Centralizer construction of the Yangian of the queer Lie superalgebra

Abstract

Summary. Consider the complex matrix Lie superalgebra gl N|N with the standard generators Eij where i , j = ±1 , . . . , ± N . Define an involutive automorphism η of gl N|N by η (Eij) = E−i,−j . The queer Lie superalgebra qN is the fixed point subalgebra in gl N|N relative to η . Consider the twisted polynomial current Lie superalgebra g = { X(t) ∈ gl N|N [t] : η (X(t)) = X(−t) } .

On the finite W-algebra for the Lie superalgebra Q(N) in the non-regular case

“…Proof. It follows from [9] (Proposition 1.6) that one can define an anti-homomorphism ω : Y (Q(n)) → U(Q(n)) by…”

On Kostant's theorem for the Lie superalgebra Q(n)

“…New generators and relations. We will need the following realization of Q(n) given by M. Nazarov and S. Sergeev in [21]. Let the indices i, j run through −n, .…”

“…Super-Yangian Y (Q(n)) was studied by M. Nazarov and A. Sergeev [21]. Recall that Y (Q(n)) is the associative unital superalgebra over C with the countable set of generators T…”

“…Using these generators, we prove that the associated graded algebra Gr K W χ with respect to the Kazhdan filtration is isomorphic to S(g χ ) (the symmetric algebra of the annihilator g χ of χ in g) (Conjecture 2.8 and Corollary 4.9). Furthermore, we prove that W χ is isomorphic to a quotient of the super-Yangian of Q(1) defined by M. Nazarov and A. Sergeev ([20,21]) (Theorem 6.1). Finally, we construct n even and n odd generators in W χ , such that all even generators commute and generate the polynomial subalgebra of rank n in W χ , and the commutators of odd generators lie in the center of W χ (Theorem 5.13).…”

An Analogue of the Perelomov-Popov Formula for the Lie Superalgebra q(N)

“…These are the Casimir elements for the Lie superalgebra q(N) as introduced in [3]. It was shown in [4] that these elements generate the center of U(q(N)) .…”