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The quantized walled Brauer algebra and mixed tensor space. (English) Zbl 1368.17017
Summary: In this paper we investigate a multi-parameter deformation \(\mathfrak B_{r,s}^n(a,\lambda,\delta)\) of the walled Brauer algebra which was previously introduced by R. Leduc [“A two-parameter version of the centralizer algebra of the mixed tensor representation of the general linear group and quantum general linear group.” Thesis, University of Wisconsin-Madison (1994)]. We construct an integral basis of \(\mathfrak B_{r,s}^n(a,\lambda,\delta)\) consisting of oriented tangles which is in bijection with walled Brauer diagrams. Moreover, we study a natural action of \(\mathfrak B_{r,s}^n(q)=\mathfrak B_{r,s}^n(q^{-1}-q,q^n,[n]_q)\) on mixed tensor space and prove that the kernel is free over the ground ring \(R\) of rank independent of \(R\). As an application, we prove one side of Schur-Weyl duality for mixed tensor space: the image of \(\mathfrak B_{r,s}^n(q)\) in the \(R\)-endomorphism ring of mixed tensor space is, for all choices of \(R\) and the parameter \(q\), the endomorphism algebra of the action of the (specialized via the Lusztig integral form) quantized enveloping algebra \(U\) of the general linear Lie algebra \(\mathfrak{gl}_n\) on mixed tensor space. Thus, the \(U\)-invariants in the ring of \(R\)-linear endomorphisms of mixed tensor space are generated by the action of \(\mathfrak B_{r,s}^n(q)\).

17B37 Quantum groups (quantized enveloping algebras) and related deformations
20C08 Hecke algebras and their representations
20G43 Schur and \(q\)-Schur algebras
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[1] Benkart, G; Chakrabarti, M; Halverson, T; Leduc, R; Lee, C; Stroomer, J, Tensor product representations of general linear groups and their connections with Brauer algebras, J. Algebra, 166, 529-567, (1994) · Zbl 0815.20028
[2] Birman, J; Wenzl, H, Braids, link polynomials and a new algebra, Trans. Am. Math. Soc., 313, 249-273, (1989) · Zbl 0684.57004
[3] Brundan, J; Stroppel, C, Highest weight categories arising from khovanov’s diagram algebra II: koszulity, Transform. Groups, 15, 1-45, (2010) · Zbl 1205.17010
[4] Brundan, J; Stroppel, C, Highest weight categories arising from khovanov’s diagram algebra I: cellularity, Mosc. Math. J., 11, 685-722, (2011) · Zbl 1275.17012
[5] Brundan, J; Stroppel, C, Highest weight categories arising from khovanov’s diagram algebra III: category \(\mathcal{O}\), Representat. Theory, 15, 170-243, (2011) · Zbl 1261.17006
[6] Brundan, J; Stroppel, C, Highest weight categories arising from khovanov’s diagram algebra V: the general linear supergroup, J. Eur. Math. Soc., 14, 373-419, (2012) · Zbl 1243.17004
[7] Brundan, J; Stroppel, C, Gradings on walled Brauer algebras and khovanov’s arc algebra, Adv. Math., 231, 709-773, (2012) · Zbl 1326.17006
[8] Cox, A; Visscher, M, Diagrammatic Kazhdan-Lusztig theory for the (walled) Brauer algebra, J. Algebra, 340, 151-181, (2011) · Zbl 1269.20037
[9] Cox, A; Visscher, M; Doty, S; Martin, P, On the blocks of the walled Brauer algebra, J. Algebra, 320, 169-212, (2008) · Zbl 1196.20004
[10] DeConcini, C; Procesi, C, A characteristic free approach to invariant theory, Adv. Math., 21, 330-354, (1976) · Zbl 0347.20025
[11] Dipper, R; Donkin, S, Quantum GL_{\(n\)}, Proc. Lond. Math. Soc., 63, 165-211, (1991) · Zbl 0734.20018
[12] Dipper, R; Doty, S, Rational Schur algebras, Representat. Theory, 12, 58-82, (2008) · Zbl 1185.20052
[13] Dipper, R; James, G, Representations of Hecke algebras of general linear groups, Proc. Lond. Math. Soc., 52, 20-52, (1986) · Zbl 0587.20007
[14] Dipper, R; James, G, The \(q\)-Schur algebra, Proc. Lond. Math. Soc., 59, 23-50, (1989) · Zbl 0711.20007
[15] Dipper, R., Doty, S., Stoll, F.: Quantized mixed tensor space and Schur-Weyl duality. To appear in Algebra Number Theory · Zbl 1290.17012
[16] Donkin, S.: On Schur algebras and related algebras VI: some remarks on rational and classical Schur algebras. Preprint · Zbl 1334.20042
[17] Du, J; Parshall, B; Scott, L, Quantum Weyl reciprocity and tilting modules, Commun. Math. Phys., 195, 321-352, (1998) · Zbl 0936.16008
[18] Goodman, FM; Graber, J, Cellularity and the Jones basic construction, Adv. Appl. Math., 46, 312-362, (2011) · Zbl 1266.20005
[19] Green, JA, Polynomial representations of GL_{\(n\)}, (1980), Berlin
[20] Green, RM, \(q\)-Schur algebras as quotients of quantized enveloping algebras, J. Algebra, 185, 660-687, (1996) · Zbl 0862.17007
[21] Härterich, M, Murphy bases of generalized temperly-Lieb algebras, Arch. Math., 72, 337-345, (1999) · Zbl 0945.20006
[22] Freyd, P; Yetter, D; Hoste, J; Lickorish, WBR; Millett, K; Oceanu, A, A new polynomial invariant of knots and links, Bull. Am. Math. Soc., 12, 239-246, (1985) · Zbl 0572.57002
[23] Hong, J., Kang, S.: Introduction to Quantum Groups and Crystal Bases. American Math. Soc., Providence (2002) · Zbl 1134.17007
[24] Jantzen, J.C.: Lectures on Quantum Groups. American Math. Soc., Providence (1995) · Zbl 0842.17012
[25] Kassel, C, Quantum groups, (1995), New York · Zbl 0808.17003
[26] Kauffman, LH, An invariant of regular isotopy, Trans. Am. Math. Soc., 318, 417-471, (1990) · Zbl 0763.57004
[27] Kemper, G.: Invariants of Hopf algebras. The curves seminar at Queen’s, vol. XIII. In: Queen’s Papers in Pure and Applied Math., vol. 119, pp. 37-61 (2000) · Zbl 1243.17004
[28] Koike, K, On the decomposition of tensor products of the representations of classical groups: by means of universal characters, Adv. Math., 74, 57-86, (1989) · Zbl 0681.20030
[29] Kosuda, M; Murakami, J, Centralizer algebras of the mixed tensor representations of the quantum algebra \(U\)_{\(q\)}(gl(\(n\),ℂ)), Osaka J. Math., 30, 475-507, (1993) · Zbl 0806.17012
[30] Leduc, R.: A two-parameter version of the centralizer algebra of the mixed tensor representation of the general linear group and quantum general linear group. Thesis, University of Wisconsin-Madison (1994) · Zbl 1275.17012
[31] Lusztig, G, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Am. Math. Soc., 3, 257-296, (1990) · Zbl 0695.16006
[32] Morton, H.R., Wassermann, A.J.: A basis for the Birman-Wenzl algebra. Preprint 1989, updated January 2000. Available online at http://www.liv.ac.uk/ su14/papers/WM.ps.gz (2000) · Zbl 1205.17010
[33] Schur, I.: Über die rationalen Darstellungen der allgemeinen linearen Gruppe (1927). Reprinted in: Schur, I. (ed.) Gesammelte Abhandlungen III, pp. 68-85. Springer, Berlin (1973) · Zbl 0815.20028
[34] Tange, R.: A bideterminant basis for a reductive monoid. arXiv:1002.4642 (2010). Preprint · Zbl 1251.05179
[35] Turaev, V, Operator invariants of tangles and R-matrices, Izvestija AN SSSR Ser. Math., 53, 1073-1107, (1989)
[36] Wang, J; Koenig, S, Cyclotomic extensions of diagram algebras, Commun. Algebra, 36, 1739-1757, (2008) · Zbl 1148.16013
[37] Weimer, F.: A tangle analogon for Leduc’s deformation of the walled Brauer algebra. Diploma thesis, Universität Stuttgart (2005) · Zbl 0572.57002
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