Special Tortkara Superalgebras

• Special Tortkara superalgebras are Z₂-graded algebras that embed into the supercommutator structure of a Zinbiel superalgebra, extending classical Tortkara theory.
• They are constructed using super shuffle products in free Zinbiel superalgebras and distinguished by an anti-invariance criterion under a specific linear map.
• A notable insight is the existence of exceptional homomorphic images in two-generator cases, revealing parity-sensitive obstructions absent in the non-super setting.

Searching arXiv for the cited papers to ground the article in current sources. Special Tortkara superalgebras are Tortkara superalgebras that admit an embedding into the supercommutator algebra of a Zinbiel superalgebra. In the formulation developed in "On Zinbiel and Tortkara superalgebras" (Bouarroudj et al., 15 Aug 2025), the subject is defined over a field $\mathbb K$ of characteristic $0$, with all structures $\mathbb Z_2$-graded. The theory extends the ordinary speciality problem for Tortkara algebras to the graded setting, but the super case exhibits behavior absent from the classical one: in contrast to the non-super case, some homomorphic images of special Tortkara superalgebras on two generators are exceptional (Bouarroudj et al., 15 Aug 2025). This places special Tortkara superalgebras at the intersection of Zinbiel superalgebra combinatorics, graded commutator constructions, and speciality-versus-exceptionality phenomena already familiar in ordinary Tortkara theory (Dzhumadil'daev et al., 2018).

1. Foundational definitions

A superalgebra is a $\mathbb Z_2$-graded $\mathbb K$-vector space

$A=A_{\bar 0}\oplus A_{\bar 1}$

with bilinear multiplication. For a homogeneous element $a\in A_{\bar i}$, its parity is denoted $|a|=\bar i$. The standard super-anticommutator and supercommutator are

$$\{a,b\}:=ab+(-1)^{|a||b|}ba, \qquad [a,b]:=ab-(-1)^{|a||b|}ba$$

for homogeneous $a,b$ (Bouarroudj et al., 15 Aug 2025).

A multiplication is super anti-commutative if

$0$0

This is the sign convention relevant for Tortkara superalgebras (Bouarroudj et al., 15 Aug 2025). The paper defines a Zinbiel superalgebra by the identity

$0$1

equivalently,

$0$2

for all homogeneous $0$3 (Bouarroudj et al., 15 Aug 2025).

A Tortkara superalgebra is defined by the Grassmann envelope method. If $0$4 is a superalgebra and $0$5 is the Grassmann algebra, its Grassmann envelope is

$0$6

Then $0$7 is a Tortkara superalgebra if $0$8 is a Tortkara algebra (Bouarroudj et al., 15 Aug 2025). The paper states that this is equivalent to super anti-commutativity together with the Tortkara super-identity

$0$9

where

$\mathbb Z_2$0

for homogeneous $\mathbb Z_2$1 (Bouarroudj et al., 15 Aug 2025).

Within this class, a Tortkara superalgebra $\mathbb Z_2$2 is called special if there exists a Zinbiel superalgebra $\mathbb Z_2$3 such that $\mathbb Z_2$4 is a super-subalgebra of

$\mathbb Z_2$5

where the multiplication is the supercommutator

$\mathbb Z_2$6

If no such embedding exists, $\mathbb Z_2$7 is called exceptional (Bouarroudj et al., 15 Aug 2025). This is exactly parallel to the classical Tortkara setting, where special Tortkara algebras are realized inside commutator algebras of Zinbiel algebras (Dzhumadil'daev et al., 2018).

2. Zinbiel superalgebras as the source of speciality

The central structural mechanism is that every Zinbiel superalgebra yields a special Tortkara superalgebra by passage to the supercommutator. The paper proves that if $\mathbb Z_2$8 is a Zinbiel superalgebra, then $\mathbb Z_2$9 is supercommutative and associative, while $\mathbb Z_2$0 satisfies the Tortkara super-identity (Bouarroudj et al., 15 Aug 2025). Thus the supercommutator of a Zinbiel superalgebra is always a Tortkara superalgebra, and this construction provides the entire source of special Tortkara superalgebras in the sense of the paper (Bouarroudj et al., 15 Aug 2025).

This mirrors a standard fact in the ordinary theory: every Zinbiel algebra with the commutator multiplication gives a Tortkara algebra (Dzhumadil'daev et al., 2018). In the non-super setting this realization underlies the notion of special Tortkara algebra and the construction of the free special object $\mathbb Z_2$1 inside $\mathbb Z_2$2 (Dzhumadil'daev et al., 2018). The super theory adopts the same template, but with parity-sensitive signs inserted throughout (Bouarroudj et al., 15 Aug 2025).

The paper also records a parallel statement for the symmetric part. If $\mathbb Z_2$3 is a Zinbiel superalgebra, then under the super-anticommutator

$\mathbb Z_2$4

the algebra becomes supercommutative and associative (Bouarroudj et al., 15 Aug 2025). This situates Zinbiel superalgebras between two derived structures: the symmetric part gives a supercommutative associative algebra, while the skew-symmetric part gives a Tortkara superalgebra (Bouarroudj et al., 15 Aug 2025). A plausible implication is that speciality questions for Tortkara superalgebras are inseparable from the internal decomposition theory of Zinbiel superalgebras.

Another source of examples arises from Rota–Baxter operators. If $\mathbb Z_2$5 is a supercommutative associative superalgebra and $\mathbb Z_2$6 is an even Rota–Baxter operator, then

$\mathbb Z_2$7

defines a Zinbiel superalgebra (Bouarroudj et al., 15 Aug 2025). Hence the supercommutator

$\mathbb Z_2$8

produces a class of special Tortkara superalgebras (Bouarroudj et al., 15 Aug 2025). The paper further states that if $\mathbb Z_2$9 is Zinbiel and $\mathbb K$0 is an even Rota–Baxter operator, then

$\mathbb K$1

is again Zinbiel, and inductively

$\mathbb K$2

Each such product yields, by supercommutator, another special Tortkara superalgebra (Bouarroudj et al., 15 Aug 2025).

3. Free special Tortkara superalgebras

If $\mathbb K$3 is a set of homogeneous generators, the free Zinbiel superalgebra $\mathbb K$4 contains a distinguished supersubalgebra $\mathbb K$5, defined as the supersubalgebra of $\mathbb K$6 generated by $\mathbb K$7 (Bouarroudj et al., 15 Aug 2025). This $\mathbb K$8 is the free special Tortkara superalgebra on $\mathbb K$9. The paper explicitly notes that $A=A_{\bar 0}\oplus A_{\bar 1}$0 is not the free Tortkara superalgebra in the abstract variety; it is free only in the subclass of special Tortkara superalgebras (Bouarroudj et al., 15 Aug 2025).

The construction relies on an explicit model of the free Zinbiel superalgebra. Let $A=A_{\bar 0}\oplus A_{\bar 1}$1, and consider the tensor superalgebra

$A=A_{\bar 0}\oplus A_{\bar 1}$2

The paper defines a super shuffle product. If

$A=A_{\bar 0}\oplus A_{\bar 1}$3

then

$A=A_{\bar 0}\oplus A_{\bar 1}$4

and

$A=A_{\bar 0}\oplus A_{\bar 1}$5

The Zinbiel product is then given by

$A=A_{\bar 0}\oplus A_{\bar 1}$6

and this yields a free Zinbiel superalgebra with basis

$A=A_{\bar 0}\oplus A_{\bar 1}$7

consisting of left-normed monomials (Bouarroudj et al., 15 Aug 2025).

This basis is the technical backbone of the speciality theory. The ordinary paper "On the speciality of Tortkara algebras" (Dzhumadil'daev et al., 2018) likewise realizes $A=A_{\bar 0}\oplus A_{\bar 1}$8 inside the free Zinbiel algebra and proves that skew-right-commutative elements form a basis of the free special Tortkara algebra. In the super case, the same strategy survives but requires sign corrections at every transposition and shuffle (Bouarroudj et al., 15 Aug 2025).

4. The criterion for Tortkara elements

A major theorem gives a recognition criterion for membership in $A=A_{\bar 0}\oplus A_{\bar 1}$9. The authors define a linear map

$a\in A_{\bar i}$0

on basis monomials by

$a\in A_{\bar i}$1

$a\in A_{\bar i}$2

and, for $a\in A_{\bar i}$3,

$a\in A_{\bar i}$4

(Bouarroudj et al., 15 Aug 2025). For a homogeneous element $a\in A_{\bar i}$5 of degree $a\in A_{\bar i}$6, one sets

$a\in A_{\bar i}$7

Elements of the form

$a\in A_{\bar i}$8

are called super skew-right-commutative, or super skew-rcom, elements (Bouarroudj et al., 15 Aug 2025).

The central theorem states that if $a\in A_{\bar i}$9 is homogeneous of degree greater than $|a|=\bar i$0, then

$|a|=\bar i$1

Equivalently, the elements of the free special Tortkara superalgebra are exactly the homogeneous Zinbiel elements that are anti-invariant under $|a|=\bar i$2 (Bouarroudj et al., 15 Aug 2025). The paper explicitly describes this as a super analogue of the “Lie criterion” from the non-super literature (Bouarroudj et al., 15 Aug 2025). In the ordinary setting, the corresponding statement is that $|a|=\bar i$3 is a Lie element in the free Zinbiel algebra if and only if $|a|=\bar i$4 (Dzhumadil'daev et al., 2018). The super paper adopts the same formal pattern, but the target class is not Lie elements in general; it is the class of Tortkara elements inside the free Zinbiel superalgebra (Bouarroudj et al., 15 Aug 2025).

The proof mechanism combines combinatorics and induction. The paper outlines the following steps: supercommutators of generators produce super skew-rcom elements; products and commutators of such elements remain in their span; every element of $|a|=\bar i$5 therefore satisfies $|a|=\bar i$6; and conversely every $|a|=\bar i$7-anti-invariant monomial $|a|=\bar i$8 lies in $|a|=\bar i$9 (Bouarroudj et al., 15 Aug 2025). For degree $$\{a,b\}:=ab+(-1)^{|a||b|}ba, \qquad [a,b]:=ab-(-1)^{|a||b|}ba$$0, the paper gives the decomposition

$$\{a,b\}:=ab+(-1)^{|a||b|}ba, \qquad [a,b]:=ab-(-1)^{|a||b|}ba$$1

which provides the base step showing that degree-$$\{a,b\}:=ab+(-1)^{|a||b|}ba, \qquad [a,b]:=ab-(-1)^{|a||b|}ba$$2 super skew-rcom elements are Tortkara elements (Bouarroudj et al., 15 Aug 2025).

The paper also records an alternative proof suggested by the referee: transfer the problem to an ordinary Zinbiel algebra on lifted generators $$\{a,b\}:=ab+(-1)^{|a||b|}ba, \qquad [a,b]:=ab-(-1)^{|a||b|}ba$$3 via the Grassmann envelope, invoke the known ordinary result there, and then transfer back while tracking Grassmann signs (Bouarroudj et al., 15 Aug 2025). This suggests that a substantial portion of the free special theory is inherited from the ordinary case through the Grassmann-envelope formalism, though the nontriviality of sign bookkeeping remains decisive.

5. Speciality, quotients, and the exceptional phenomenon

The speciality problem becomes most delicate for quotients. The paper uses the following criterion: if $$\{a,b\}:=ab+(-1)^{|a||b|}ba, \qquad [a,b]:=ab-(-1)^{|a||b|}ba$$4 is an ideal of $$\{a,b\}:=ab+(-1)^{|a||b|}ba, \qquad [a,b]:=ab-(-1)^{|a||b|}ba$$5, and $$\{a,b\}:=ab+(-1)^{|a||b|}ba, \qquad [a,b]:=ab-(-1)^{|a||b|}ba$$6 is the ideal of the free Zinbiel superalgebra $$\{a,b\}:=ab+(-1)^{|a||b|}ba, \qquad [a,b]:=ab-(-1)^{|a||b|}ba$$7 generated by $$\{a,b\}:=ab+(-1)^{|a||b|}ba, \qquad [a,b]:=ab-(-1)^{|a||b|}ba$$8, then

$$\{a,b\}:=ab+(-1)^{|a||b|}ba, \qquad [a,b]:=ab-(-1)^{|a||b|}ba$$9

(Bouarroudj et al., 15 Aug 2025). This is the super analogue of Cohn’s criterion. In the ordinary theory, an analogous criterion underlies the proof that every homomorphic image of a free special Tortkara algebra on two generators is special (Dzhumadil'daev et al., 2018).

The super case diverges sharply from that classical picture. The paper states that in the ordinary case, Dzhumadil’daev–Ismailov–Mashurov had proved that every homomorphic image of a free special Tortkara algebra on two generators is special (Bouarroudj et al., 15 Aug 2025, Dzhumadil'daev et al., 2018). By contrast, the super paper proves the theorem:

There exists a homomorphic image of $a,b$0 which is exceptional (Bouarroudj et al., 15 Aug 2025).

The counterexample is explicit. Take generators $a,b$1 with parities

$a,b$2

Let $a,b$3 be the ideal of $a,b$4 generated by

$a,b$5

Now consider

$a,b$6

The paper shows that $a,b$7, where $a,b$8 is the ideal of $a,b$9 generated by $0$00, because

$0$01

It also shows that $0$02 by the criterion theorem, but $0$03, since there do not exist $0$04 such that

$0$05

Therefore

$0$06

and the quotient

$0$07

is exceptional (Bouarroudj et al., 15 Aug 2025).

This is the defining new phenomenon of the super setting. The paper emphasizes that the failure is not a cosmetic byproduct of superization. Rather, the parity assignment

$0$08

permits identities and cancellations producing an obstruction element that has no counterpart in the ordinary two-generator theory (Bouarroudj et al., 15 Aug 2025). A plausible implication is that any super analogue of a general Cohn-type theorem must depend on parity data, not merely on the number of generators.

6. Low-dimensional structure and classification context

The paper also classifies all Tortkara superalgebras of dimensions $0$09 and $0$10, although not specifically by speciality versus exceptionality (Bouarroudj et al., 15 Aug 2025). This classification provides structural context for the speciality problem, especially because many low-dimensional examples remain close to Lie or Malcev behavior.

For dimension $0$11, the paper lists three isomorphism classes with $0$12: $0$13 and two classes with $0$14: $0$15 It states that all of these are Malcev type, hence Lie superalgebras (Bouarroudj et al., 15 Aug 2025).

For dimension $0$16, with $0$17, the paper lists the classes

$0$18

and states that the Malcev ones are

$0$19

while

$0$20

are not Lie (Bouarroudj et al., 15 Aug 2025). With $0$21, the classes

$0$22

are all listed as Malcev (Bouarroudj et al., 15 Aug 2025).

A later classification paper on $0$23-dimensional anticommutative superalgebras over $0$24 gives an independent low-dimensional perspective. It identifies the $0$25-dimensional Tortkara superalgebras of type $0$26 as

$0$27

and of type $0$28 as

$0$29

with explicit multiplication tables and geometric decomposition into irreducible components (Abdelwahab et al., 15 Feb 2026). That paper does not define or determine which of these are special or exceptional (Abdelwahab et al., 15 Feb 2026). This suggests that low-dimensional classification and speciality theory remain partly orthogonal problems: the former organizes the ambient graded varieties, while the latter concerns realizability inside Zinbiel supercommutator algebras.

The low-dimensional data do not produce the exceptional quotient in the super speciality theorem. The paper explicitly notes that the exceptional quotient is a constructed homomorphic image of the free special object $0$30, not a low-dimensional classification example (Bouarroudj et al., 15 Aug 2025). This indicates that exceptionality in the super setting is fundamentally a quotient-theoretic obstruction rather than merely a feature of small multiplication tables.

7. Relation to ordinary Tortkara theory and open direction

The ordinary theory provides the immediate prototype. In "On the speciality of Tortkara algebras" (Dzhumadil'daev et al., 2018), the free special Tortkara algebra $0$31 is realized inside the free Zinbiel algebra, skew-right-commutative elements form a basis, and the criterion

$0$32

is established (Dzhumadil'daev et al., 2018). That paper also proves the Tortkara analogues of Shirshov’s and Cohn’s theorems: the free Tortkara algebra on two generators is special, every two-generated Tortkara algebra is special, and every homomorphic image of a free special Tortkara algebra on two generators is special; this fails for three generators (Dzhumadil'daev et al., 2018).

The super paper adopts the same conceptual architecture but overturns the two-generator quotient theorem. Its principal contribution to the speciality problem is therefore not merely a superized restatement of ordinary results, but the demonstration that the super quotient theory is strictly subtler than the ordinary one (Bouarroudj et al., 15 Aug 2025). The paper’s own final synthesis identifies five main contributions: the definition of Tortkara superalgebras via Grassmann envelopes; proof that supercommutators of Zinbiel superalgebras are Tortkara; construction of free Zinbiel superalgebras via super shuffle products; the criterion

$0$33

and the existence of an exceptional homomorphic image of $0$34 (Bouarroudj et al., 15 Aug 2025).

The paper does not formally state open problems, but it clearly suggests several: characterize which homomorphic images of $0$35 remain special in the super setting; determine whether there is a broader super-Cohn theorem under additional parity assumptions; classify exceptional Tortkara superalgebras more systematically; and study speciality in higher dimensions and for other generator parities (Bouarroudj et al., 15 Aug 2025). These are suggestions explicitly identified in the paper’s synthesis rather than solved results.

In this sense, special Tortkara superalgebras form a graded analogue of a classical special-exceptional theory, but one in which parity-sensitive identities alter the behavior of quotients in an essential way. The free-object combinatorics still come from Zinbiel structures and the $0$36-anti-invariance criterion, yet the super setting introduces a genuine embeddability obstruction absent from the non-super two-generator case (Bouarroudj et al., 15 Aug 2025).