In mathematics, the Zhu algebra and the closely related C2-algebra, introduced by Yongchang Zhu in his PhD thesis, are two associative algebras canonically constructed from a given vertex operator algebra. Many important representation theoretic properties of the vertex algebra are logically related to properties of its Zhu algebra or C2-algebra.
Definitions Let V = ⨁ n ≥ 0 V ( n ) {\displaystyle V=\bigoplus _{n\geq 0}V_{(n)}} be a graded vertex operator algebra with V ( 0 ) = C 1 {\displaystyle V_{(0)}=\mathbb {C} \mathbf {1} } and let Y ( a , z ) = ∑ n ∈ Z a n z − n − 1 {\displaystyle Y(a,z)=\sum _{n\in \mathbb {Z} }a_{n}z^{-n-1}} be the vertex operator associated to a ∈ V {\displaystyle a\in V} . Define C 2 ( V ) ⊂ V {\displaystyle C_{2}(V)\subset V} to be the subspace spanned by elements of the form a − 2 b {\displaystyle a_{-2}b} for a , b ∈ V {\displaystyle a,b\in V} . An element a ∈ V {\displaystyle a\in V} is homogeneous with wt a = n {\displaystyle \operatorname {wt} a=n} if a ∈ V ( n ) {\displaystyle a\in V_{(n)}} . There are two binary operations on V {\displaystyle V} defined by a ∗ b = ∑ i ≥ 0 ( wt a i ) a i − 1 b {\displaystyle a*b=\sum _{i\geq 0}{\binom {\operatorname {wt} a}{i}}a_{i-1}b} , a ∘ b = ∑ i ≥ 0 ( wt a i ) a i − 2 b {\displaystyle a\circ b=\sum _{i\geq 0}{\binom {\operatorname {wt} a}{i}}a_{i-2}b} for homogeneous elements and extended linearly to all of V {\displaystyle V} . Define O ( V ) ⊂ V {\displaystyle O(V)\subset V} to be the span of all elements a ∘ b {\displaystyle a\circ b} . The algebra A ( V ) := V / O ( V ) {\displaystyle A(V):=V/O(V)} with the binary operation induced by ∗ {\displaystyle *} is an associative algebra called the Zhu algebra of V {\displaystyle V} . The algebra R V := V / C 2 ( V ) {\displaystyle R_{V}:=V/C_{2}(V)} with multiplication a ⋅ b = a − 1 b mod C 2 ( V ) {\displaystyle a\cdot b=a_{-1}b\mod C_{2}(V)} is called the C2-algebra of V {\displaystyle V} .
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