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Trace and center of the twisted Heisenberg category

Michael Reeks

June 4, 2018

Michael Reeks Trace and center of the twisted Heisenberg category

Trace decategorification

The trace (or zeroth Hochschild homology) of a C-linear additive category C: Tr(C) :=

• ⊕x∈ob(C) EndC(X)
• Span{fg − gf },

where f and g run through all pairs of morphisms f : x → y and g : y → x with x, y ∈ Ob(C). If C is monoidal, then Tr(C) is an algebra.

Michael Reeks Trace and center of the twisted Heisenberg category

Prototype: Heisenberg category

[Cautis-Lauda-Licata-Sussan 2015] show Tr(H) ∼ = W1+∞/C − 1, w0,0. [Kvinge, Licata, Mitchell 2017] show the center, EndH(✶), of the Heisenberg category is isomorphic to the shifted symmetric functions Λ∗.

Michael Reeks Trace and center of the twisted Heisenberg category

Objective

Twisted Heisenberg algebra htw: associative algebra with generators hm/2, m ∈ 2Z + 1 subject to

• h n

2 , h m 2 ] = n

2δn,−m. Categorified by twisted Heisenberg category Htw[Cautis-Sussan, 2015]: htw ⊂ K0(Htw). Conjecturally isomorphic. We will describe Tr (and center) of Htw.

Michael Reeks Trace and center of the twisted Heisenberg category

Twisted Heisenberg category

Twisted Heisenberg category Htw: generating objects: P = Q = Think: P is induction and Q is restriction on modules for the Sergeev algebra (finite Hecke-Clifford algebra). Morphisms: P P[1] Q[1] Q P P P P Q P id P Q id P Q id Q P id

Michael Reeks Trace and center of the twisted Heisenberg category

New relations - empty dots

Empty dots correspond to generators ci of Clifford algebra Cℓn. Tr(Htw) is Z/2Z-graded where empty dots have degree 1. = c2

i = 1

= − cicj = −cjci = cisi = sici+1

Michael Reeks Trace and center of the twisted Heisenberg category

Dot interactions

Define: := Empty dots and solid dots on different strands commute. = Empty dots and solid dots on the same strand anticommute. = − xici = −cixi Dots, hollow dots, and crossings generate the degenerate affine Hecke-Clifford algebra Hc

n.

Michael Reeks Trace and center of the twisted Heisenberg category

W1+∞

W1+∞: unique nontrivial central extension of Lie algebra of differential operators on the circle. Connected to gl∞. Important in 2D quantum field theory and integrable systems. [Kac, Wang, Yan, 1998] define a certain subalgebra W − of W1+∞ fixed by degree-preserving anti-involution.

Michael Reeks Trace and center of the twisted Heisenberg category

W -algebra W −

Denote D = t∂t. W − =

• tjg(D + (j − 1)/2)|g odd polynomial
• ,

if j even

• tjg(D + (j − 1)/2)|g even polynomial
• ,

if j odd W − is generated by t−1, D3, and t±2(D ∓ 1).

Michael Reeks Trace and center of the twisted Heisenberg category

Fock space representations

W − Tr(Htw)0 C[t−1, t−2, . . .] Fock space C

• ,

, . . .

• acts on

acts on ∼ ∼

Identify images of generators of each algebra in the Fock space.

Michael Reeks Trace and center of the twisted Heisenberg category

Isomorphism

Define an algebra map Φ : Tr(Htw)0 → W − by mapping

• →

√ 2t−1     +     → −2 √ 2t2(D ∓ 1)

• +

  2   → 2D3 and extending algebraically. Theorem (O˘ guz-Reeks 2017) The map Φ is an algebra isomorphism Tr(Htw)0 → W −.

Michael Reeks Trace and center of the twisted Heisenberg category

Centers

The center of a monoidal category C is the algebra EndC(✶). The center of Htw is the algebra of closed diagrams: α(3,2) =

Michael Reeks Trace and center of the twisted Heisenberg category

Center of the twisted Heisenberg category

It can be shown that Z(Htw) ∼ = C[d0, d2, . . .], where dk := k Multiplication is inhomogeneous: α(5,1) = α(5) + l. o. t. α(1) Expect associated graded object to correspond to power sum

Michael Reeks Trace and center of the twisted Heisenberg category

The algebra Γ

Let Γ ⊂ Λ be the subalgebra of symmetric functions generated by {p2n+1|n ∈ N}. Γ has many interesting bases: pλ = pλ + l.o.t. inhomogenous power sums Q∗

λ = Qλ + l.o.t.

factorial Schur Q-functions g↑

k, g↓ k+1

moments of probability measures on Schur’s graph

Michael Reeks Trace and center of the twisted Heisenberg category

Center

Theorem (Kvinge, O˘ guz, Reeks) The center EndHtw (✶) of the twisted Heisenberg category is isomorphic as an algebra to Γ. Γ pµ g↑

k

g↓

k+1

diagram in EndH(✶) · · · µ 2ℓµ 1 2k 2k

Michael Reeks Trace and center of the twisted Heisenberg category

Selected references

[Cautis-Lauda-Licata-Sussan] W -algebras from Heisenberg categories, Comm. Math. Phys., 2015. [Kac-Wang-Yan] Quasifinite representations of classical Lie subalgebras of W1+∞, Adv. Math., 1998. [Khovanov] Heisenberg algebra and a graphical calculus,

• Fund. Math., 2010.

[Kvinge-O˘ guz-Reeks] The center of the twisted Heisenberg category, factorial Schur Q-functions, and transition functions

• n the Schur graph, 2017.

[O˘ guz-Reeks] Trace of the twisted Heisenberg category,

• Comm. Math. Phys., 2017.

Michael Reeks Trace and center of the twisted Heisenberg category