Hopf structure on the universal enveloping of a super Lie algebra

Question

The universal enveloping algebra of a Lie algebra has a canonically defined Hopf algebra structure. Is the same true of the universal enveloping of a super Lie algebra? A presentation in terms of the universal properties would be of most interest.

Answer 1

No, it is not true unless the odd part is zero or the characteristic is 2. More precisely, there is no natural Hopf algebra structure outside these conditions, but there may be some odd examples. It is a Hopf superalgebra instead, which is what Peter May means in his answer.

You can see this on the easiest example: $L_0=0$, $L_1=F$, the field. Its universal enveloping is $F[x]/(x^2)$. It is a Hopf algebra if and only if the characteristic of $F$ is 2.

In general, you have difficulty with comultiplying the odd element $x\in L_1$ because $2x^2=[x,x]$. If you try $\Delta (y)=1\otimes y + y \otimes 1$ for $y\in L_0$ as suggested by Gods, no formula for $\Delta (x)$ will accommodate this relation.

PS Note that $C[x]/(x^2)$ is the cohomology $H^\ast (SU_2, C)$. The cohomology of a compact Lie group is the original example of "a Hopf algebra" studied by Hopf. However, the modern terminology follows Sweedler's 1969 book, which I also follow in my answer and urge everyone to follow as well.

PPS There is no difference between associative ($Z_2$-graded) algebras and associative superalgebras. Why is there a difference between Hopf algebras and superalgebras? It happens because the interaction axiom between multiplication and comultiplication requires braiding. Take any associative superalgebra $A=A_0\oplus A_1$. Then $A\otimes^{sup} A = A \otimes A$ as a vector space but their algebra structures are different. Given homogeneous $w,x,y,z$, the products are different: $$ (w \otimes^{sup} x) \cdot (y \otimes^{sup} z) = (-1)^{|x||y|}wy \otimes^{sup} xz, \ \ (w \otimes x) \cdot (y \otimes z) = wy \otimes xz. $$ Now the universal enveloping $U=U(L)$ is an associative $Z_2$-graded algebra or superalgebra. For each $x\in L$, the God-given $\Delta (x) = 1\otimes x + x \otimes 1$ extends to an algebra homomorphism $$ \Delta: U \rightarrow U\otimes^{sup}U $$ but not to an algebra homomorphism to $U\otimes U$. This is why it is Hopf superalgebra. It has the obvious universal property:

the unique algebra map $U \rightarrow U\otimes^{sup}U$ given on $L$ by $x\mapsto 1\otimes x + x \otimes 1$.

Answer 2

This is true. In other language, if I understand rightly, a super Lie algebra is just a graded Lie algebra with grading over {0,1} (even and odd), with the standard sign conventions as in algebraic topology. The treatment of graded Lie algebras and their universal enveloping algebras in Section 22.1 of ``More concise algebraic topology'' by Kate Ponto and myself works verbatim. We understood grading to be over the integers there, but the mathematics works equally well with other gradings. Chapters 20-24 of that book are a purely algebraic modernized treatment of bialgebras and Hopf algebras, with focus on (graded) Lie algebras and restricted Lie algebras. Aside from some topological examples, these chapters are independent of the earlier ones.

@{Bugs Bunny} I wrote "with the standard sign conventions as in algebraic topology''. Those conventions long precede the "super'' language, and in fact long precede Milnor and Moore's basic 1965 paper "On the structure of Hopf algebras". They were already standard when I started as a graduate student in 1960. For us, graded Lie algebras and graded Hopf algebras, whether Z or Z/2 graded, have always meant what people very much later started calling ``super''. I vividly remember Irving Kaplansky, certainly not an algebraic topologist, sneering at the word "super" since he too had always understood graded structures the way we do. The fundamental structure theorem for graded Lie algebras is Poincaré-Birkhoff-Witt, which guarantees that the associated graded of the universal enveloping algebra with respect to the Lie filtration is a free graded commutative algebra, polynomial on even degree generators tensor exterior on odd degree generators (just polynomial in characteristic 2), and that is a perfectly good Hopf algebra. The original notion of a Hopf algebra was meant to describe the structure seen by the real homology of a compact Lie group, which of course is an exterior algebra on odd degree elements. Nobody can define those away as not Hopf algebras!

(Bastille day edit): Ok, there is no real mathematical argument, but an argument about terminology. I don't have access to a copy of Sweedler's 1969 book, but my recollection is that he focused on classical ungraded algebra. In fact, the first paragraph of its Math Review reads "The book is mainly concerned with Hopf algebras which are not graded" and ends with "The theory of graded Hopf algebras is not treated in depth". So that reference seems spurious. The definitive 1965 paper of Milnor and Moore "On the structure of Hopf algebras'' has both historical priority and very common present usage.

Historically, I think this is an example of the common phenomenon of people rediscovering known mathematics and giving it a (super) new name. The tower of Babel was constructed that way. It spawns ambiguous language: the same term, in this case "Hopf algebra", is given two meanings. Graded Hopf algebras, with signs, are never going to be renamed "super'' by those in the fields in which they originated: to them (or us), they are just plain Hopf algebras, always have been, always will be. The adjective super is superfluous. (Graded Hopf algebras without signs might unglamorously be called unsigned Hopf algebras. They are structurally very much less well-behaved mathematical objects). And on that ground, my answer of ``yes'' to the original question remains valid. I'm sure the reference I gave provides exactly what Nadia wants.

Answer 3

The notion of hopf algebras has slowly emerged from the work of topologists in the late '30's and '40's on the cohomology of compact Lie groups and their homogeneous spaces. Initially the term had been used in the "graded" or "signed" sense (for example, Milnor and Moore's seminal paper refers to $\mathbb{Z}$-graded hopf algebras and I think this is the case in Mac Lane's "Homology" book as well).
As far as I can understand, the "present day definition" (i.e. the "unsigned" and "ungraded" as presented in most hopf algebra textbooks after the '70's - that is after Sweedler's textbook) has been formulated during the early '60's and is mainly due to works of Cartier and Dieudonne. (the introduction of A primer of Hopf algebras includes interesting details on that point) However, lots of people from different schools have contributed to the evolution of this notion; lots of authors have kept using the term for signed or graded objects long after that. So it does not come as a surprise that there are still discussions on the "correct" form of the definition.

It is not my purpose -in this answer- to argue on what is the "correct" definition but rather to try to provide some terminology, unifying the above descriptions: That is the language of braided groups, in the sense this notion has been introduced and used since the mid'90's after the Majid's school of hopf algebras and quantum groups.
Imo, one of the advantages of this language is that it helps in clarifying that the "grading" and the "signs" are not the same thing (the same grading may refer to different sign rules).

$\bullet$ Some introductory remarks on graded Lie structures, their "signs", their colors and their universal properties:

Let $G$ a countable, abelian group and a function $\theta: G \times G \rightarrow \mathbb{C}^{*}$ satisfying (for all $a, b, c \in G$) \begin{equation} \begin{array}{c} \theta(a+b, c) = \theta(a,c) \theta(b,c) \\ \theta(a, b+c) = \theta(a,b) \theta(a,c) \\ \theta(a,b) \theta(b,a) = 1 \end{array} \end{equation} The function $\theta$ is called a color map on $G$ or a commutation factor for $G$. It is a symmetric bicharacter on $G$. Given such a function and a $G$-graded, complex v.s. $L = \oplus_{g \in G} L_{g}$, then $L$ will be called a $\theta$-colored, $G$-graded Lie algebra or a $(G, \theta)$-Lie algebra, for short, If $L$ is further equipped with a bilinear, non-associative multiplication $\langle .., .. \rangle$ which repsects the grading, is $\theta$-antisymmetric and satisfies the $\theta$-Jacobi, identity i.e. $$ \begin{array}{c} \langle L_{a}, L_{b} \rangle \subseteq L_{a+b} \\ \langle x, y \rangle = - \theta(a,b) \langle y, x \rangle \\ \theta(c,a) \langle x, \langle y, z \rangle \rangle + \theta(b,c) \langle z, \langle x, y \rangle \rangle + \theta(a,b) \langle y, \langle z, x \rangle \rangle = 0 \\ \end{array} $$ for all $x \in L_{a}$, $y \in L_{b}$, $z \in L_{c}$ and for all $a, b, c \in G$.

Similarly to the ordinary Lie algebra case, the universal enveloping algebra (UEA) of the $(G, \theta)$-Lie algebra $L$ is a pair $(\mathbb{U}(L), i_{\mathbb{U}})$, where $\mathbb{U}(L)$ is a $G$-graded, associative algebra and $i_{\mathbb{U}}$ an homogeneous linear map of zero degree (i.e.: a $G$-graded v.s. homomorphism) $i_{\mathbb{U}}: L \rightarrow \mathbb{U}(L)$, which -by definition- satisfies $$i_{\mathbb{U}}(\langle x, y \rangle) = i_{\mathbb{U}}(x)i_{\mathbb{U}}(y) - \theta(a,b)i_{\mathbb{U}}(y)i_{\mathbb{U}}(x)$$ $\mathbb{U}(L)$ is defined to be the quotient of the tensor algebra $\mathbb{T}(L)$ with the homogeneous -wrt to the $G$-grading- ideal $Ι(L)$ generated from all elements of the form $\langle x, y \rangle - xy + \theta(a,b) yx$ for all homogeneous elements $x,y$ of $L$ and $i_{\mathbb{U}}$ is the composition $L \hookrightarrow \mathbb{T}(L) \twoheadrightarrow \mathbb{U}(L)$.

As a consequence of the above $\mathbb{U}(L) = \mathbb{T}(L)/Ι(L)$ has the following universal property:

If $\mathcal{Α}_{gr}$ is an associative, $G$-graded algebra and $f_{L} : L \rightarrow \mathcal{Α}_{gr}$ is a $G$-graded v.s. homomorphism (equivalently: an homogeneous linear map of zero degree) which furthermore satisfies $$ f_{L}( \langle x, y \rangle ) = \langle f_{L}(x), f_{L}(y) \rangle = f_{L}(x)f_{L}(y) - \theta(a,b) f_{L}(y)f_{L}(x) $$ for all homogeneous elements $x \in L_{a}$, $y \in L_{b}$, then there is a unique homomorphism of associative $G$-graded algebras (equivalently: homogeneous assoc algebra homomorphism of zero degree) $f : \mathbb{U}(L) \rightarrow \mathcal{Α}_{gr}$ which extends the linear map $f_{L}$, such that $$f \circ i_{\mathbb{U}} = f_{L}$$ $f$ is fully defined by its values on the generators of $\mathbb{U}(L)$, i.e. from the values of $f_{L}$ on the elements of $L$.

The usual Poincare-Birkhoff-Witt theorem generalizes as well.

$\bullet$ On the Hopf structure of the UEA $U(L)$, of the $\theta$-colored, $G$-graded Lie algebra:

If we equip the v.s. $\mathbb{U}(L) \otimes \mathbb{U}(L)$ with an associative product defined by \begin{equation} (x \otimes y)(z \otimes w) = \theta(a,b) xz \otimes yw \end{equation} for all homogeneous elements $y \in L_{a}$ and $z \in L_{b}$, then the $G$-graded v.s. $\mathbb{U}(L) \otimes \mathbb{U}(L)$ becomes an associative, $G$-graded algebra. We will denote this by $\mathbb{U}(L) \underline{\otimes} \mathbb{U}(L)$ and call it ($G$-graded), $\theta$-braided tensor product algebra or $(G, \theta)$-tensor product algebra.

The UEA $\mathbb{U}(L)$ of the $(G, \theta)$-Lie algebra $L$, is not a hopf algebra -well at least not in the "ordinary" (ungraded) sense (here "ordinary" should be taken to mean the modern day definition of hopf algebras as this is presented in most of the hopf algebra textbooks which have appeared after the '70's). Let me try to shed some light in this point:
$\mathbb{U}(L)$ is equipped with a "comultiplication"
$$ \underline{\Delta} : \mathbb{U}(L) \rightarrow \mathbb{U}(L) \underline{\otimes} \mathbb{U}(L) $$ which is a homomorphism of assoc., $G$-graded algebras (equivalently: a homogeneous homomorphism of assoc algebras of degree $0$) i.e. $$ \underline{\Delta}(ab) = \sum \theta\big(deg(a_{2}), deg(b_{1})\big) a_{1}b_{1} \otimes a_{2}b_{2} = \underline{\Delta}(a) \underline{\Delta}(b) $$ for all $a,b \in \mathbb{U}(L)$, with $\underline{\Delta}(a) = \sum a_{1} \otimes a_{2}$, $\underline{\Delta}(b) = \sum b_{1} \otimes b_{2}$, and $a_{2}$, $b_{1}$ homogeneous. The product $\underline{\Delta}(a) \underline{\Delta}(b)$ in the rhs of the above is understood to be in the $\mathbb{U}(L) \underline{\otimes} \mathbb{U}(L)$ algebra.
Due to the universal property of the UEA, $\underline{\Delta}$ is uniquely defined by its values on the elements of $L$ i.e. on the generators of $\mathbb{U}(L)$ $$ \underline{\Delta}(x) = 1 \otimes x + x \otimes 1 $$ Similarly, $\mathbb{U}(L)$ is equipped with the "antipode" $\underline{S} : U(L) \rightarrow U(L)$ which is no more an algebra antihomomorphism (as in the "ordinary" hopf algebra case) but a twisted or braided antihomomorphism of $G$-graded algebras, in the sense that: $$ \underline{S}(ab) = \theta\big(deg(a), deg(b)\big) \underline{S}(b)\underline{S}(a) $$ and $deg(a) = deg(\underline{S}(a))$ for all homog elements $a,b \in \mathbb{U}(L)$. Again the universal property of the UEA, ensures us that the antipode is uniquely defined by its values on the elements of $L$ i.e. on the generators of the UEA $\mathbb{U}(L)$: $$ \underline{S}(x) = -x $$ If the above are complemented with the counit $\underline{\varepsilon}(x) = 0 $ for all $x \in \mathbb{U}(L)$ then we get a $G$-graded, $\theta$-braided Hopf algebra or -for short- a $(G,\theta)$-hopf algebra.

$\bullet$ The relation with ordinary Hopf algebras and super-Hopf algebras:

The notion of $G$-graded, $\theta$-braided Hopf algebras as defined above, generalizes the definition of ordinary hopf algebras and the various $\mathbb{Z}$ or $\mathbb{Z}_2$-graded (super) Hopf algebras:

• If $G=\mathbb{Z}_{2}$ and the color function $\theta$ is taken to be $\theta(a,b) = (-1)^{ab}$, then a $(G, \theta)$-Lie algebra is a lie superalgebra (or a $\mathbb{Z}_2$-graded Lie algebra) and its UEA is a $(\mathbb{CZ}_{2},\theta)$-hopf algebra which is nothing different than the hopf superalgebra (known under this name mainly in the mathematical physics literature dealing with SUSY algebras) referred to in the OP and also in Bugs Bunny's answer.
• If $G$ and $\theta$ are trivial we get the ordinary definition of Hopf algebras.

An interesting remark here has to do with the fact that the same grading group $G$ may give rise to different $G$-graded, $\theta$-braided Hopf algebras (for different choices of the color function $\theta$). (imo this imposes interesting and tractable classification problems for such structures).

According to present day terminology (here i am mainly following the terminology as used by the Majid's school) such structures are called braided groups or hopf algebras in the braided monoidal Category ${}_{\mathbb{CZ}_2}\mathcal{M}$ of $\mathbb{CZ}_2$-modules (remember that the $\mathbb{CZ}_2$-modules are exactly the $\mathbb{Z}_2$-graded v.s. or "super"-v.s.). The braiding $\theta$ can be shown to be "generated" through a "1-1" correspondence, with the non-trivial, quasitriangular structure (i.e. the non-trivial $R$-matrix) of the $C\mathbb{Z_2}$ group hopf algebra: $$R=\frac{1}{2}\big(1\otimes 1+1\otimes g+g\otimes 1-g\otimes g\big)$$ (More generally, under the term braided groups we frequently mean hopf algebras in the braided monoidal Category ${}_{H}\mathcal{M}$ of $H$-modules, where $H$ is any quasitriangular hopf algebra -and not necessarily a group algebra).

$\bullet$ Representations of hopf algebras vs "super"-representations of hopf superalgebras:

It has been shown that given a $(\mathbb{CZ}_{2},\theta)$-hopf algebra $H$ (or: a "super"-hopf algebra $H$) we can form the smash product hopf algebra $H\star \mathbb{CZ}_2$ by adjoining an extra generator $g$ to $H$. This is an ordinary hopf algebra and the construction is functorial in the sense that there is an equivalence of categories $${}_{H}\underline{\mathcal{M}} \thicksim {}_{H \star CZ_{2}}\mathcal{M}$$ between the Category ${}_{H}\underline{\mathcal{M}}$ of super-reps of $H$ and the Category ${}_{H \star CZ_{2}}\mathcal{M}$ of representations of the (ordinary) smash product Hopf algebra $H \star CZ_{2}$.
The converse procedure can also be done: these are the Bosonization and Transmutation techniques developed in the early 90's (and partially based on the previously known idea of Radford's biproduct).

Finally -hoping that the above are somewhat helpful for the OP- some references:

• Foundations of Quantum Group theory, S. Majid (see mainly chapter 9),
• Hopf algebras and their actions on rings, S. Montgomery (see mainly chapter 10)

If you are further interested on the terminology, maybe you can find some interest in section 3.1 of

• Gradings, Braidings, Representations, Paraparticles: Some Open Problems (and the references therein).