重观说

 大凡物之可久者，不以先得而尽，不以人先而穷。四时行焉，百物生焉，天何言哉？而人以其心观之、思之、咏之，斯不穷矣。

今之治数学者，闻智能之器能算、能证、能穷搜，则惶然曰："器既为之，人复何为？"余谓此非数学之危，乃首功之见蔽之也。

夫数学岂以先得一名、先证一理、先解一题为极哉？今之俗，一题既解，功归首解；一理既明，名属先证。后人虽深思别解，更造语言，发其未发之美，皆若无所增。呜呼！此以名录尽鸟，以登顶尽山，以证明尽理也。

观有独，有共。独观者，一人之涵泳也；共观者，百世之传承也。

余观鸟，旧识者未尝不复观。非不知世人已见之也，爱其四时之变：雄羽春妍，冬雀蓬然成球，鸣止各异，飞栖异情。鸟不以人识而减其色，时不以岁迁而止其变。若以"已有人见"而弃之，则鸟之生趣绝矣。余治数学，旧理亦未尝不复思。非不知前人已证之也，爱其可重铸、可重言、可重观。同一理也，在代数之言则质，在几何之言则明，在范畴之言则通。言殊而理一，境变而味新。若以"已有人证"而弃之，则理之深趣亦绝矣。此独观之乐也。

天下名山，先我而登者众矣。然山之所以名，非徒以高也。千载之下，迁客骚人登临凭吊，歌咏满壁，文章盈卷。山以人传，人以山寄。彼之题咏，非夺吾游，乃召吾游；彼之登临，非尽山，乃开山。庐山自晋宋以来，慧远、陶潜、谢灵运相继吟咏，至唐有李白之瀑布，白居易之桃花，宋有苏轼之西林壁。诸公非不知前贤已有题咏，各以一时之怀，寄一山之感，未尝以"已有李白"而搁笔，亦未尝以"已有苏轼"而贬诗。天宝间，高适、岑参、储光羲、杜甫、薛据同登大雁塔，各赋一诗，四诗并传，而薛据之作独佚。然诗可佚，登不可夺；名可失，观不可夺也。此共观之功也。

盖山不自言，而人言之；山不自文，而人文之。容千载之咏而不倦，传百代之情而不竭，此山之所以为名山也。

今智能之器，诚能速算、博证、穷搜。其证一难题，则告我曰：此思可通；其立一说，则告我曰：此路可行。既知其可通，则学者从容涵泳，不惧徒劳。譬如山未有人登，则望之者疑其不可攀，惧其无径，相与却步；及一人登焉，告人曰"此山可登"，于是后人欣然治装，循径而上。先登者之功，不在尽山，在示其可登也。此非大好事乎？

或曰："器既证之，人复证之，不亦赘乎？"应之曰："在名则是，在道则非。"

何谓也？名者，首功也，署名也，录目也。一题既解，则首解者居其功；一理既证，则先证者享其名。后人虽复解之，于名录无所增，于功簿无所加。名有涯而相夺，故曰：在名则是。道者，会心也，涵泳也，重铸也，玩味也，教化也。理非死物，待思而后活；言非一途，待择而后安。同一理也，一人证之，众人思之，其义愈广，其味愈深。道无涯而相生，故曰：在道则非。庄生曰："名者，实之宾也。"夫诗有先后，而工拙不系乎先后；理有先证，而深浅岂系乎先证哉？后人复证之，非复其旧也，乃入其史而再观之。

然重观非皆可贵也。其判有二：新与不新，视人而言；得与不得，视己而言。独观求于己，无新可也，有得斯可：学者自证一理，及见其已在书中，证虽旧，会心则真。共观传于人，则不徒曰有得，当言使人何所多得。前人题咏，若已说尽矣，而登者以一时之怀、一己之目，见其所未见，言其所未言，此使人多得者也。若徒袭其语、换其韵，山在纸上而不在目中，于己无会，于人无益，此不得之重观也。数学亦然。更造其言而理为之显，别辟其径而路为之通，一证既出而旧理若新，此使人多得者也；易其符而不易其思，换其言而不换其见，虽百证而不异一证，此两无所得者也。古人论文，不忌同题，而韩愈曰："惟陈言之务去。"同题者，山也；陈言者，不得之重观也。山可再登，陈言不可再陈。故前所谓"在道则非"者，谓其有得者也；两无所得者，在名固赘，在道亦赘。

重观之贵，本在会心与传承。冬雀再观，非为新见；旧理再思，非为新峰。然理与山又有异者：山不以咏而增其峰，理则或以重言而生新境。刘徽注《九章》，本为释旧，而割圆之术出焉；陈省身重证高斯—博内，本为求其内蕴，而陈类出焉；格罗腾迪克重述黎曼—罗赫，非徒广之，乃更造其言，以K群及其运算重构之，而K理论之端启焉；劳维尔重观康托尔之对角，本为明旧，而不动点之理出焉，罗素、哥德尔、塔斯基之说皆归一律。此数事者，非皆一朝而成，其后犹有百家之力；然其端，皆自重观启之。是有得之重观，其大者非徒咏山，实亦益山也。益山既久，则新峰起焉，而后来者复有先登之名。故名者，道之余也；今日之首功，多自昨日之重观积之。

或曰："一理可重观，诚然。然器既尽证一域之名题，则此域于后学，尚有可为乎？"应之曰：此今人所最惑，而余所最欲辨者也。夫一域者，非群题之聚而已。有其法，有其构，有其言，有其与他域之相连。首证毕其一事，而余事未毕，且或因之而易为：路既通，则可及者广；理既立，则可问者多。后学重观而更造其言、重立其构，即是创作，不必待新定理出而后追认之。前所谓新峰者，其余获也；新见新组，其本获也。故不可以名题之尽，断一域之尽——名题者，山之一峰，非山也。至于人之去留，则视其学有所养否、其评有所录否，此人事也，不可以理断也。

然名非可废也。首证者或独得其神，其署名所以志其劳、护其劳，此固当有〔一〕。所可病者，非名之有，乃名之独据其评也。夫题咏之有评，尚矣。钟嵘《诗品》，分上中下；萧统《文选》，择其菁华。所品者，工拙也，气象也，境界也；未尝问其先后，未尝问其题之新旧。同一庐山，李白得其势，白居易得其时，苏轼得其理，不以先咏而独尊，不以后咏而见黜。题咏满壁，而传者什一；山不拒其登，评不纵其滥，容与择并行，此题咏之评也。诗之所以有品，以诗之不稀也：人皆能诗，故不问其有无，而问其工拙。数学之评则异是，以证之稀也：一理之证，或数十年而不得，故一证既出，问其有无而已，不暇问其工拙。是以所重者，首证也；所计者，篇之数也，引之数也，刊之等也。一理既证，后人重证之，投之而多不受，曰：此已知也。一学既立，后人重述之，陈之而少见计，曰：此非新也。注疏、教本、通释，非无奖之者，然不与首证齿。工拙非不问也，先后常先之；深浅非不较也，多寡常掩之。此证稀之世之制也。今则异矣。器一夕而出百证，证不复稀，而工者稀。同一理也，其证有直有曲，有显有晦，有通于他域者，有止于一题者，高下判然。证既如诗之满壁，评不可不如诗之有品。昔以稀而贵首，今以滥而贵工。使仍以首评之，则先出者独录，虽拙亦录；后出者皆黜，虽工亦黜——是评名录也，非评数学也。使钟嵘以此品诗，则庐山止一诗，大雁塔止一咏，李杜之后无诗矣。

或曰："异日智器亦能得其神，亦能造自然之言，亦能辨数学之美，则奈何？"应之曰："无伤于道也。"夫前人能得其神，未尝废后人之思；前人能为至文，未尝废后人之游。李杜既咏名山，而山不为李杜所有；怀尔斯既证费马，而理不为怀尔斯所尽。器之能思，犹他人之能思也。彼有所见，何妨我复见之？彼有所言，何妨我复言之？器可为舟，亦可为梯，然游者自游，登者自登。人借器以游，则所见愈远；若因器而废人，则失其所以为学矣。

或曰："志道固善矣。然学者亦须衣食，而世之爵禄，皆以名予之。名既为器所夺，道将何托？"应之曰："此惧与前惧异。惧人先我一证，名之惧也，首功之见蔽之耳；惧学之无所养、群之不自主，道之惧也，不可以前说解之。此诚可忧，非一篇所能决也。然请言其大者。"世之以首功予爵禄，非古也，亦非必也。郑玄注经，刘徽注算，其所为者注也，非作也，而其名与作者并传；后世之设学官、立博士，所取者传经之人，非造经之人。首功之为币，特近世之制耳。制可变也，然不自变。首功既贱，币宜别铸；铸与不铸，在人不在器。或谓器多出则人择之，器博证则人断之，器能言则人教之，衣食可系于此。是未必然也。器若能得其神、辨其美，则择也、断也、教也，器亦将能之。人之所以犹当择、当断、当教，非以人之必胜器也，以数学者人之事，而吾辈愿其常为人之事也。世固有需其劳而久不酬其劳者，此不可以理推而免也。故志道者之衣食，非理之所能保，唯择之所能定：愿以何养人，愿养何等人，此吾辈所自决，器不能代也。

然则何以择？曰：其事有二。一曰改其评。数学之评，非天之所设，吾辈之所立也；吾辈立之，则吾辈可改之。愿自今以往，评数学者如评诗：品以工拙，不独品以先后；录以深浅，不独录以多寡。一理之重证，一学之重述，一书之注疏，一门之教化，皆当称其实而录之，恒而不偶。非谓重观皆录也，录其有得者，如诗之录其工者。刊物受"已知"之作，而问其言之工否；学官取"非新"之人，而问其明之深否。二曰养其人。评虽改矣，器之述若亦深，器之教若亦工，则新评之所归，仍可在器，而人未必得所托也。故改评之外，吾辈尚当自言其所愿：所以设学、所以养士、所以久聚而共讲者，非独取其所出之文，亦以贵夫能亲见其理之人，与夫使此等人生生不已之地。此非评之所能定，愿之所在也。夫如是，则重观者有所归，志道者有所托——托于吾辈之愿，非托于理之必然也。智器愈出，则重观愈多，得与不得杂陈，择之责愈重，而评愈不可缓。此非一人之事，治数学者当共图之。不改其评而徒畏其器，犹畏影而疾走也，走愈疾而影愈随，不知处阴以休之耳。

若以名尽数学，则智器足以夺人；若以道观数学，则智器适足助人。故曰：首功者，开山之功；重观者，成山之功。开山者一，成山者百。无开山则山不始，无成山则山不名；而后之开山者，未有不自成山中出者也。

名山之不朽，不在石，在其诗文；数学之不朽，不在术，在其思理。纵使器亦能赏山，山不因此而不可复游。四时之景不同，其乐亦无穷也。器虽巧，岂能尽人之乐哉？

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注

〔一〕博雷尔、塞尔一九五八年述黎曼—罗赫，开卷即曰：此格氏之得也，吾侪整理而书之耳。此署名所以志劳、护劳之一例；亦见整理阐释之功，数学中非全无承认者。

Representable Functors, Group-Like Elements, and Primitives in Hopf Algebras

Primitive Elements: The Lie Algebra Hidden Inside a Hopf AlgebraSome Immediate Properties of Primitive ElementsPrimitive Elements Form a Lie AlgebraPrimitive Elements Define a FunctorThe Universal Enveloping Algebra Is Its Left AdjointRepresentability and the Monoidal Origin of Group-Like and Primitive ElementsGroup-Like Elements Are RepresentablePrimitive Elements Are RepresentableA General Principle Behind the Two RepresentationsStrong Monoidality and the Group-Like MultiplicationPrimitive Elements and the Additive GroupThe Two Strong Monoidal Functors

 

Primitive Elements: The Lie Algebra Hidden Inside a Hopf Algebra

A Hopf algebra carries both multiplication and comultiplication:

$$m:H\otimes H\to H,\mathrm{\Delta }:H\to H\otimes H.$$

It is precisely the interaction between these two structures that allows both groups and Lie algebras to arise naturally inside a Hopf algebra.

We are already familiar with the group-like elements:

$$G(H)=\{g\in H\mid \mathrm{\Delta }(g)=g\otimes g,\epsilon (g)=1\}.$$

They form a group under multiplication.

In parallel, an element $x\in H$$x\in H$ is called primitive if

$$\mathrm{\Delta }(x)=x\otimes 1+1\otimes x.$$

We write

$$P(H)=\mathrm{Prim}(H)=\{x\in H\mid \mathrm{\Delta }(x)=x\otimes 1+1\otimes x\}.$$

At first sight, this merely looks like a way of selecting a special class of elements from a Hopf algebra. In fact, the primitive elements automatically form a Lie algebra, and this construction defines a functor

$$P:{\mathbf{Hopf}}_k⟶{\mathbf{Lie}}_k.$$

More importantly, it is the right adjoint of the universal enveloping algebra functor:

$$U⊣P.$$

In characteristic zero, this adjunction has a particularly strong property:

$$P(U(\mathfrak{g}))=\mathfrak{g}.$$

Thus, if we first pass from a Lie algebra to its universal enveloping algebra and then extract the primitive elements, no new primitive directions appear.

The goal of this article is to explain this functor and prove this statement.

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Some Immediate Properties of Primitive Elements

Let $x\in P(H)$$x\in P(H)$. Starting from

$$\mathrm{\Delta }(x)=x\otimes 1+1\otimes x,$$

apply $\epsilon \otimes \mathrm{id}$$\varepsilon\otimes\operatorname{id}$. We obtain

$$x=\epsilon (x)1+x,$$

and hence

$$\epsilon (x)=0.$$

Using the antipode identity

$$m(S\otimes \mathrm{id})\mathrm{\Delta }=\eta \epsilon ,$$

we similarly obtain

$$S(x)+x=0,$$

so that

$$S(x)=-x.$$

Thus a primitive element satisfies

$$\mathrm{\Delta }(x)=x\otimes 1+1\otimes x,\epsilon (x)=0,S(x)=-x.$$

This should be compared with a group-like element:

$$\mathrm{\Delta }(g)=g\otimes g,\epsilon (g)=1,S(g)=g^{-1}.$$

The group-like condition is multiplicative, while the primitive condition is linear.

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Primitive Elements Form a Lie Algebra

Every associative algebra $A$$A$ becomes a Lie algebra under the commutator bracket

$$[x,y]=xy-yx.$$

Now let $x,y\in P(H)$$x,y\in P(H)$. Since $\mathrm{\Delta }$$\Delta$ is an algebra homomorphism,

$$\begin{array}{rl}\mathrm{\Delta }([x,y])& =\mathrm{\Delta }(xy-yx)\\ & =\mathrm{\Delta }(x)\mathrm{\Delta }(y)-\mathrm{\Delta }(y)\mathrm{\Delta }(x).\end{array}$$

Substituting

$$\mathrm{\Delta }(x)=x\otimes 1+1\otimes x,\mathrm{\Delta }(y)=y\otimes 1+1\otimes y,$$

we get

$$\begin{array}{rl}\mathrm{\Delta }([x,y])& =(x\otimes 1+1\otimes x)(y\otimes 1+1\otimes y)\\ & -(y\otimes 1+1\otimes y)(x\otimes 1+1\otimes x).\end{array}$$

After expanding, the mixed terms cancel, leaving

$$\mathrm{\Delta }([x,y])=[x,y]\otimes 1+1\otimes [x,y].$$

Therefore

$$[x,y]\in P(H).$$

Hence $P(H)$$P(H)$ is a Lie subalgebra of the commutator Lie algebra $H_{\mathrm{Lie}}$$H_{\mathrm{Lie}}$:

$$P(H)\subseteq H_{\mathrm{Lie}}.$$

Notice that no cocommutativity assumption on $H$$H$ is needed.

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Primitive Elements Define a Functor

Let

$$f:H\to K$$

be a morphism of Hopf algebras.

If $x\in P(H)$$x\in P(H)$, then

$$\begin{array}{rl}{\mathrm{\Delta }}_K(f(x))& =(f\otimes f){\mathrm{\Delta }}_H(x)\\ & =(f\otimes f)(x\otimes 1+1\otimes x)\\ & =f(x)\otimes 1+1\otimes f(x).\end{array}$$

Therefore

$$f(x)\in P(K).$$

Since $f$$f$ is also an algebra homomorphism,

$$f([x,y])=[f(x),f(y)].$$

Thus restriction gives a Lie algebra homomorphism

$$P(f):P(H)\to P(K).$$

Hence we obtain a functor

$$P:{\mathbf{Hopf}}_k\to {\mathbf{Lie}}_k.$$

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The Universal Enveloping Algebra Is Its Left Adjoint

Let $\mathfrak{g}$$\mathfrak g$ be a Lie algebra. Its universal enveloping algebra $U(\mathfrak{g})$$U(\mathfrak g)$ carries a canonical Hopf algebra structure determined on $x\in \mathfrak{g}$$x\in\mathfrak g$ by

$$\mathrm{\Delta }(x)=x\otimes 1+1\otimes x,$$

$$\epsilon (x)=0,$$

and

$$S(x)=-x.$$

Therefore the canonical map

$$i:\mathfrak{g}\to U(\mathfrak{g})$$

actually lands in the primitive elements:

$$\mathfrak{g}⟶P(U(\mathfrak{g})).$$

More importantly, there is a natural bijection

$${\mathrm{Hom}}_{\mathbf{Hopf}}(U(\mathfrak{g}),H)\cong {\mathrm{Hom}}_{\mathbf{Lie}}(\mathfrak{g},P(H)).$$

Thus

$$U⊣P.$$

Let us see why.

Suppose we are given a Lie algebra homomorphism

$$f:\mathfrak{g}\to P(H).$$

Forgetting the coalgebra structure for the moment, the universal property of $U(\mathfrak{g})$$U(\mathfrak g)$ gives a unique algebra homomorphism

$$\tilde{f}:U(\mathfrak{g})\to H.$$

Since $f(x)$$f(x)$ is primitive,

$${\mathrm{\Delta }}_H(f(x))=f(x)\otimes 1+1\otimes f(x).$$

Hence for every $x\in \mathfrak{g}$$x\in\mathfrak g$,

$${\mathrm{\Delta }}_H\tilde{f}(x)=(\tilde{f}\otimes \tilde{f}){\mathrm{\Delta }}_{U(\mathfrak{g})}(x).$$

Both sides are algebra homomorphisms from $U(\mathfrak{g})$$U(\mathfrak g)$ to $H\otimes H$$H\otimes H$, and they agree on the generators $\mathfrak{g}$$\mathfrak g$. Therefore they agree everywhere.

The counit and antipode are handled similarly.

Thus $\tilde{f}$$\widetilde f$ is automatically a Hopf algebra morphism.

Equivalently:

The primitive condition is exactly the extra condition needed for a Lie algebra map $\mathfrak{g}\to H_{\mathrm{Lie}}$$\mathfrak g\to H_{\mathrm{Lie}}$ to extend to a Hopf algebra map $U(\mathfrak{g})\to H$$U(\mathfrak g)\to H$.

Representability and the Monoidal Origin of Group-Like and Primitive Elements

The group-like and primitive-element constructions admit another categorical interpretation. Both are representable, and both interact strongly with tensor products.

These two facts describe different aspects of the construction:

• representability tells us that a group-like or primitive element can be regarded as a morphism from a universal probe;

• strong monoidality explains how the multiplication of a Hopf algebra induces algebraic structure on the collection of such elements.

It is useful to examine these two ideas separately.

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Group-Like Elements Are Representable

For a Hopf algebra $H$$H$, let

$$G(H)=\{g\in H\mid \mathrm{\Delta }(g)=g\otimes g,\epsilon (g)=1\}.$$

The elements of $G(H)$$G(H)$ form a group under the multiplication of $H$$H$.

Consider the infinite cyclic group $\mathbb{Z}$$\mathbb Z$. For every group $\mathrm{\Gamma }$$\Gamma$, choosing a group homomorphism

$$\mathbb{Z}\to \mathrm{\Gamma }$$

is equivalent to choosing the image of $1\in \mathbb{Z}$$1\in\mathbb Z$, hence to choosing an arbitrary element of $\mathrm{\Gamma }$$\Gamma$. Therefore

$$|\mathrm{\Gamma }|\cong {\mathrm{Hom}}_{\mathbf{Grp}}(\mathbb{Z},\mathrm{\Gamma }).$$

Now use the adjunction

$$k[-]:\mathbf{Grp}\rightleftarrows {\mathbf{HopfAlg}}_k:G.$$

We obtain

$$\begin{array}{rl}|G(H)|& \cong {\mathrm{Hom}}_{\mathbf{Grp}}(\mathbb{Z},G(H))\\ & \cong {\mathrm{Hom}}_{{\mathbf{HopfAlg}}_k}(k[\mathbb{Z}],H).\end{array}$$

Thus the underlying-set-valued group-like functor is represented by

$$k[\mathbb{Z}].$$

In other words,

$$|G(H)|\cong {\mathrm{Hom}}_{{\mathbf{HopfAlg}}_k}(k[\mathbb{Z}],H).$$

If $z$$z$ denotes the canonical generator of $k[\mathbb{Z}]$$k[\mathbb Z]$, then

$$\mathrm{\Delta }(z)=z\otimes z,\epsilon (z)=1,S(z)=z^{-1}.$$

A Hopf algebra morphism

$$f:k[\mathbb{Z}]\to H$$

is completely determined by $f(z)$$f(z)$, and the Hopf compatibility forces $f(z)$$f(z)$ to be group-like.

Thus $k[\mathbb{Z}]$$k[\mathbb Z]$ is the universal Hopf algebra containing one group-like element.

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Primitive Elements Are Representable

Now consider

$$P(H)=\{x\in H\mid \mathrm{\Delta }(x)=x\otimes 1+1\otimes x\}.$$

Let $\mathfrak{a}=kx$$\mathfrak a=kx$ be the one-dimensional abelian Lie algebra.

For every Lie algebra $\mathfrak{g}$$\mathfrak g$, a Lie algebra morphism

$$\mathfrak{a}\to \mathfrak{g}$$

is completely determined by the image of $x$$x$, and this image may be any element of $\mathfrak{g}$$\mathfrak g$. Hence

$$|\mathfrak{g}|\cong {\mathrm{Hom}}_{{\mathbf{Lie}}_k}(\mathfrak{a},\mathfrak{g}).$$

Using the adjunction

$$U:{\mathbf{Lie}}_k\rightleftarrows {\mathbf{HopfAlg}}_k:P,$$

we obtain

$$\begin{array}{rl}|P(H)|& \cong {\mathrm{Hom}}_{{\mathbf{Lie}}_k}(\mathfrak{a},P(H))\\ & \cong {\mathrm{Hom}}_{{\mathbf{HopfAlg}}_k}(U(\mathfrak{a}),H).\end{array}$$

Since $\mathfrak{a}$$\mathfrak a$ is one-dimensional and abelian,

$$U(\mathfrak{a})\cong k[t],$$

with

$$\mathrm{\Delta }(t)=t\otimes 1+1\otimes t,\epsilon (t)=0,S(t)=-t.$$

Therefore

$$|P(H)|\cong {\mathrm{Hom}}_{{\mathbf{HopfAlg}}_k}(k[t],H).$$

A Hopf algebra morphism

$$f:k[t]\to H$$

is determined by $f(t)$$f(t)$, and Hopf compatibility says precisely that

$$\mathrm{\Delta }(f(t))=f(t)\otimes 1+1\otimes f(t).$$

Thus $k[t]$$k[t]$ is the universal Hopf algebra containing one primitive element.

We therefore obtain the parallel picture

$$\begin{array}{rl}\text{group-like element}& ⟷k[\mathbb{Z}]\to H,\\ \text{primitive element}& ⟷k[t]\to H.\end{array}$$

The representing objects themselves reflect the two defining equations:

$$\mathrm{\Delta }(z)=z\otimes z,$$

and

$$\mathrm{\Delta }(t)=t\otimes 1+1\otimes t.$$

The first is multiplicative, while the second is its infinitesimal, additive analogue.

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A General Principle Behind the Two Representations

Both calculations are instances of the same elementary categorical fact.

Suppose

$$L:\mathcal{C}\rightleftarrows \mathcal{D}:R$$

is an adjunction, and suppose a Set-valued functor on $\mathcal{C}$$\mathcal C$ is represented by an object $A$$A$:

$$F(X)\cong {\mathrm{Hom}}_{\mathcal{C}}(A,X).$$

Then

$$F(R(Y))\cong {\mathrm{Hom}}_{\mathcal{D}}(L(A),Y).$$

Thus representability is transported across the adjunction.

For group-like elements, the generic element of a group is represented by

$$\mathbb{Z},$$

and the left adjoint sends it to

$$k[\mathbb{Z}].$$

For primitive elements, the generic element of a Lie algebra is represented by the one-dimensional abelian Lie algebra

$$\mathfrak{a},$$

and the left adjoint sends it to

$$U(\mathfrak{a})=k[t].$$

Hence the representing Hopf algebras are not accidental. They are the images of the generic one-generator objects under the corresponding left adjoints.

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Strong Monoidality and the Group-Like Multiplication

Representability tells us what an individual group-like element is. To understand why group-like elements form a group, it is useful to move one categorical level down and regard a Hopf algebra as a monoid object in coalgebras equipped with an antipode.

Let

$$G_0:({\mathbf{Coalg}}_k,\otimes ,k)⟶(\mathbf{Set},\times ,\ast )$$

be the group-like-element functor.

It is strong symmetric monoidal:

$$G_0(C\otimes D)\cong G_0(C)\times G_0(D).$$

Under this isomorphism,

$$(g,h)⟼g\otimes h.$$

Indeed, if $g$$g$ and $h$$h$ are group-like, then

$$\mathrm{\Delta }(g\otimes h)=(g\otimes h)\otimes (g\otimes h).$$

Now a bialgebra $H$$H$ is precisely a monoid object in the monoidal category of coalgebras. Its multiplication and unit are coalgebra morphisms

$$m:H\otimes H\to H,\eta :k\to H.$$

A strong monoidal functor sends monoid objects to monoid objects. Therefore $G_0$$G_0$ sends the multiplication of $H$$H$ to

$$G_0(H)\times G_0(H)\cong G_0(H\otimes H)\stackrel{G_0(m)}{\to }{G}_0(H).$$

Explicitly,

$$(g,h)⟼g\otimes h⟼gh.$$

Thus the multiplication on group-like elements is not an additional construction. It is the Hopf multiplication transported through the strong monoidal functor $G_0$$G_0$.

The strong monoidal argument first gives a monoid. When $H$$H$ is a Hopf algebra, the antipode satisfies

$$S(g)=g^{-1}$$

for every group-like element $g$$g$, so this monoid is in fact a group.

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Primitive Elements and the Additive Group

There is a completely parallel construction for primitive elements, but with an important change in the target monoidal structure.

For a coaugmented coalgebra $C$$C$, let

$$P_0(C)=\{x\in C\mid \mathrm{\Delta }(x)=x\otimes 1+1\otimes x\}.$$

Then

$$P_0:({\mathbf{Coalg}}_k^{\mathrm{coaug}},\otimes ,k)⟶({\mathbf{Vect}}_k,\oplus ,0)$$

is strong symmetric monoidal:

$$P_0(C\otimes D)\cong P_0(C)\oplus P_0(D).$$

The comparison map is

$$(x,y)⟼x\otimes 1+1\otimes y.$$

Now take a Hopf algebra $H$$H$. Its multiplication

$$m:H\otimes H\to H$$

is a morphism of coaugmented coalgebras. Applying $P_0$$P_0$ gives

$$P_0(H\otimes H)\stackrel{P_0(m)}{\to }{P}_0(H).$$

Using strong monoidality, this becomes

$$P(H)\oplus P(H)⟶P(H).$$

What operation is this?

Starting with $(x,y)$$(x,y)$,

$$(x,y)⟼x\otimes 1+1\otimes y,$$

and then applying multiplication gives

$$m(x\otimes 1+1\otimes y)=x+y.$$

Therefore the multiplication transported through $P_0$$P_0$ is

$$(x,y)⟼x+y.$$

The unit becomes

$$0,$$

and the antipode acts on primitive elements by

$$S(x)=-x.$$

Thus the group structure produced on primitive elements is precisely their additive group structure:

$$(P(H),+,0,-).$$

If we further compose with the underlying-set functor

$$U:({\mathbf{Vect}}_k,\oplus ,0)⟶(\mathbf{Set},\times ,\ast ),$$

then

$$U\circ P_0$$

is again strong symmetric monoidal, and it sends a Hopf algebra to the ordinary additive group underlying its primitive vector space.

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The Two Strong Monoidal Functors

We can now place the two constructions side by side:

$$G_0:(\mathbf{Coalg},\otimes )⟶(\mathbf{Set},\times ),$$

with

$$G_0(C\otimes D)\cong G_0(C)\times G_0(D),$$

and

$$P_0:({\mathbf{Coalg}}^{\mathrm{coaug}},\otimes )⟶(\mathbf{Vect},\oplus ),$$

with

$$P_0(C\otimes D)\cong P_0(C)\oplus P_0(D).$$

For the same Hopf multiplication

$$m:H\otimes H\to H,$$

the first functor produces

$$(g,h)⟼gh,$$

while the second produces

$$(x,y)⟼x+y.$$

Thus

$$\begin{array}{rl}G(H)& \text{ sees the multiplicative group carried by group-like points},\\ P(H)& \text{ sees the additive group carried by infinitesimal points}.\end{array}$$

This is the categorical form of the passage from a group to its infinitesimal linearization.

There is, however, one further piece of structure on $P(H)$$P(H)$.

The additive group law on $P(H)$$P(H)$ is explained entirely by the strong monoidality of $P_0$$P_0$. The Lie bracket

$$[x,y]=xy-yx$$

is additional information. It comes from the noncommutativity of the multiplication of $H$$H$, together with the fact that primitive elements are closed under commutators.

Hence the two structures on $P(H)$$P(H)$ have conceptually different origins:

$$\begin{array}{rl}\text{strong monoidality}& ⇝x+y,\\ \text{noncommutative multiplication}& ⇝[x,y].\end{array}$$

This distinction mirrors ordinary Lie theory. The differential of group multiplication at the identity is

$$d{m}_{(e,e)}:\mathfrak{g}\oplus \mathfrak{g}\to \mathfrak{g},$$

and is simply

$$(X,Y)⟼X+Y.$$

The Lie bracket is not this first derivative; it records a higher-order failure of commutativity.

In the Hopf-algebraic picture, group-like elements retain the multiplicative structure itself, while primitive elements retain its infinitesimal additive structure, with the commutator supplying the additional Lie bracket.