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 ) End C ( 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 ) ∼ = W 1+ ∞ / � C − 1 , w 0 , 0 � . [Kvinge, Licata, Mitchell 2017] show the center, End H ( ✶ ), 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 h tw : associative algebra with generators h m / 2 , m ∈ 2 Z + 1 subject to 2 ] = n � h n 2 , h m 2 δ n , − m . Categorified by twisted Heisenberg category H tw [Cautis-Sussan, 2015]: h tw ⊂ K 0 ( H tw ) . Conjecturally isomorphic. We will describe Tr (and center) of H tw . Michael Reeks Trace and center of the twisted Heisenberg category
Twisted Heisenberg category Twisted Heisenberg category H tw : generating objects: P = Q = Think: P is induction and Q is restriction on modules for the Sergeev algebra (finite Hecke-Clifford algebra). Morphisms: P [1] Q P P Q Q P P id id Q Q id id P P Q [1] P P P Michael Reeks Trace and center of the twisted Heisenberg category
New relations - empty dots Empty dots correspond to generators c i of Clifford algebra C ℓ n . Tr( H tw ) is Z / 2 Z -graded where empty dots have degree 1. c 2 = i = 1 = − c i c j = − c j c i = c i s i = s i c i +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. = − x i c i = − c i x i Dots, hollow dots, and crossings generate the degenerate affine Hecke-Clifford algebra H c n . Michael Reeks Trace and center of the twisted Heisenberg category
W 1+ ∞ W 1+ ∞ : 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 W 1+ ∞ fixed by degree-preserving anti-involution. Michael Reeks Trace and center of the twisted Heisenberg category
W -algebra W − Denote D = t ∂ t . � t j g ( D + ( j − 1) / 2) | g odd polynomial � , if j even W − = t j g ( D + ( j − 1) / 2) | g even polynomial � � , if j odd W − is generated by t − 1 , D 3 , and t ± 2 ( D ∓ 1). Michael Reeks Trace and center of the twisted Heisenberg category
Fock space representations W − Tr( H tw ) 0 acts on acts on � � ∼ ∼ C [ t − 1 , t − 2 , . . . ] Fock space C , , . . . 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( H tw ) 0 → W − by mapping � � √ 2 t − 1 �→ √ + 2 t 2 ( D ∓ 1) �→ − 2 � � 2 �→ 2 D 3 + and extending algebraically. Theorem (O˘ guz-Reeks 2017) The map Φ is an algebra isomorphism Tr( H tw ) 0 → W − . Michael Reeks Trace and center of the twisted Heisenberg category
Centers The center of a monoidal category C is the algebra End C ( ✶ ) . The center of H tw 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 ( H tw ) ∼ = C [ d 0 , d 2 , . . . ] , where k d k := Multiplication is inhomogeneous: + l. o. t. = α (5) α (1) α (5 , 1) Michael Reeks Trace and center of the twisted Heisenberg category Expect associated graded object to correspond to power sum
The algebra Γ Let Γ ⊂ Λ be the subalgebra of symmetric functions generated by { p 2 n +1 | n ∈ N } . Γ has many interesting bases: p λ = p λ + l.o.t. inhomogenous power sums Q ∗ λ = Q λ + 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 End H tw ( ✶ ) of the twisted Heisenberg category is isomorphic as an algebra to Γ . g ↑ g ↓ Γ p µ k k +1 2 k 2 k diagram · · · 1 µ 2 ℓµ in End H ( ✶ ) 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 W 1+ ∞ , 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 on 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
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