Derived identities of differential algebras

TL;DR

This paper explores how identities in differential algebras translate into identities for associated binary operations, revealing a connection with operad compositions and Novikov algebras.

Contribution

It establishes a method to derive identities for operations defined via derivations in differential algebras, linking them to operad compositions and Novikov algebra structures.

Findings

Identities for operations $ riangleright$ and $ riangleleft$ are derived from differential algebra identities.

The operations satisfy relations of the operad Var∘Nov, connecting to Novikov algebras.

No additional identities hold beyond those derived from the operad composition.

Abstract

Suppose $A$ is a not necessarily associative algebra with a derivation $d$. Then $A$ may be considered as a system with two binary operations $\succ $ and $\prec $ defined by $x\succ y = d(x)y$, $x\prec y = xd(y)$, $x,y\in A$. Suppose $A$ satisfies some multi-linear polynomial identities. We show how to find the identities that hold for operations $\prec $ and $\succ $. It turns out that if $A$ belongs to a variety governed by an operad Var then $\succ $ and $\prec $ satisfy the defining relations of the operad Var$\circ $Nov, where $\circ $ is the Manin white product of operads, Nov is the operad of Novikov algebras. Moreover, there are no other identities that hold for operations $\succ $, $\prec $ on an arbitrary differential Var-algebra.

Equations

$$a\prec b=a\cdot T(b),a\succ b=T(a)\cdot b,a,b\in A.$$

$$\mathrm{Var}(n)=M_n(X)/M_n(X)\cap T_{\mathrm{Var}}(X),$$

$$\mu (\mathrm{id},\mu )=[x_1[x_2{x}_3]],\mu (\mu ,\mathrm{id})=[[x_1{x}_2]x_3],$$

$$\mu (\mathrm{id},\mu )-\mu (\mathrm{id},\mu {)}^{(12)}=\mu (\mu ,\mathrm{id}).$$

$$\nu (\nu ,\mathrm{id})=\nu (\mathrm{id},\nu ).$$

$$(x_1{x}_2)x_3-x_1(x_2{x}_3)=(x_2{x}_1)x_3-x_2(x_1{x}_3),$$

$$(x_1{x}_2)x_3=(x_1{x}_3)x_2.$$

$$\begin{array}{c}\nu (\nu ,\mathrm{id})-\nu (\mathrm{id},\nu )=\nu ({\nu }^{(12)},\mathrm{id})-{\nu }^{(12)}(\nu ,\mathrm{id}{)}^{(23)},\\ \nu (\nu ,\mathrm{id})=\nu (\nu ,\mathrm{id}{)}^{(23)}.\end{array}$$

$$\begin{array}{c}{m}_1=[[x_1\prec x_2]\prec x_3]=[[x_1{x}_2]x_3]\otimes (x_1{x}_2)x_3,\\ m_2=[x_1\prec [x_2\prec x_3]]=[x_1[x_2{x}_3]]\otimes x_1(x_2{x}_3),\end{array}$$

$$(x_1\succ x_2)\prec x_3-x_1\succ (x_2\prec x_3)=0,$$

$$(x_1\prec x_2)\prec x_3-x_1\prec (x_2\succ x_3)+(x_1\prec x_2)\succ x_3-x_1\succ (x_2\succ x_3)=0.$$

$$\begin{array}{c}(x_1\prec x_2)\prec x_3=(x_1{x}_2)x_3\otimes (x_1{x}_2)x_3,\\ x_1\succ (x_2\succ x_3)=x_1(x_2{x}_3)\otimes (x_3{x}_2)x_1,\\ (x_1\succ x_2)\prec x_3=(x_1{x}_2)x_3\otimes (x_2{x}_1)x_3,\\ x_1\succ (x_2\prec x_3)=x_1(x_2{x}_3)\otimes (x_2{x}_3)x_1,\\ (x_1\prec x_2)\succ x_3=(x_1{x}_2)x_3\otimes x_3(x_1{x}_2),\\ x_1\prec (x_2\succ x_3)=x_1(x_2{x}_3)\otimes x_1(x_3{x}_2),\\ x_1\prec (x_2\prec x_3)=x_1(x_2{x}_3)\otimes x_1(x_2{x}_3),\\ (x_1\succ x_2)\succ x_3=(x_1{x}_2)x_3\otimes x_3(x_2{x}_1)\end{array}$$

$$x\prec y=xd(y),x\succ y=d(x)y,x,y\in A.$$

$$x^{(n)}\cdot x^{(m)}=\left(\genfrac{}{}{0ex}{}{n+m-1}{n}\right)x^{(n+m-1)}.$$

$$(a\otimes f)\prec (b\otimes g)=ab\otimes f\cdot g,(a\otimes f)\succ (b\otimes g)=ab\otimes g\cdot f.$$

$$A\to \hat{A},$$

$$a\mapsto \sum _{s\ge 0}{d}^s(a)\otimes x^{(s)},$$

$$\begin{gathered} (a\prec b)=ad(b)\mapsto \sum _{s\ge 0}{d}^s(ad(b))\otimes x^{(s)}=\sum _{s,t\ge 0}\left(\genfrac{}{}{0ex}{}{s+t}{s}\right)d^s(a)d^{t+1}(b)\otimes x^{(s+t)} \\ =\sum _{s,t\ge 0}{d}^s(a)d^{t+1}(b)\otimes x^{(s)}\cdot x^{(t+1)}=\left(\sum _{s\ge 0}{d}^s(a)\otimes x^{(s)}\right)\prec \left(\sum _{t\ge 0}{d}^t(a)\otimes x^{(t)}\right). \end{gathered}$$

$$(u\otimes f)\prec (v\otimes g)=(uv\otimes fg),(u\otimes f)\succ (v\otimes g)=(uv\otimes gf)$$

$$D(ab)=D(a)b+aD(b)+\lambda ab,a,b\in A,$$

Full text

1. Introduction

Let $A$ be an algebra over a field $\Bbbk$, i.e., a linear space equipped with a binary linear operation (multiplication) $\cdot:A\otimes A\to A$. Suppose $T$ is a linear operator on $A$, and let $\prec$ and $\succ$ be two new linear maps $A\otimes A\to A$ defined by

[TABLE]

Denote the system $(A,\prec,\succ)$ by $A^{(T)}$. If the initial algebra $A$ satisfies a polynomial identity then what could be said about $A^{(T)}$? The answer is known if $T$ is a Rota—Baxter operator [1], [4], [2] or averaging operator [5]. In these cases, the identities of $A^{(T)}$ may be obtained by means of categoric procedures (black and white Manin products of operads [3]).

The purpose of this note is to consider the case when $T$ is a derivation (or generalized derivation). It is well-known [7] that a commutative (and associative) algebra $A$ with a derivation $d$ induces Novikov algebra structure on $A^{(d)}$, assuming $(a\succ b)=(b\prec a)$. Conversely, if an identity holds on $A^{(d)}$ for an arbitrary commutative algebra $A$ with a derivation $d$ then this identity is a consequence of Novikov identities.

In this paper, we generalize this observation for an arbitrary variety $\mathop{Var}\nolimits$ of algebras. Namely, if an identity $f$ holds on $A^{(d)}$ for every $\mathop{Var}\nolimits$-algebra $A$ with a derivation $d$ then we say $f$ is a derived identity of $\mathop{Var}\nolimits$. For $\mathop{Var}\nolimits=\mathop{As}\nolimits$, some derived identities were found in [8]. We show that for a multi-linear variety $\mathop{Var}\nolimits$ the set of derived identities coincides with the set of relations on the operad $\mathop{Var}\nolimits\circ\mathop{Nov}\nolimits$, where $\mathop{Var}\nolimits$ and $\mathop{Nov}\nolimits$ are the operads governing the varieties $\mathop{Var}\nolimits$ and $\mathop{Nov}\nolimits$, respectively, $\circ$ is the Manin white product of operads.

Calculation of Manin products is a relatively simple linear algebra problem, it is based on finding intersections of vector spaces. Therefore, our result provides an easy way for finding a complete list of derived identities for an arbitrary binary operad.

2. White product of operads

Suppose $\mathop{Var}\nolimits$ is a multi-linear variety of algebras, i.e., a class of all algebras that satisfy a given family of multi-linear identities (over a field $\Bbbk$ of characteristic zero, every variety is multi-linear). Fix a countable set of variables $X=\{x_{1},x_{2},\dots\}$ and denote

[TABLE]

where $M_{n}(X)$ is the space of all (non-associative) multi-linear polynomials of degree $n$ in $x_{1},\dots,x_{n}$, $T_{\mathop{Var}\nolimits}(X)$ is the T-ideal of all identities that hold in $\mathop{Var}\nolimits$.

The collection of spaces $(\mathop{Var}\nolimits(n))_{n\geq 1}$ forms a symmetric operad relative to the natural composition rule and symmetric group action (see, e.g., [9]). We will denote this operad by the same symbol $\mathop{Var}\nolimits$. Every algebra $A\in\mathop{Var}\nolimits$ may be considered as a morphism of multi-categories $\mathop{Var}\nolimits\to\mathop{Hom}\nolimits_{\Bbbk}$, where $\mathop{Hom}\nolimits_{\Bbbk}$ stands for the multi-category of linear spaces over $\Bbbk$ and multi-linear maps. The operad $\mathop{Var}\nolimits$ is a binary one, i.e., it is generated (as a symmetric operad) by the elements of $\mathop{Var}\nolimits(2)$.

Example 1**.**

The operad $\mathop{Lie}\nolimits$ governing the variety of Lie algebras is generated by 1-dimensional space $\mathop{Lie}\nolimits(2)=\Bbbk\mu$, $\mu^{(12)}=-\mu$. The space $\mathop{Lie}\nolimits(1)$ is also 1-dimensional, it is spanned by the identity $\mathop{id}\nolimits$ of the operad. If we identify $\mathop{id}\nolimits$ with $x_{1}$ and $\mu$ with $[x_{1}x_{2}]$ then

[TABLE]

so the Jacobi identity may be expressed as

[TABLE]

Example 2**.**

The operad $\mathop{As}\nolimits$ governing the variety of associative algebras is generated by 2-dimensional space $\mathop{As}\nolimits(2)$ spanned by $\nu=(x_{1}x_{2})$ and $\nu^{(12)}=(x_{2}x_{1})$. Associativity relations form an $S_{3}$-submodule in $M_{3}(X)$ spanned by

[TABLE]

Example 3**.**

Novikov algebra is a linear space with a multiplication satisfying the following axioms:

[TABLE]

The corresponding operad is generated by 2-dimensional $\mathop{Nov}\nolimits(2)$ spanned by $\nu=(x_{1}x_{2})$ and $\nu^{(12)}=(x_{2}x_{1})$. Defining identities of the variety $\mathop{Nov}\nolimits$ may be expressed as

[TABLE]

Let $\mathop{Var}\nolimits_{1}$ and $\mathop{Var}\nolimits_{2}$ be two operads. Then the family of spaces $(\mathop{Var}\nolimits_{1}(n)\otimes\mathop{Var}\nolimits_{2}(n))_{n\geq 1}$ is an operad relative to the natural (componentwise) composition and symmetric group action (known as the Hadamard product of $\mathop{Var}\nolimits_{1}$ and $\mathop{Var}\nolimits_{2}$). Even if $\mathop{Var}\nolimits_{1}$ and $\mathop{Var}\nolimits_{2}$ were binary operads, their Hadamard product $\mathop{Var}\nolimits_{1}\otimes\mathop{Var}\nolimits_{2}$ may be non-binary. The sub-operad of $\mathop{Var}\nolimits_{1}\otimes\mathop{Var}\nolimits_{2}$ generated by $\mathop{Var}\nolimits_{1}(2)\otimes\mathop{Var}\nolimits_{2}(2)$ is known as Manin white product of $\mathop{Var}\nolimits_{1}$ and $\mathop{Var}\nolimits_{2}$, it is denoted $\mathop{Var}\nolimits_{1}\circ\mathop{Var}\nolimits_{2}$ [3].

Example 4**.**

The operad $\mathop{Lie}\nolimits\circ\mathop{Nov}\nolimits$ is isomorphic to the operad governing the class of all algebras (magmatic operad).

Indeed, both $\mathop{Lie}\nolimits$ and $\mathop{Nov}\nolimits$ are quadratic operad, and so is $\mathop{Lie}\nolimits\circ\mathop{Nov}\nolimits$ [3]. Let $\mu$ and $\nu$ be the generators of $\mathop{Lie}\nolimits$ and $\mathop{Nov}\nolimits$. It is enough to find the defining identities of $\mathop{Lie}\nolimits\circ\mathop{Nov}\nolimits$ that are quadratic with respect to $\mu$ and $\nu$.

Identify $\mu\otimes\nu$ with $[x_{1}\prec x_{2}]$, then $\mu\otimes\nu^{(12)}=-(\mu\otimes\nu)^{(12)}$ corresponds to $-[x_{2}\prec x_{1}]$. Hence, $(\mathop{Lie}\nolimits\circ\mathop{Nov}\nolimits)(n)$ is an image of the space $M_{n}(X)$ of all multi-linear non-associative polynomials of degree $n$ in $X=\{x_{1},x_{2},\dots\}$ relative to the operation $[\cdot\prec\cdot]$. Calculating the compositions $\mu(\mu,\mathop{id}\nolimits)\otimes\nu(\nu,\mathop{id}\nolimits)$ and $\mu(\mathop{id}\nolimits,\mu)\otimes\nu(\mathop{id}\nolimits,\nu)$, we obtain

[TABLE]

It remains to find the intersection of the $S_{3}$-submodule generated by $m_{1}$ and $m_{2}$ in $M_{3}(X)\otimes M_{3}(X)$ with the kernel of the projection $M_{3}(X)\otimes M_{3}(X)\to\mathop{Lie}\nolimits(3)\otimes\mathop{Nov}\nolimits(3)$. Straightforward calculation shows the intersection is zero. Hence, the operation $[\cdot\prec\cdot]$ satisfies no identities.

Example 5**.**

The operad $\mathop{As}\nolimits\circ\mathop{Nov}\nolimits$ generated by 4-dimensional space $(\mathop{As}\nolimits\circ\mathop{Nov}\nolimits)(2)$ spanned by $(x_{1}\succ x_{2})$, $(x_{2}\succ x_{1})$, $(x_{1}\prec x_{2})$, $(x_{2}\prec x_{1})$ relative to the following identities:

[TABLE]

This result may be checked with a straightforward computation, as in Example 4. Namely, one has to find the intersection of the $S_{3}$-submodule in $M_{3}(X)\otimes M_{3}(X)$ generated by

[TABLE]

with the kernel of the projection $M_{3}(X)\otimes M_{3}(X)\to\mathop{As}\nolimits(3)\otimes\mathop{Nov}\nolimits(3)$.

3. Derived identities

Given a (non-associative) algebra $A$ with a derivation $d$, denote by $A^{(d)}$ the same linear space $A$ considered as a system with two binary linear operations of multiplication $(\cdot\prec\cdot)$ and $(\cdot\succ\cdot)$ defined by

[TABLE]

Let $\mathop{Var}\nolimits$ be a multi-linear variety of algebras. As above, denote by $T_{\mathop{Var}\nolimits}(X)$ the T-ideal of all identities in a set of variables $X$ that hold on $\mathop{Var}\nolimits$. A non-associative polynomial $f(x_{1},\dots,x_{n})$ in two operations of multiplication $\prec$ and $\succ$ is called a derived identity of $\mathop{Var}\nolimits$ if for every $A\in\mathop{Var}\nolimits$ and for every derivation $d\in\mathop{Der}\nolimits(A)$ the algebra $A^{(d)}$ satisfies the identity $f(x_{1},\dots,x_{n})=0$. Obviously, the set of all derived identities is a T-ideal of the algebra of non-associative polynomials in two operations.

For example, $(x\succ y)=(y\prec x)$ is a derived identity of $\mathop{Com}\nolimits$. Moreover, the operation $(\cdot\prec\cdot)$ satisfies the axioms of Novikov algebras (1) and (2). It was actually shown in [7] that the entire T-ideal of derived identities of $\mathop{Com}\nolimits$ is generated by these identities.

For the variety of associative algebras, it was mentioned in [8] that (3) and (4) are derived identities of $\mathop{As}\nolimits$.

Remark 1** (c.f. [6]).**

For a multi-linear variety $\mathop{Var}\nolimits$, the free differential $\mathop{Var}\nolimits$-algebra generated by a set $X$ with one derivation is nothing but $\mathop{Var}\nolimits\langle d^{\omega}X\rangle$, where $d^{\omega}X=\{d^{s}(x)\mid s\geq 0,x\in X\}$.

Indeed, consider the free magmatic algebra $F(X;d)=M\langle d^{\omega}X\rangle$ and define a linear map $d:F(X;d)\to F(X;d)$ in such a way that $d(d^{s}(x))=d^{s+1}(x)$, $d(uv)=d(u)v+ud(v)$. Since $\mathop{Var}\nolimits$ is defined by multi-linear relations, the T-ideal $T_{\mathop{Var}\nolimits}(d^{\omega}X)$ is $d$-invariant. Hence, $U=F(X;d)/T_{\mathop{Var}\nolimits}(d^{\omega}X)$ is a $\mathop{Var}\nolimits$-algebra with a derivation $d$. Since all relations from $T_{\mathop{Var}\nolimits}(d^{\omega}X)$ hold in every differential $\mathop{Var}\nolimits$-algebra, $U$ is free.

Let $\mathcal{N}_{\mathop{Var}\nolimits}$ be the class of differential $\mathop{Var}\nolimits$-algebras with one locally nilpotent derivation, i.e., for every $A\in\mathcal{N}_{\mathop{Var}\nolimits}$ and for every $a\in A$ there exists $n\geq 1$ such that $d^{n}(a)=0$.

The following statement shows why the variety generated by $\mathcal{N}_{\mathop{Var}\nolimits}$ coincides with the class of all differential $\mathop{Var}\nolimits$-algebras.

Lemma 1**.**

Suppose $f=f(x_{1},\dots,x_{n})$ is a multi-linear identity that holds on $A^{(d)}$ for all $A\in\mathcal{N}_{\mathop{Var}\nolimits}$. Then $f$ is a derived identity of $\mathop{Var}\nolimits$.

Proof.

Consider the free differential $\mathop{Var}\nolimits$-algebra $U_{n}$ generated by the set $X=\{x_{1},x_{2},\dots\}$ with one derivation $d$ modulo defining relations $d^{n}(x)=0$, $x\in X$.

Then $U_{n}\in\mathcal{N}_{\mathop{Var}\nolimits}$. As a differential $\mathop{Var}\nolimits$-algebra, $U_{n}$ is a homomorphic image of the free magma $M\langle d^{\omega}X\rangle$. Denote by $I_{n}$ the kernel of the homomorphism $M\langle d^{\omega}X\rangle\to U_{n}$. As an ideal of $M\langle d^{\omega}X\rangle$, $I_{n}$ is a sum of two ideals: $I_{n}=T_{\mathop{Var}\nolimits}(d^{\omega}X)+N_{n}$, where $N$ is generated by $d^{n+t}(x)$, $x\in X$, $t\geq 0$. Note that the last relations form a $d$-invariant subset of $M\langle d^{\omega}X\rangle$, so the ideal $N_{n}$ is $d$-invariant.

If a polynomial $F\in M\langle d^{\omega}X\rangle$ belongs to $I_{n}$ and its degree in $d$ is less than $n$ then $F$ belongs to $T_{\mathop{Var}\nolimits}(d^{\omega}X)$. If $f=f(x_{1},\dots,x_{n})$ is a multi-linear identity in $U_{n}^{(d)}$ then the image $F$ of $f$ in $M\langle d^{\omega}X\rangle$ has degree $n-1$ in $d$, so $F\in T_{\mathop{Var}\nolimits}(d^{\omega})$. Hence, $f$ is a derived identity of $\mathop{Var}\nolimits$. ∎

Theorem 1**.**

If a multi-linear identity $f$ holds on the variety governed by the operad $\mathop{Var}\nolimits\circ\mathop{Nov}\nolimits$ then $f$ is a derived identity of $\mathop{Var}\nolimits$.

Proof.

For every $A\in\mathcal{N}_{\mathop{Var}\nolimits}$ we may construct an algebra in the variety $\mathop{Var}\nolimits\circ\mathop{Nov}\nolimits$ as follows. Consider the linear space $H$ spanned by elements $x^{(n)}$, $n\geq 0$, with multiplication

[TABLE]

This is a Novikov algebra (in characteristic 0, this is just the ordinary polynomial algebra). Denote $\hat{A}=A\otimes\Bbbk[x]$ and define operations $\prec$ and $\succ$ on $\hat{A}$ by

[TABLE]

The algebra $\hat{A}$ obtained belongs to the variety governed by the operad $\mathop{Var}\nolimits\circ\mathop{Nov}\nolimits$. There is an injective map

[TABLE]

which preserves operations $\prec$ and $\succ$. Indeed, apply (5) to $a\prec b$, $a,b\in A$:

[TABLE]

Therefore, $A^{(d)}$ is a subalgebra of a $(\mathop{Var}\nolimits\circ\mathop{Nov}\nolimits)$-algebra, so all defining identities of $\mathop{Var}\nolimits\circ\mathop{Nov}\nolimits$ hold on $A^{(d)}$. Lemma 1 completes the proof. ∎

Theorem 2**.**

Let $f$ be a multi-linear derived identity of $\mathop{Var}\nolimits$. Then $f$ holds on all ($\mathop{Var}\nolimits\circ\mathop{Nov}\nolimits$)-algebras.

Proof.

Let $X=\{x_{1},x_{2},\dots\}$, and let $f(x_{1},\dots,x_{n})$ be a non-associative multi-linear polynomial in two operations of multiplication $\prec$ and $\succ$. By definition, $f$ is an identity of the algebra $\mathop{Var}\nolimits\langle X\rangle\otimes\mathop{Nov}\nolimits\langle X\rangle$ with operations

[TABLE]

if and only if $f$ is a defining identity of $\mathop{Var}\nolimits\circ\mathop{Nov}\nolimits$.

It is well-known that $\mathop{Nov}\nolimits\langle X\rangle$ is a subalgebra of $C^{(d)}$ for the free differential commutative algebra $C$ generated by $X$ with a derivation $d$ [7]. Therefore, $\mathop{Var}\nolimits\langle X\rangle\otimes\mathop{Nov}\nolimits\langle X\rangle$ is a subalgebra of $(\mathop{Var}\nolimits\langle X\rangle\otimes C)^{(d)}$, where $d(u\otimes f)=u\otimes d(f)$ for $u\in\mathop{Var}\nolimits\langle X\rangle$, $f\in C$. Hence, $\mathop{Var}\nolimits\langle X\rangle\otimes\mathop{Nov}\nolimits\langle X\rangle$ is embedded into a differential $\mathop{Var}\nolimits$-algebra. ∎

Remark 2**.**

In this paper, we consider algebras with one binary operation, so $\mathop{Var}\nolimits$ has only one generator. However, Theorems 1, 2 are easy to generalize for algebras with multiple binary operations, i.e., for an arbitrary binary operad $\mathop{Var}\nolimits$.

Remark 3**.**

In the definition of a derived identity, a derivation $d$ of an algebra $A$ may be replaced with generalized derivation, i.e., a linear map $D:A\to A$ such that

[TABLE]

where $\lambda$ is a fixed scalar in $\Bbbk$. Indeed, it is enough to note that $D-\lambda\mathop{id}\nolimits$ is an ordinary derivation.