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Khovanov–Seidel type A algebra

Definition

Fix an integer $m \geq 1$ . The doubled line quiver $Γ_{m}$ is the finite quiver of Integral path ring of a finite quiver with vertices $0, 1, \dots, m$ and, for every $0 \leq i \leq m - 1$ , exactly two arrows between the consecutive vertices $i$ and $i + 1$ , one in each direction. Its directed paths of length $l$ are therefore the tuples $(i_{1} ∣ i_{2} ∣ \dots ∣ i_{l + 1}), i_{j} \in {0, 1, \dots, m}, ∣ i_{j + 1} - i_{j} ∣ = 1,$ and the product of the path ring $Z Γ_{m}$ is the left-to-right concatenation of Integral path ring of a finite quiver.

The algebra. For $m \geq 1$ , let $I_{m} ⊴ Z Γ_{m}$ be the two-sided ideal generated by the elements $(i - 1 ∣ i ∣ i + 1), (i + 1 ∣ i ∣ i - 1), (i ∣ i + 1 ∣ i) - (i ∣ i - 1 ∣ i), (0 ∣ 1 ∣ 0),$ where the first three families are over all $0 < i < m$ ; when $m = 1$ the middle family is empty and only $(0 ∣ 1 ∣ 0)$ is imposed. The Khovanov–Seidel type A algebra is the quotient ring $A_{m} : = Z Γ_{m} / I_{m}$ (The quotient ring $R / I$ with $(r + I) (s + I) = r s + I$ ), whose elements are written as cosets of paths.

Internal degree. The internal degree of a path is the integer $\deg (v_{0} ∣ v_{1} ∣ \dots ∣ v_{l}) : = # {j : 0 \leq j < l, v_{j + 1} = v_{j} - 1},$ the number of steps in which the path moves down; so a vertex and an ascending arrow $(i ∣ i + 1)$ have degree $0$ , while a descending arrow $(i + 1 ∣ i)$ has degree $1$ . This number is additive under concatenation, $\deg (p q) = \deg (p) + \deg (q)$ .

Grading. Write $(Z Γ_{m})_{d}$ for the free abelian group spanned by the paths of degree $d$ , so that $Z Γ_{m} = ⨁_{d \geq 0} (Z Γ_{m})_{d}$ and $(Z Γ_{m})_{a} (Z Γ_{m})_{b} \subseteq (Z Γ_{m})_{a + b}$ for all $a, b$ . The quotient $A_{m}$ is graded, $A_{m} = ⨁_{d \geq 0} (A_{m})_{d}, (A_{m})_{d} : = ((Z Γ_{m})_{d} + I_{m}) / I_{m},$ with $(A_{m})_{a} (A_{m})_{b} \subseteq (A_{m})_{a + b}$ and $1_{A_{m}} \in (A_{m})_{0}$ , as proved below. Elements of $(A_{m})_{d}$ are called homogeneous of degree $d$ , and homogeneous elements of the two lowest degrees are written $e_{i} : = (i), u_{i} : = (i ∣ i + 1), d_{i} : = (i + 1 ∣ i) (0 \leq i \leq m - 1),$ so that $\deg e_{i} = 0 = \deg u_{i}$ and $\deg d_{i} = 1$ . In keeping with the tuple notation of the path ring, the left entry of $(i + 1 ∣ i)$ is its source, so $(i + 1 ∣ i)$ is the arrow from $i + 1$ to $i$ ; likewise $(i - 1 ∣ i ∣ i + 1)$ is the two-step path $i - 1 \to i \to i + 1$ and $(i ∣ i - 1 ∣ i)$ is the return $i \to i - 1 \to i$ .

Facts & Assumptions

Given: An integer $m \geq 1$ , the doubled line quiver $Γ_{m}$ with its path ring $Z Γ_{m}$ , the two-sided ideal $I_{m}$ generated by the displayed elements, and the internal degree of paths.

[F1]

$Z Γ_{m}$ is the free abelian group on the directed paths of $Γ_{m}$ , with left-to-right concatenation product, unit $\sum_{i = 0}^{m} (i)$ , and the endpoint formulas $(u) p = p$ if $s (p) = u$ and $0$ otherwise, $p (v) = p$ if $t (p) = v$ and $0$ otherwise (Integral path ring of a finite quiver).

[L2]

For a two-sided ideal $I$ of a ring $R$ the quotient $R / I$ is a ring with multiplication $(r + I) (s + I) = r s + I$ , defined on cosets of the additive quotient group (The quotient ring $R / I$ with $(r + I) (s + I) = r s + I$ ).

Proof

technique · direct

1.1

Additivity of the internal degree. Every path is a concatenation of its consecutive arrows, and a step moves down exactly when it contributes $1$ to the count; splitting a path at a vertex therefore splits the set of steps, so $\deg (p q) = \deg (p) + \deg (q)$ for composable paths, and $\deg$ is additive over concatenations of any finite number of arrows. In particular $\deg$ is a function on the path set, and each path has one degree.

algebra

2.1

The path ring is a graded ring for this degree. Put $(Z Γ_{m})_{d}$ for the subgroup spanned by the paths of degree $d$ . Distinct paths form a basis of $Z Γ_{m}$ by [F1], so $Z Γ_{m} = ⨁_{d \geq 0} (Z Γ_{m})_{d}$ ; the product of paths of degrees $a$ and $b$ is $0$ or a path of degree $a + b$ by step 1.1, so $(Z Γ_{m})_{a} (Z Γ_{m})_{b} \subseteq (Z Γ_{m})_{a + b}$ by bilinearity; and $1 = \sum_{i} (i)$ lies in degree $0$ .

step 1.1F1

2.2

Each relation generator is homogeneous. The generator $(i - 1 ∣ i ∣ i + 1)$ is a path of degree $0$ , the generator $(i + 1 ∣ i ∣ i - 1)$ is a path of degree $2$ , the two sides of $(i ∣ i + 1 ∣ i) - (i ∣ i - 1 ∣ i)$ are paths of degree $1$ , and $(0 ∣ 1 ∣ 0)$ is a path of degree $1$ . Hence every generator lies in a single summand $(Z Γ_{m})_{d}$ for its own $d$ .

step 1.1

3.1

The ideal is homogeneous. The two-sided ideal generated by homogeneous elements of a graded ring is homogeneous: the set of finite sums $\sum_{j} r_{j} s_{j} t_{j}$ with $s_{j}$ among the generators, $r_{j}, t_{j} \in Z Γ_{m}$ , is a two-sided ideal, and decomposing $r_{j}$ and $t_{j}$ into homogeneous components expresses each such element as a sum of elements lying in single summands of the form $(Z Γ_{m})_{d^{'}} (Z Γ_{m})_{d} (Z Γ_{m})_{d^{''}}$ ; hence $I_{m} = ⨁_{d} (I_{m} \cap (Z Γ_{m})_{d})$ .

step 2.1step 2.2

4.1

The quotient is a graded ring. By step 3.1 each $(A_{m})_{d} : = ((Z Γ_{m})_{d} + I_{m}) / I_{m}$ is the image of $(Z Γ_{m})_{d}$ in the quotient, the sum $\sum_{d} (A_{m})_{d}$ is direct because the sum decomposition of $Z Γ_{m}$ is direct modulo the homogeneous ideal, and $A_{m} = ⨁_{d \geq 0} (A_{m})_{d}$ . Products satisfy $(A_{m})_{a} (A_{m})_{b} \subseteq (A_{m})_{a + b}$ by step 2.1 and surjectivity of the quotient map, and the image of $1_{Z Γ_{m}}$ lies in $(A_{m})_{0}$ . By [L2] the quotient is a ring, so $A_{m}$ is a ring graded by internal degree with $1 \in (A_{m})_{0}$ .

step 2.1step 3.1L2

5.1

Conclusion and conventions. The operations of $A_{m}$ and its grading are those displayed, the generators of $I_{m}$ hold as equalities in $A_{m}$ , and the elements $e_{i}, u_{i}, d_{i}$ are homogeneous of degrees $0, 0, 1$ . For $m = 1$ the families of relations indexed by $0 < i < m$ are empty, and $A_{1}$ is the quotient of $Z Γ_{1}$ by $(0 ∣ 1 ∣ 0)$ alone.

step 4.1algebra∎

Depends on

Used by

Finite graded Aₘ-modules, internal shifts and the vertex projectives Definition
Signed totalization of graded Aₘ-bimodule actions Definition
The Khovanov–Seidel bimodule maps βᵢ and γᵢ Definition
The vertex modules Sᵢ and their prime quotients Definition
The algebra A₂ and its vertex projectives Example
The 4m+1 path basis Lemma
The Khovanov-Seidel grid resolutions of the vertex modules Lemma
Corner computations: the Uᵢ satisfy the Temperley-Lieb relations Theorem
Finite homological dimension of the finite graded Khovanov-Seidel module category Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, §1b, printed pp. 3-4 (standard reference, not scraped)