s{\hspace{0.5pt}}(u)\) is a function such that
Hence these solutions are antisymmetric.
Now let \(\eta \) be the involutive automorphism of \(\mathfrak {gl}_{\,n|n}\) defined in Sect. 2. The fixed point subalgebra of \(\mathfrak {gl}_{\,n|n}\) relative to \(\eta \) is the queer Lie superalgebra \(\mathfrak {q}_{{\hspace{0.5pt}}n}\,\). The automorphism \(\eta \) preserves the subalgebra \(\mathfrak {sl}_{\,n|n}\) and descends to \(\mathfrak {psl}_{\,n|n}\,\). In our Sect. 5 we show that the function \(s{\hspace{0.5pt}}(u)\) in [9] can be so chosen that
and that
is another solution of classical Yang–Baxter equation for \(\mathfrak {g}=\mathfrak {psl}_{\,n|n}\,\).
It immediately follows from (3) and (4) that
By comparing the definition (4) with (5) and by using (2) we see that our solution \(r{\hspace{0.5pt}}(u,v)\) is antisymmetric as well.
2 General Conventions
We shall use the following general conventions. Let \(\textrm{A}\) and \(\textrm{B}\) be any associative \(\mathbb {Z}_{{\hspace{0.5pt}}2}\)-graded algebras. Their tensor product \(\textrm{A}\otimes \textrm{B}\) is also an associative \(\mathbb {Z}_2\)-graded algebra such that for any homogeneous elements \(X,X^\prime \in \textrm{A}\) and \(Y,Y^\prime \in \textrm{B}\)
Furthermore, for any two \(\mathbb {Z}_2\)-graded modules U and V over \(\textrm{A}\) and \(\textrm{B}\), respectively, the vector space \(U\otimes V\) is a \(\mathbb {Z}_2\)-graded module over \(\textrm{A}\otimes \textrm{B}\) such that for any homogeneous elements \(x\in U\) and \(y\in V\)
Now let the indices i, j, k, l run through \({\hspace{0.5pt}}\pm 1{{\hspace{0.5pt}},\hspace{1.0pt}\ldots {\hspace{0.5pt}},{\hspace{0.5pt}}}{\hspace{-0.5pt}}\pm n\,\). Put \({\bar{\imath }\,}=0\) if \(i>0\) and \({\bar{\imath }\,}=1\) if \(i<0\,\). Consider the \(\mathbb {Z}_{{\hspace{0.5pt}}2}\)-graded vector space \({\mathbb {C}}^{{\hspace{0.5pt}}n|n}{\hspace{0.5pt}}\). Let \(e_i\in {\mathbb {C}}^{{\hspace{0.5pt}}n|n}\) be an element of the standard basis. The \(\mathbb {Z}_{{\hspace{0.5pt}}2}\)-grading on \({\mathbb {C}}^{{\hspace{0.5pt}}n|n}\) is defined by \(\deg e_i={\bar{\imath }\,}\,\).
Let \(E_{{\hspace{0.5pt}}ij}\in {\text {End}}\hspace{1.0pt}{\mathbb {C}}^{{\hspace{0.5pt}}n|n}\) be the standard matrix unit, defined by \(E_{{\hspace{0.5pt}}ij}\,e_k=\delta _{{\hspace{0.5pt}}jk}\,e_i\,\). The associative algebra \({\text {End}}\hspace{1.0pt}{\mathbb {C}}^{{\hspace{0.5pt}}n|n}\) is \(\mathbb {Z}_{{\hspace{0.5pt}}2}\)-graded by setting \(\deg E_{{\hspace{0.5pt}}ij}={\bar{\imath }\,}+{\hspace{0.5pt}}{\bar{\jmath }\,}\,\). Hence \({\mathbb {C}}^{{\hspace{0.5pt}}n|n}\) is a \(\mathbb {Z}_2\)-graded module over \({\text {End}}\hspace{1.0pt}{\mathbb {C}}^{{\hspace{0.5pt}}n|n}\,\). For any positive integer m we can also identify the tensor product \(({\text {End}}\hspace{1.0pt}{\mathbb {C}}^{{\hspace{0.5pt}}n|n})^{\otimes {\hspace{0.5pt}}m}\) with the algebra \({\text {End}}\hspace{1.0pt}(({\mathbb {C}}^{{\hspace{0.5pt}}n|n})^{\otimes {\hspace{0.5pt}}m})\) acting on the vector space \(({\mathbb {C}}^{{\hspace{0.5pt}}n|n})^{\otimes {\hspace{0.5pt}}m}\) by repeatedly using conventions (6) and (7).
The supertrace \(\textrm{str}\) is a linear function \({\text {End}}\hspace{1.0pt}{\mathbb {C}}^{{\hspace{0.5pt}}n|n}\rightarrow {\mathbb {C}}\) defined by setting
This definition implies that for any homogeneous elements \(X{\hspace{0.5pt}},Y\in {\text {End}}\hspace{1.0pt}{\mathbb {C}}^{{\hspace{0.5pt}}n|n}\)
Further, we can define an involutive automorphism \(\eta \) of \({\text {End}}\hspace{1.0pt}{\mathbb {C}}^{{\hspace{0.5pt}}n|n}\) by mapping
This automorphism is the conjugation by the involutive odd element of \({\text {End}}\hspace{1.0pt}{\mathbb {C}}^{{\hspace{0.5pt}}n|n}\)
We have
in \({\text {End}}\hspace{1.0pt}{\mathbb {C}}^{{\hspace{0.5pt}}n|n}\,\). Here the square brackets indicate the supercommutator. We will also consider each \(E_{{\hspace{0.5pt}}ij}\) as an element of the Lie superalgebra \(\mathfrak {gl}_{\,n|n}\,\). The special linear Lie superalgebra \(\mathfrak {sl}_{\,n|n}\) is the subalgebra of \(\mathfrak {gl}_{\,n|n}\) defined as the kernel of the function \(\textrm{str}\,\). The centre of \(\mathfrak {gl}_{\,n|n}\) is spanned by the element
The subalgebra \(\mathfrak {sl}_{\,n|n}\) contains this element. The quotient of the Lie superalgebra \(\mathfrak {sl}_{\,n|n}\) by the one-dimensional subspace spanned by this element is denoted by \(\mathfrak {psl}_{\,n|n}\,\). According to [8] the Lie superalgebra \(\mathfrak {psl}_{\,n|n}\) is simple if and only if \(n>1\,\). By using (9) we obtain that the Lie bracket on \(\mathfrak {psl}_{\,1|1}\) is just zero.
By (9) our \(\eta \) is also an involutive automorphism of the Lie superalgebra \(\mathfrak {gl}_{\,n|n}\,\). This automorphism preserves its subalgebra \(\mathfrak {sl}_{\,n|n}\,\). The queer Lie superalgebra \(\mathfrak {q}_{{\hspace{0.5pt}}n}\) is the fixed point subalgebra of \(\mathfrak {gl}_{\,n|n}\) by \(\eta \,\).
3 Theta Functions
Fix any complex number \(\tau \) with a positive imaginary part. For any real numbers a and b consider the Hermite theta function
The above series converges to a holomorphic function of the complex variable \(u\,\). All zeroes of this function are simple and form the subset
see [7, pp. 196–199]. The numbers a and b here are called characteristics. For \(a=b=0\) the series (10) is the Jacobi theta function. It follows from (10) that
By changing m to \(m+1\) in (10) and by using the first equation in (11) we obtain
respectively. Further, by changing m to \(-{\hspace{0.5pt}}m\) in (10) we obtain the parity relation
Now let g and h be any integers not simultaneously divisible by \(2{\hspace{0.5pt}}n\) and by n, respectively. Consider the function of the complex variable u
This function is meromorphic. It has simple poles at every point on the lattice
Due to the chosen normalisation the residue of the pole of \(\varphi _{{\hspace{0.5pt}}g,h}(u)\) at \(u=0\) is \(1\,\).
It immediately follows from the relations (12) that
Thus \(\varphi _{{\hspace{0.5pt}}g,h}(u)\) depends on the integers g and h only modulo \(2{\hspace{0.5pt}}n\) and n, respectively. From now on g and h will run not through \(\mathbb {Z}\,\), but through the additive groups \(\mathbb {Z}_{{\hspace{0.5pt}}2{\hspace{0.5pt}}n}=\mathbb {Z}{\hspace{0.5pt}}/{\hspace{0.5pt}}2{\hspace{0.5pt}}n\,\mathbb {Z}\) and \(\mathbb {Z}_{{\hspace{0.5pt}}n}=\mathbb {Z}{\hspace{0.5pt}}/{\hspace{0.5pt}}n\,\mathbb {Z}\), respectively.
Let \(\varepsilon =e^{\,\pi {\hspace{0.5pt}}{\mathrm i}{\hspace{0.5pt}}/n}\) be a primitive root of unity of the \(2{\hspace{0.5pt}}n\,\). Direct calculation using (11) yields the periodicity properties
Another direct calculation using (12) and (13) yields the parity relation
4 Commuting Automorphisms
Consider the following two elements of the algebra \({\text {End}}\hspace{1.0pt}{\mathbb {C}}^{{\hspace{0.5pt}}n|n}\,\),
and
These elements are invertible and of \(\,\mathbb {Z}_2{\hspace{0.5pt}}\)-degree zero. They satisfy the relations
By (8) we get
Let us define two automorphisms \(\alpha \) and \(\beta \) of the algebra \({\text {End}}\hspace{1.0pt}{\mathbb {C}}^{{\hspace{0.5pt}}n|n}\) by setting
for \(X\in {\text {End}}\hspace{1.0pt}{\mathbb {C}}^{{\hspace{0.5pt}}n|n}\,\). These automorphisms commute and are of degrees \(2{\hspace{0.5pt}}n\) and n, respectively. Here \(\alpha {\hspace{0.5pt}}(X)=\beta {\hspace{0.5pt}}(X)=X\) if and only if X is a linear combination of
Since the elements A and B are of \(\mathbb {Z}_2{\hspace{0.5pt}}\)-degree \(0\,\), for each \(X\in {\text {End}}\hspace{1.0pt}{\mathbb {C}}^{{\hspace{0.5pt}}n|n}\) we have
Let us now regard \(\alpha \) and \(\beta \) as automorphisms of the Lie superalgebra \(\mathfrak {gl}_{\,n|n}\,\). It follows from the first equation in (16) that \(\alpha ^{\,2{\hspace{0.5pt}}n}=\beta ^{\,n}=1{\hspace{0.5pt}}\). The eigenvalues of \(\alpha \) and \(\beta \) are \(\varepsilon ^{\,g}\) and \(\varepsilon ^{\,2{\hspace{0.5pt}}h}\) where g and h range over \(\mathbb {Z}_{{\hspace{0.5pt}}2{\hspace{0.5pt}}n}\) and \(\mathbb {Z}_{{\hspace{0.5pt}}n}\), respectively. Let \(\mathfrak {gl}_{\,n|n}^{{\hspace{0.5pt}}\,g,h}\) be the joint eigenspace of \(\alpha \) and \(\beta \) corresponding to \(\varepsilon ^{\,g}\) and \(\varepsilon ^{\,2{\hspace{0.5pt}}h}\,\). By (18)
Consider the Casimir element of the tensor square \(\mathfrak {gl}_{\,n|n}\!\otimes \mathfrak {gl}_{\,n|n}\)
This element is invariant by \(\mathfrak {gl}_{\,n|n}\) as for any indices k, l by using (9) we get
Note that
It follows from (20) that the Casimir element t is invariant by both \(\alpha {\hspace{0.5pt}}\otimes \alpha \) and \(\beta \otimes \beta \,\). Therefore t belongs to the direct sum of subspaces
Let \(t_{{\hspace{0.5pt}}g,h}\) be the projection of element t to the direct summand (22). By (17),(21)
Further, let \(\sigma \) be the linear transformation of \(\mathfrak {gl}_{\,n|n}\!\otimes \mathfrak {gl}_{\,n|n}\) defined by setting
for homogeneous elements X and \(Y\,\). By the definition of t we have \(\sigma (t)=t\,\). So
We do not need explicit formulas for every projection \(t_{{\hspace{0.5pt}}g,h}\,\). We only note that
where
5 Classical Yang–Baxter Equation
The central element 1 of the Lie superalgebra \(\mathfrak {gl}_{\,n|n}\) is contained in the eigenspace \(\mathfrak {gl}_{\,n|n}^{{\hspace{0.5pt}}\,0,0}\,\). Therefore for any \((g,h)\ne (0,0)\) the element \(t_{{\hspace{0.5pt}}g,h}\) can be identified with its image in \(\mathfrak {pgl}_{\,n|n}\!\otimes {\hspace{0.5pt}}\mathfrak {pgl}_{\,n|n}\,\). By using this identification, introduce a function of u
Observe that the function \(s{\hspace{0.5pt}}(u)\) takes all its values in the subspace
Changing the summation indices g, h to \(-g,-h\), respectively, and then using (15), (24) proves that \(s{\hspace{0.5pt}}(u)\) indeed satisfies the condition (2) for \(\mathfrak {g}=\mathfrak {psl}_{\,n|n}\,\). Using (15), (23) proves that \(s{\hspace{0.5pt}}(u)\) satisfies (3). Here we regard \(\eta \) as an automorphism of the Lie algebra \(\mathfrak {psl}_{\,n|n}\,\). The automorphisms \(\alpha \) and \(\beta \) of \(\mathfrak {gl}_{\,n|n}\) preserve \(\mathfrak {sl}_{\,n|n}\) and descend to \(\mathfrak {psl}_{\,n|n}\) too. By the periodicity properties (14) of the function \(\varphi _{g,h}(u)\)
and
By (25) the image in \(\mathfrak {pgl}_{\,n|n}\otimes \,\mathfrak {pgl}_{\,n|n}\) of the element \(t_{\,0,0}\) is zero. Hence the image of the element t is
Denote this image by \(p\,\). The residue of the function \(s{\hspace{0.5pt}}(u)\) at \(u=0\) equals \(p\,\).
Now consider the function \(r{\hspace{0.5pt}}(u,v)\) as defined by (4). We already observed in Sect. 1 that (2) and (3) imply the antisymmetry property (1). Let us show that \(r{\hspace{0.5pt}}(u,v)\) is indeed a solution of the classical Yang–Baxter equation for \(\mathfrak {g}=\mathfrak {psl}_{\,n|n}\,\). We will employ general arguments from [4, Sect. 5].
First observe that due to (4) and to the first equalities in (26) and (27) we get
Similarly, due to (5) and to the second equalities in (26) and (27) we get
Let \(f{\hspace{0.5pt}}(u,v,w)\) be the left-hand side of the classical Yang–Baxter equation for our \(r{\hspace{0.5pt}}(u,v)\) with \(\mathfrak {g}=\mathfrak {psl}_{\,n|n}\,\). Since \(\alpha \) and \(\beta \) are Lie algebra automorphisms, we have
and
Choose any values of v and w such that \(v-w,v+w\notin \mathcal {L}\,\). We will prove that then \(f{\hspace{0.5pt}}(u,v,w)\) is a holomorphic function of u in the whole \({\mathbb {C}}\,\). By (29) and (30) this function is bounded and hence a constant. This constant is then an element of \(\mathfrak {psl}_{\,n|n}\!\otimes {\hspace{0.5pt}}\mathfrak {psl}_{\,n|n}\!\otimes {\hspace{0.5pt}}\mathfrak {psl}_{\,n|n}\) invariant by \(\alpha \) and \(\beta \) applied in the first tensor factor. However the only element of \(\mathfrak {psl}_{\,n|n}\) invariant by both \(\alpha \) and \(\beta \) is zero. Therefore our function must be zero.
If \(u\pm v\in \mathcal {L}\), then \(u\pm w\not \in \mathcal {L}\) since \(v\pm w\not \in \mathcal {L}\,\). Then the function \(r_{{\hspace{0.5pt}}13}(u,w)\) has no pole. Consider the first two of the three summands of \(f{\hspace{0.5pt}}(u,v,w)\,\). By the definition (4) their sum can be written as
By multiplying (31) by \(u-v\) and then setting \(u=v\) we get
Here we employ (20) and the definition of p as the image of t in \(\mathfrak {pgl}_{\,n|n}\!\otimes {\hspace{0.5pt}}\mathfrak {pgl}_{\,n|n}\) . Therefore (31) has no pole at \(u-v=0\,\). It now follows from (26), (27), (28) that the summand (31) has no pole whenever \(u-v\in \mathcal {L}\,\).
By multiplying the summand (32) by \(u+v\) and then setting \(u=-\,v\) we get
So (32) has no pole at \(u+v=0\,\). It follows from (26), (27), (28) that (32) has no pole whenever \(u+v\in \mathcal {L}\,\).
Thus the function \(f{\hspace{0.5pt}}(u,v,w)\) has no pole whenever \(u\pm v\in \mathcal {L}\,\). Further, by using the antisymmetry property (1) the function \(-{\hspace{0.5pt}}f{\hspace{0.5pt}}(u,v,w)\) can be written as
Similarly to the above argument we can show that the latter function has no pole when \(u\pm w\in \mathcal {L}\,\). So \(r{\hspace{0.5pt}}(u,v)\) is a solution of the classical Yang–Baxter equation.
Following [2, 3, 6, 10] it would be interesting to find a solution of the quantum Yang–Baxter equation corresponding to our \(r{\hspace{0.5pt}}(u,v)\,\). For the rational solutions of the classical equation constructed in [11] this was already done in the same work.
References
Avan, J.: Graded Lie algebras in the Yang–Baxter equation. Phys. Lett. B 245, 491–496 (1990)
Baxter, R.: One-dimensional anisotropic Heisenberg chain. Ann. Phys. 70, 323–337 (1972)
Belavin, A.: Dynamical symmetry of integrable quantum systems. Nucl. Phys. B 180, 189–200 (1981)
Belavin, A., Drinfeld, V.: Solutions of the classical Yang–Baxter equation for simple Lie algebras. Funct. Anal. Appl. 16, 159–180 (1982)
Belavin, A., Drinfeld, V.: Classical Yang–Baxter equation for simple Lie algebras. Funct. Anal. Appl. 17, 220–221 (1983)
Cherednik, I.: On the properties of factorized \(S\) matrices in elliptic functions. Sov. J. Nucl. Phys. 36, 320–324 (1983)
Hurwitz, A., Courant, R.: Funktionentheorie. Springer, Berlin (1929)
Kac, V.: Lie superalgebras. Adv. Math. 26, 8–96 (1977)
Leites, D., Serganova, V.: Solutions of the classical Yang–Baxter equation for simple superalgebras. Theor. Math. Phys. 58, 16–24 (1984)
Matushko, M., Zotov, A.: Anisotropic spin generalization of elliptic Macdonald-Ruijsenaars operators and \(R\)-matrix identities. Ann. Henri Poincaré 24, 3373–3419 (2023)
Nazarov, M.: Yangians of the strange Lie superalgebras. Lect. Notes Math. 1510, 90–97 (1992)
Nazarov, M.: Yangian of the queer Lie superalgebra. Commun. Math. Phys. 208, 195–223 (1999)
Acknowledgements
I thank Evgeny Sklyanin for helpful discussions on the theory of elliptic functions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Nikolai Kitanine.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Nazarov, M. An Elliptic Solution of the Classical Yang–Baxter Equation Associated with the Queer Lie Superalgebra. Ann. Henri Poincaré (2024). https://doi.org/10.1007/s00023-024-01449-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00023-024-01449-8