Summary
Consider the complex matrix Lie superalgebra \( \mathfrak{g}\mathfrak{l}_{\left. N \right|N} \) with the standard generators E ij where i, j = ±1, . . . , ± N. Define an involutive automorphism η of \( \mathfrak{g}\mathfrak{l}_{\left. N \right|N} \) by η(E ij) = E −i,−j . The queer Lie superalgebra qN is the fixed point subalgebra in \( \mathfrak{g}\mathfrak{l}_{\left. N \right|N} \) relative to η. Consider the twisted polynomial current Lie superalgebra
. The enveloping algebra U(\( \mathfrak{g} \) ) of the Lie superalgebra g has a deformation, called the Yangian of qN. For each M = 1,2, . . . , denote by A M N the centralizer of qM ⊂ q N+M in the associative superalgebra U(q N+M). In this article we construct a sequence of surjective homomorphisms U(qN) ← A 1 N ← A 2 N ← . . . . We describe the inverse limit of the sequence of centralizer algebras A 1 N , A 2 N , . . . in terms of the Yangian of qN.
To Professor Anthony Joseph on the occasion of his 60th birthday
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References
V. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math.Dokl. 32 (1985), 254–258.
V. Drinfeld, A new realization of Yangians and quantized affine algebras, Soviet Math.Dokl. 36 (1988), 212–216.
T. Józefiak, Semisimple superalgebras, Lecture Notes Math., Vol. 1352 (1988), 96–113.
S. Lang, Algebra, Addison-Wesley, Reading MA, 1965.
A. Molev, Finite-dimensional irreducible representations of twisted Yangians, J. Math. Phys. 39 (1998), 5559–5600.
J. Milnor and J. Moore, On the structure of Hopf algebras, Ann. of Math. 81 (1965), 211–264.
A. Molev and G. Olshanski, Centralizer construction for twisted Yangians, Selecta Math. 6 (2000), 269–317.
M. Nazarov, Young’s symmetrizers for projective representations of the symmetric group, Adv.Math. 127 (1997), 190–257.
M. Nazarov, Capelli identities for Lie superalgebras, Ann. Scient. Éc.Norm. Sup. 30 (1997), 847–872.
M. Nazarov, Yangian of the queer Lie superalgebra, Commun. Math. Phys. 208 (1999), 195–223.
M. Nazarov, Representations of twisted Yangians associated with skew Young diagrams, Selecta Math. 10 (2004), 71–129.
G. Olshanski, Extension of the algebra U(g) for infinite-dimensional classical Lie algebras g, and the Yangians Y (gl(m)), Soviet Math. Dokl. 36 (1988), 569–573.
G. Olshanski, Representations of infinite-dimensional classical groups, limits of enveloping algebras, and Yangians, Adv. Soviet Math. 2 (1991), 1–66.
I. Penkov, Characters of typical irreducible finite-dimensional q(n)-modules, Funct. Anal. Appl. 20 (1986), 30–37.
A. Sergeev, The centre of enveloping algebra for Lie superalgebra Q(n, ℂ), Lett.Math. Phys. 7 (1983), 177–179.
A. Sergeev, The tensor algebra of the identity representation as a module over the Lie superalgebras GL(n, m) and Q(n), Math. Sbornik 51 (1985), 419–427.
T. Wall, Graded Brauer groups, J. Reine Angew. Math. 213 (1964), 187–199.
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Nazarov, M., Sergeev, A. (2006). Centralizer construction of the Yangian of the queer Lie superalgebra. In: Bernstein, J., Hinich, V., Melnikov, A. (eds) Studies in Lie Theory. Progress in Mathematics, vol 243. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4478-4_17
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DOI: https://doi.org/10.1007/0-8176-4478-4_17
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