Quantum walled Brauer-Clifford superalgebras.

*(English)*Zbl 1342.17002It is an interesting idea to get a new family of superalgebras by using quantum deformation theory and Schur-Weyl duality theory of the representation theory. By duality theory, one algebra appears as the centralizer of the other acting on a common representation space. Many important algebras have been constructed as centralizer algebras in this way. For example, the group algebra \(\mathbb{C}S_k\) of the symmetric group \(S_k\) appears as the centralizer of the \(gl(n)\)-action on \(V^{\otimes k}\) where \(V=C^n\) in the natural representation of the general linear Lie algebra \(gl(n)\) and Hecke-Clifford superalgebras are the centralizer algebras of the action of corresponding Lie (super) algebras or Quantum (super) algebras on the tensor powers of their natural representations.

There are further generalization of Schur-Weyl duality on mixed tensor powers. Let \(V^{r,s}=V^{\otimes r}\otimes(V^*)^{\otimes S}\) be the mixed tensor space of the natural representation \(V\) of \(gl(n)\) and its dual space \(V^*\). The centralizer algebra of the \(gl(n)\)-action on \(V^{r,s}\) is the walled Brauer algebra \(B_{r,s}(n)\). By replacing \(gl(n)\) by the quantum enveloping algebra \(U_q(gl(n))\) and \(V=C^n\) by \(V_q=C(q)^n\), we obtain as the centralizer algebra the quantum walled Brauer algebra. Super versions of the above constructions have been investigated in [C. L. Shader and D. Moon, Commun. Algebra 30, No. 2, 839–857 (2002; Zbl 1035.17013); ibid. 35, No. 3, 781–806 (2007; Zbl 1155.17005)] with the following substitution: Replace \(gl(n)\) by \(gl(m/n)\), \(C^n\) by \(C^{(m/n)}\), \(U_q(gl(n))\) by \(U_q(gl(m/n))\) and \(C(q)^n\) by \(C(q)^{(m/n)}\).

The Lie superalgebra \(q(n)\) is commonly referred to as the queer Lie superalgebra. Its natural representation is the superspace \(V=C^{(m/n)}\). The corresponding centralizer algebra \(\text{End}_{q(n)}(V^{\otimes r})\) was studied by A. N. Sergeev in [Math. USSR, Sb. 51, 419–427 (1985); translation from Mat. Sb., Nov. Ser. 123(165), No. 3, 422–430 (1984; Zbl 0573.17002)], and it is often referred to as Sergeev algebra.

Olshanskii introduced the quantum queer superalgebra \(U_q(q(n))\) and established an analogue of Schur-Weyl duality. He showed that there is a subjective algebra homomorphism \(\rho^r_{n,q}: HC_r(q)\to \text{End}_{q(n)}(V^{\otimes r})\) where \(HC_r(q)\) is the Hecke-Clifford superalgebra, a quantum version of the Sergeev algebra.

Moreover, \(\rho^r_{n,q}\) is an isomorphism when \(n\geq r\). Jung and Kang considered a super version of the walled Brauer algebra. For the mixed tensor space \(V^{r,s}=V^{\otimes r}\otimes(V^*)^{\otimes S}\). They introduced two versions of the Brauer-Clifford superalgebra. The first is constructed using \((r,s)\)-superdiagrams, and the second is defined by generators and relations. They showed that these two definitions are equivalent and that there is a subjective algebra homomorphism \(\rho^{r,s}_n:BC_{r,s}\to \text{End}_{q(n)}(V^{r,s})\), which is an isomorphism whenever \(n\geq r+s\).

The purpose of this paper is to combine the constructions of Olshanskii and Jung and Kang to determine the centralizer algebra of the \(U_q(q(n))\)-action of the mixed space \(V_q^{r,s}=V_q^{\otimes r}\otimes(V_q^*)^{\otimes S}\).

The paper consists of five sections with a hard calculations mixed between Knots, diagrams and algebra. One must read the paper more than one time. In section one and two they give the walled Brauer-Clifford superalgebra and \((r,s)\)-bead diagram algebra. In section three and four they defined and studied the quantum walled Brauer-Clifford superalgebra and the \((r,s)\)-bead tangle algebras

In section five theory introduce and define the notion of \(q\)-Schur superalgebra of type \(Q\) and its dual.

I think that the calculations and the algebraic structure in this paper open a way to more applications of Schur-Weyl duality and quantum deformation.

There are further generalization of Schur-Weyl duality on mixed tensor powers. Let \(V^{r,s}=V^{\otimes r}\otimes(V^*)^{\otimes S}\) be the mixed tensor space of the natural representation \(V\) of \(gl(n)\) and its dual space \(V^*\). The centralizer algebra of the \(gl(n)\)-action on \(V^{r,s}\) is the walled Brauer algebra \(B_{r,s}(n)\). By replacing \(gl(n)\) by the quantum enveloping algebra \(U_q(gl(n))\) and \(V=C^n\) by \(V_q=C(q)^n\), we obtain as the centralizer algebra the quantum walled Brauer algebra. Super versions of the above constructions have been investigated in [C. L. Shader and D. Moon, Commun. Algebra 30, No. 2, 839–857 (2002; Zbl 1035.17013); ibid. 35, No. 3, 781–806 (2007; Zbl 1155.17005)] with the following substitution: Replace \(gl(n)\) by \(gl(m/n)\), \(C^n\) by \(C^{(m/n)}\), \(U_q(gl(n))\) by \(U_q(gl(m/n))\) and \(C(q)^n\) by \(C(q)^{(m/n)}\).

The Lie superalgebra \(q(n)\) is commonly referred to as the queer Lie superalgebra. Its natural representation is the superspace \(V=C^{(m/n)}\). The corresponding centralizer algebra \(\text{End}_{q(n)}(V^{\otimes r})\) was studied by A. N. Sergeev in [Math. USSR, Sb. 51, 419–427 (1985); translation from Mat. Sb., Nov. Ser. 123(165), No. 3, 422–430 (1984; Zbl 0573.17002)], and it is often referred to as Sergeev algebra.

Olshanskii introduced the quantum queer superalgebra \(U_q(q(n))\) and established an analogue of Schur-Weyl duality. He showed that there is a subjective algebra homomorphism \(\rho^r_{n,q}: HC_r(q)\to \text{End}_{q(n)}(V^{\otimes r})\) where \(HC_r(q)\) is the Hecke-Clifford superalgebra, a quantum version of the Sergeev algebra.

Moreover, \(\rho^r_{n,q}\) is an isomorphism when \(n\geq r\). Jung and Kang considered a super version of the walled Brauer algebra. For the mixed tensor space \(V^{r,s}=V^{\otimes r}\otimes(V^*)^{\otimes S}\). They introduced two versions of the Brauer-Clifford superalgebra. The first is constructed using \((r,s)\)-superdiagrams, and the second is defined by generators and relations. They showed that these two definitions are equivalent and that there is a subjective algebra homomorphism \(\rho^{r,s}_n:BC_{r,s}\to \text{End}_{q(n)}(V^{r,s})\), which is an isomorphism whenever \(n\geq r+s\).

The purpose of this paper is to combine the constructions of Olshanskii and Jung and Kang to determine the centralizer algebra of the \(U_q(q(n))\)-action of the mixed space \(V_q^{r,s}=V_q^{\otimes r}\otimes(V_q^*)^{\otimes S}\).

The paper consists of five sections with a hard calculations mixed between Knots, diagrams and algebra. One must read the paper more than one time. In section one and two they give the walled Brauer-Clifford superalgebra and \((r,s)\)-bead diagram algebra. In section three and four they defined and studied the quantum walled Brauer-Clifford superalgebra and the \((r,s)\)-bead tangle algebras

In section five theory introduce and define the notion of \(q\)-Schur superalgebra of type \(Q\) and its dual.

I think that the calculations and the algebraic structure in this paper open a way to more applications of Schur-Weyl duality and quantum deformation.

Reviewer: Ahmed Hegazi (Mansoura)

##### MSC:

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

05E10 | Combinatorial aspects of representation theory |

20G43 | Schur and \(q\)-Schur algebras |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

##### Keywords:

quantum walled Brauer-Clifford superalgebra; Hecke algebra; knot theory; quantum deformation; bead tangle algebra; centralizer algebra; queer Lie superalgebra Q-Schur superalgebra**OpenURL**

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