Abstract
In this paper, we introduce the Harish-Chandra homomorphism for the quantum superalgebra \(\mathrm {U}_q({\mathfrak {g}})\) associated with a simple basic Lie superalgebra \({\mathfrak {g}}\) and give an explicit description of its image. We use it to prove that the center of \(\mathrm {U}_q({\mathfrak {g}})\) is isomorphic to a subring of the ring \(J({\mathfrak {g}})\) of exponential super-invariants in the sense of Sergeev and Veselov, establishing a Harish-Chandra type theorem for \(\mathrm {U}_q({\mathfrak {g}})\). As a byproduct, we obtain a basis of the center of \(\mathrm {U}_q({\mathfrak {g}})\) with the aid of quasi-R-matrix.
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Notes
In general, the center of the Lie superalgebra and quantum superalgebra is \({\mathbb {Z}}_2\)-graded [8, Sect. 2.2]. Similar to the basic Lie superalgebra case, the center of \(\mathrm {U}_q({\mathfrak {g}})\) consists of only even elements. However, the center contains odd part is also interesting in some aspects; e.g., the skew center of generalized quantum groups [3].
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Acknowledgements
We would like to express our debt to Shun-Jen Cheng, Hiroyuki Yamane, Hechun Zhang, and Ruibin Zhang for many insightful discussions. We are very grateful to referees for their insightful comments which helped us to improve the paper considerably. This paper was partially written up during the second author visit to Institute of Geometry and Physics, USTC, in Summer 2021, from which we gratefully acknowledge the support and excellent working environment where most of this work was completed. Y. Wang is partially supported by the National Natural Science Foundation of China (Nos. 11901146 and 12071026), and the Fundamental Research Funds for the Central Universities JZ2021HGTB0124. Y. Ye is partially supported by the National Natural Science Foundation of China (Nos. 11971449 and 12131015).
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Appendices
Appendix A. Dynkin Diagrams in Distinguished Root Systems
The Dynkin diagrams in the distinguished root systems of a simple basic Lie superalgebra of type A-G are listed below, where r is the number of nodes and s is the element of \(\tau \). Note that the form of Dynkin diagrams in the distinguished root systems is quite uniform in the literature.
\(\boxed {A(m,n) ~{\mathbf{case }}:}\) Let \({\mathfrak {h}}^*\) be a vector space spanned by \(\{\varepsilon _i-\varepsilon _{i+1},\varepsilon _{m+1}-\delta _1 ,\delta _j-\delta _{j+1}| 1\leqslant i\leqslant m, 1\leqslant j\leqslant n \}\) satisfies
We equip the dual \({\mathfrak {h}}^*\) with a bilinear form \((\cdot ,\cdot )\) such that
The distinguished fundamental system \(\Pi =\{\alpha _1,\ \ldots ,\ \alpha _{m+n+1}\}\) is given by
The Dynkin diagram associated with \(\Pi \) is depicted as follows:
In this case \(r=m+n+1\), \(s=m+1\). The distinguished positive system \(\Phi ^+=\Phi _{{\bar{0}}}^+\cup \Phi _{{\bar{1}}}^+\) corresponding to the distinguished Borel subalgebra for A(m, n) is
The Weyl group \(W\cong {\mathfrak {S}}_{m+1}\times {\mathfrak {S}}_{n+1}\).
\(\boxed {B(m,n) ~\mathbf{case: }}\) Let \({\mathfrak {h}}^*\) be a vector space with basis \(\{\varepsilon _i, \delta _j| 1\leqslant i\leqslant m, 1\leqslant j\leqslant n\}\). We equip the dual \({\mathfrak {h}}^*\) with a bilinear form \((\cdot ,\cdot )\) such that
The distinguished fundamental system \(\Pi =\{\alpha _1,\ \ldots ,\ \alpha _{m+n}\}\) is given by
The Dynkin diagram associated with \(\Pi \) is depicted as follows:
In this case \(r=m+n\), \(s=n+1\). The distinguished positive system \(\Phi ^+=\Phi ^+_{{\bar{0}}}\cup \Phi _{{\bar{1}}}^+\) corresponding to the distinguished Borel subalgebra is
where \(1\leqslant i<j\leqslant n, 1\leqslant k<l\leqslant m, 1\leqslant p\leqslant n, 1\leqslant q\leqslant m\). The Weyl group \(W\cong ({\mathfrak {S}}_{n}\ltimes {\mathbb {Z}}_2^n)\times ({\mathfrak {S}}_{m}\ltimes {\mathbb {Z}}_2^m)\).
\(\boxed {B(0,n) ~\mathbf{case: }}\) Let \({\mathfrak {h}}^*\) be a vector space with basis \(\{\delta _i|1\leqslant i\leqslant n \}\). We equip the dual \({\mathfrak {h}}^*\) with a bilinear form \((\cdot ,\cdot )\) such that
The distinguished fundamental system \(\Pi =\{\alpha _1,\ \ldots ,\ \alpha _{n}\}\) is given by
The Dynkin diagram associated with \(\Pi \) is depicted as follows:
In this case, \(r=s=n\). The distinguished positive system \(\Phi ^+=\Phi ^+_{{\bar{0}}}\cup \Phi _{{\bar{1}}}^+\) corresponding to the distinguished Borel subalgebra is
The Weyl group \(W\cong ({\mathfrak {S}}_{n}\ltimes {\mathbb {Z}}_2^n)\).
\(\boxed {C(n+1) ~\mathbf{case: }} \) Let \({\mathfrak {h}}^*\) be a vector space with basis \(\{\varepsilon ,\delta _i|1\leqslant i\leqslant n \}\). We equip the dual \({\mathfrak {h}}^*\) with a bilinear form \((\cdot ,\cdot )\) such that
The distinguished fundamental system \(\Pi =\{\alpha _1,\ \ldots ,\ \alpha _{n+1}\}\) is given by
The Dynkin diagram associated with \(\Pi \) is depicted as follows:
In this case \(r=n+1,s=1\). The distinguished positive system \(\Phi ^+=\Phi ^+_{{\bar{0}}}\cup \Phi _{{\bar{1}}}^+\) corresponding to the distinguished Borel subalgebra is
The Weyl group \(W\cong ({\mathfrak {S}}_{n}\ltimes {\mathbb {Z}}_2^n)\).
\(\boxed {D(m,n) ~\mathbf{case: }} \) Let \({\mathfrak {h}}^*\) be a vector space with basis \(\{\varepsilon _i, \delta _j| 1\leqslant i\leqslant m, 1\leqslant j\leqslant n\}\). We equip the dual \({\mathfrak {h}}^*\) with a bilinear form \((\cdot ,\cdot )\) such that
The distinguished fundamental system \(\Pi =\{\alpha _1,\ \ldots ,\ \alpha _{m+n}\}\) is given by
The Dynkin diagram associated with \(\Pi \) is depicted as follows:
In this case \(r=m+n\), \(s=n+1\). The distinguished positive system \(\Phi ^+=\Phi ^+_{{\bar{0}}}\cap \Phi _{{\bar{1}}}^+\) corresponding to the distinguished Borel subalgebra is
where \(1\leqslant i<j\leqslant n, 1\leqslant k<l\leqslant m, 1\leqslant p\leqslant n, 1\leqslant q\leqslant m\). The Weyl group \(W\cong ({\mathfrak {S}}_{n}\ltimes {\mathbb {Z}}_2^n)\times ({\mathfrak {S}}_{m}\ltimes {\mathbb {Z}}_2^{m-1} )\).
\(\boxed {D(2,1;\alpha ) ~\mathbf{case }:}\) Let \({\mathfrak {h}}^*\) be a vector space with basis \(\{\varepsilon _1, \varepsilon _2,\varepsilon _3\}\). We equip the dual \({\mathfrak {h}}^*\) with a bilinear form \((\cdot ,\cdot )\) with
The distinguished fundamental system
The Dynkin diagram associated with \(\Pi \) is depicted as follows:
In this case \(r=3\), \(s=1\). The distinguished positive system \(\Phi ^+=\Phi ^+_{{\bar{0}}}\cap \Phi _{{\bar{1}}}^+\) corresponding to the distinguished Borel subalgebra is
The Weyl group \(W\cong {\mathbb {Z}}_2^3\).
\(\boxed {F(4) ~{\mathbf{case }}:}\) Let \({\mathfrak {h}}^*\) be a vector space with basis \(\{\delta , \varepsilon _1, \varepsilon _2, \varepsilon _3\}\).We equip the dual \({\mathfrak {h}}^*\) with a bilinear form \((\cdot ,\cdot )\) such that
The distinguished fundamental system
The Dynkin diagram associated with \(\Pi \) is depicted as follows:
In this case \(r=4\), \(s=1\). The distinguished positive system \(\Phi ^+=\Phi ^+_{{\bar{0}}}\cap \Phi _{{\bar{1}}}^+\) corresponding to the distinguished Borel subalgebra is
The Weyl group \(W={\mathbb {Z}}_2\times ({\mathfrak {S}}_3\ltimes {\mathbb {Z}}_2^3)\).
\(\boxed {G(3) ~\mathbf{case }:}\) Let \({\mathfrak {h}}^*\) be a vector space with basis \(\{\delta , \varepsilon _1, \varepsilon _2\}\) and \(\varepsilon _3=-\varepsilon _1-\varepsilon _2\). We equip the dual \({\mathfrak {h}}^*\) with a bilinear form \((\cdot ,\cdot )\) such that
The distinguished fundamental system
The Dynkin diagram associated with \(\Pi \) is depicted as follows:
In this case \(r=3\), \(s=1\). The distinguished positive system \(\Phi ^+=\Phi ^+_{{\bar{0}}}\cap \Phi _{{\bar{1}}}^+\) corresponding to the distinguished Borel subalgebra is
The Weyl group \(W={\mathbb {Z}}_2\times D_6\), where \(D_6\) is the dihedral group of order 12.
Appendix B. Explicit Description of the Rings \(J_{\mathrm {ev}}({\mathfrak {g}})\)
Now we give the explicit description of the rings \(J_{\mathrm {ev}}({\mathfrak {g}})\) for quantum superalgebras, which is inspired by Sergeev and Veselov’s description for Lie superalgebras [42, Sects. 7, 8]. Let \(x_i={\mathbb {K}}_{-\varepsilon _i/2}\) and \(y_j={\mathbb {K}}_{-\delta _j/2}\) formally. First we need to review the rings \(J({\mathfrak {g}})\) for \({\mathfrak {g}}\) is of type A. Let
be the weights of \(\mathfrak {sl}_{m+1|n+1}\), where \(\gamma =\varepsilon _1+\cdots +\varepsilon _{m+1}-\delta _1-\cdots -\delta _{n+1}\) and \(x_i=e^{\varepsilon _i}, y_j=e^{\delta _j}\) for all possible i, j be the elements of the group ring of \({\mathbb {C}}[P_0]\). For convenience, we set \({\mathbb {C}}[x^{\pm },y^{\pm }]={\mathbb {C}}[x_1^{\pm 1},\cdots ,x_{m+1}^{\pm 1},y_1^{\pm 1},\cdots ,y_{n+1}^{\pm 1}],~ {\mathbb {Z}}[x^{\pm },y^{\pm }]={\mathbb {Z}}[x_1^{\pm 1},\cdots ,x_{m+1}^{\pm 1},y_1^{\pm 1},\cdots ,y_{n+1}^{\pm 1}]\) and then for \((m,n)\ne (1,1)\)
where
if \(a\notin {\mathbb {Z}}\);
and \({\mathbb {Z}}[x^{\pm 1},y^{\pm 1}]_0^{{\mathfrak {S}}_{m+1}\times {\mathfrak {S}}_{n+1}}\) is the quotient of the ring \({\mathbb {Z}}[x^{\pm 1},y^{\pm 1}]^{{\mathfrak {S}}_{m+1}\times {\mathfrak {S}}_{n+1}}\) by the ideal generated by \(x_1\cdots x_{m+1}-y_1\cdots y_{n+1}\).
\(J(A(n,n))=\mathop {\oplus }\limits _{i=0}^{n}J(A(n,n))_i\) for \(n\ne 1\), where for \(i\ne 0\)
and \(J(A(n,n))_0\) is the subring of \(J(\mathfrak {sl}_{n+1|n+1})_0\) consisting of elements of degree 0.
\(J(A(1,1))=\{c+(u-v)^2g |c\in {\mathbb {Z}}, g\in {\mathbb {Z}}[u,v] \}\) where \(u=\left( \frac{x_1}{x_2}\right) ^{\frac{1}{2}}+\left( \frac{x_2}{x_1}\right) ^{\frac{1}{2}},~ v=\left( \frac{y_1}{y_2}\right) ^{\frac{1}{2}}+\left( \frac{y_2}{y_1}\right) ^{\frac{1}{2}}\).
\(\boxed {A(m,n), m\ne n ~\mathbf{case: }}\) Define
and
Thus, \(J^{m|n}=\bigoplus \limits _{k\in {\mathbb {Z}}}J^{m|n}_{k}\).
For any element \(\lambda \in {\mathfrak {h}}^*\), we write \(\lambda =\mathop {\sum }\limits _{i=1}^{m+1}a_i\varepsilon _i+\mathop {\sum }\limits _{j=1}^{n+1}b_j\delta _j\), then we have
and
By direct computation, we know that
for some non-negative integers k, l. Then the algebra
for some non-negative integers k, l. So it can be viewed as a subalgebra of \(J({\mathfrak {g}})\) by \(\iota :J_{\mathrm {ev}}({\mathfrak {g}})\rightarrow J({\mathfrak {g}})\) with \({\mathbb {K}}_i\mapsto e^{-\alpha _i/2}\) and its image is coincide with \(\mathrm {Sch}(K_{\mathrm {ev}}({\mathfrak {g}}))\).
\(\boxed {A(n,n)~~ (n\ne 1)~\mathbf{case: }}\) In this case, we set
where \({\mathbb {Z}}[x^{\pm 1},y^{\pm 1}]_{0,0}\) is the quotient of the ring \({\mathbb {Z}}[x^{\pm 1},y^{\pm 1}]\) with degree 0 by the ideal \(I=\left\langle \frac{x_1\cdots x_{n+1}}{y_1\cdots y_{n+1}}-1\right\rangle \). Then we have
where \(\overrightarrow{x}=x_1x_2\cdots x_{n+1}\) and \(W={\mathfrak {S}}_{n+1}\times {\mathfrak {S}}_{n+1}\). It can be viewed as a subalgebra by \(\iota :J_{\mathrm {ev}}({\mathfrak {g}})\rightarrow J({\mathfrak {g}})\) with \({\mathbb {K}}_i\mapsto e^{-\alpha _i/2}\) and its image is coincide with \(\mathrm {Sch}(K_{\mathrm {ev}}({\mathfrak {g}}))\).
\(\boxed {A(1,1) ~\mathbf{case: }}\) We have \(J_{\mathrm {ev}}(A(1,1))=\left\{ c+(u-v)g\left| g\in {\mathbb {Z}}[u,v]\right. \right\} \) where \(u=\left( \frac{x_1}{x_2}\right) ^{\frac{1}{2}}+\left( \frac{x_2}{x_1}\right) ^{\frac{1}{2}},~ v=\left( \frac{y_1}{y_2}\right) ^{\frac{1}{2}}+\left( \frac{y_2}{y_1}\right) ^{\frac{1}{2}}\). And \(u-v={\mathbb {K}}_{1}+{\mathbb {K}}^{-1}_{1}-{\mathbb {K}}_{3}-{\mathbb {K}}_{3}^{-1}\in J_{\mathrm {ev}}(A(1,1))\), but \(u-v\notin J(A(1,1))\).
\(\boxed {B(m,n), m, n>0 ~\mathbf{case: }}\) We set \(\lambda =\mathop {\sum }\limits _{i=1}^{m}\lambda _i\varepsilon _i+\mathop {\sum }\limits _{j=1}^{n} \mu _j\delta _j\in {\mathfrak {h}}^*\), then in this case
So \(2\Lambda \cap {\mathbb {Z}}\Phi =2\Lambda \). Let \(u_i=x_i+x_i^{-1}\) and \(v_j=y_j+y_j^{-1}\) for all possible i, j, then we have \(J_{\mathrm {ev}}({\mathfrak {g}})=J({\mathfrak {g}})_0\oplus J({\mathfrak {g}})_{1/2}\), where
and
\(\boxed {B(0,n) ~\mathbf{case: }}\) In this case \(\Lambda ={\mathbb {Z}}\Phi =\Big \{\mathop {\sum }\limits _{j=1}^{n} \mu _j\delta _j\Big |\mu _j\in {\mathbb {Z}},~\forall j \Big \},\) so \(2\Lambda \cap {\mathbb {Z}}\Phi =2\Lambda \) and this algebra \(J_{\mathrm {ev}}({\mathfrak {g}})={\mathbb {Z}}[v_1,v_2,\cdots ,v_n]^{{\mathfrak {S}}_n}\), where the notation \(v_i\) are the same as above.
\(\boxed {C(n+1) ~\mathbf{case: }}\) In this case
and
So \(2\Lambda \cap {\mathbb {Z}}\Phi =\Big \{ \lambda \varepsilon +\mathop {\sum }\limits _{j=1}^{n} \mu _j\delta _j\Big |\lambda ,\mu _j\in 2{\mathbb {Z}},~\forall j \Big \}\) and the algebra
\(\boxed {D(m,n), m>1,n>0 ~\mathbf{case: }}\) Let \(\lambda =\mathop {\sum }\limits _{i=1}^{m}\lambda _i\varepsilon _i+\mathop {\sum }\limits _{j=1}^{n} \mu _j\delta _j\in {\mathfrak {h}}^*\) and \(u_i, v_j\) are as above, then
and
So
for some positive integer k. Thus the algebra \(J_{\mathrm {ev}}({\mathfrak {g}})\) is, respectively, equal to \(J({\mathfrak {g}})_0\oplus J({\mathfrak {g}})_{1/2}\) for \(m=2k\) and \(J({\mathfrak {g}})_0\) for \(m=2k+1\), where
and
\(\boxed {D(2,1,\alpha ) ~\mathbf{case: }}\) In this case,
So \(2\Lambda \cap {\mathbb {Z}}\Phi =2\Lambda \). Thus the algebra
where
and
\(\boxed {F(4) ~\mathbf{case: }}\) In this case,
and
So \(2\Lambda \cap {\mathbb {Z}}\Phi =2\Lambda \), and the algebra
where
and
with \(k=1,2\), and \(W={\mathbb {Z}}_2\times ({\mathfrak {S}}_3\ltimes {\mathbb {Z}}_2^3)\).
\(\boxed {G(3) ~\mathbf{case: }}\) In this case, \(\Lambda ={\mathbb {Z}}\Phi =\big \{ \lambda _1\varepsilon _1+\lambda _2\varepsilon _2+\mu \delta |\lambda _1,\lambda _2,\mu \in {\mathbb {Z}} \big \}\). So \(2\Lambda \cap {\mathbb {Z}}\Phi =2\Lambda \), and the algebra
where
and the notations \(u_i, v\) are the same as above.
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Luo, Y., Wang, Y. & Ye, Y. On the Harish-Chandra Homomorphism for Quantum Superalgebras. Commun. Math. Phys. 393, 1483–1527 (2022). https://doi.org/10.1007/s00220-022-04394-x
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DOI: https://doi.org/10.1007/s00220-022-04394-x